Items are labelled (a) through (m), omitting (l).
Several are not bugs, just suggestions.
Most bugs are in *gesdd.
There's one bug (g) in *bdsdc. This is the underlying cause of LAPACK bug #111.
There's one bug (m) in [cz]gesvd. I also added an INT() cast in these
assignments to silence compiler warnings. Changed:
LWORK_ZGEQRF=CDUM(1)
to:
LWORK_ZGEQRF = INT( CDUM(1) )
Where possible, I ran a test showing the wrong behavior, then a test showing the
corrected behavior. These use a modified version of the MAGMA SVD tester
(calling LAPACK), because I could adjust the lwork as needed. The last 3 columns
are the lwork type, the lwork size, and the lwork formula. The lwork types are:
doc_old as documented in LAPACK 3.6.
doc as in the attached, updated documentation.
min_old minwrk, as computed in LAPACK 3.6.
min minwrk, as computed in the attached, updated code.
min-1 minimum - 1; this should cause gesdd to return an error.
opt optimal size.
max the maximum size LAPACK will take advantage of;
some cases, the optimal is n*n + work, while the max is m*n + work.
query what gesdd returns for an lwork query; should equal opt or max.
After the lwork, occasionally there is a ! or ? error code indicating:
Error codes: ! error: lwork < min. For (min-1), this ought to appear.
? compatability issue: lwork < min_old, will fail for lapack <= 3.6.
I also tested the update routines on a wide variety of sizes and jobz, with
various lwork.
Besides fixing the bugs below, I made two significant changes.
1) Changed *gesdd from computing each routine's workspace using, e.g.:
N*ilaenv(...)
to querying each routine for its LWORK, e.g.:
CALL ZGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
LWORK_ZGEBRD_MN = INT( CDUM(1) )
This matches how *gesvd was changed in LAPACK 3.4.0.
2) Changed the Workspace: comments, which were incredibly deceptive.
For instance, in Path 2 before dbdsdc, it said
Workspace: need N + N*N + BDSPAC
since dbdsdc needs the [e] vector, [U] matrix, and bdspac.
However, that ignores that the [tauq, taup] vectors and [R] matrix
are also already allocated, though dbdsdc doesn't need them.
So the workspace needed at that point in the code is actually
Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
For clarity, I added in [brackets] what matrices or vectors were allocated,
and the order reflects their order in memory.
I may do a similar change for *gesvd eventually. The workspace comments in
MAGMA's *gesvd have already been updated as above.
================================================================================
a) Throughout, to simplify equations, let:
mn = min( M, N )
mx = max( M, N )
================================================================================
b) [sdcz]gesdd Path 4 (m >> n, job="all") has wrong minwrk formula in code:
minwrk = bdspac + mn*mn + 2*mn + mx
= 4*mn*mn + 6*mn + mx
This is an overestimate, needlessly rejecting the documented formula:
doc = 4*mn*mn + 7*mn
In complex, the correct min fails, but the doc matches the wrong minwrk.
Solution: fix code to:
minwrk = mn*mn + max( 3*mn + bdspac, mn + mx )
= mn*mn + max( 3*mn*mn + 7*mn, mn + mx )
Test cases:
m=40, ..., 100; n=20; jobz='A'
See bug (d) showing documentation is also wrong.
Also, see bug (i), complex [cz]gesdd should return -12 instead of -13.
================================================================================
bt) transposed case
[sd]gesdd Path 4t (n >> m, job="all") has a different wrong minwrk; see bug (c).
[cz]gesdd Path 4t exhibits same bug as Path 4.
Test cases:
m=20; n=40, ..., 100; jobz='A'
================================================================================
c) [sd]gesdd Path 4t (n >> m, job="all") has wrong minwrk formula in code,
different than bug (b):
minwrk = bdspac + m*m + 3*m
= 4*mn*mn + 7*mn
This formula lacks any dependence on N, so the code will fail (without
setting info; orglq calls xerbla) if N is too large, N > 3*M*M + 6*M.
Bug does not occur in complex.
Test cases:
m=20; n = 1320; jobz='A' ok with documented lwork
m=20; n > 1320; jobz='A' fails with documented lwork
Solution: as in bug (b), fix code to:
minwrk = mn*mn + max( 3*mn + bdspac, mn + mx )
= mn*mn + max( 3*mn*mn + 7*mn, mn + mx )
See bug (d) showing documentation is also wrong.
================================================================================
d) [sd]gesdd documentation lists the same minimum size for jobz='S' and 'A':
If JOBZ = 'S' or 'A', LWORK >= min(M,N)*(7 + 4*min(M,N))
However, jobz='A' actually also depends on max(M,N):
minwrk = mn*mn + max( 3*mn*mn + 7*mn, mn + mx )
This causes the formula to fail for mx > 3*mn*mn + 6*mn.
Test cases:
m > 1320; n = 20; jobz='A' fails with document lwork, even after fixing bugs (b) and (c).
m = 20; n > 1320; jobz='A' fails also.
Solution: in docs, split these two cases. This fix uses an overestimate,
so that codes using it will be backwards compatible with LAPACK <= 3.6.
If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
================================================================================
e) [sd]gesdd, Path 5, jobz='A' has wrong maxwrk formula in the code:
MAXWRK = MAX( MAXWRK, BDSPAC + 3*N )
Should be:
MAXWRK = MAX( WRKBL, BDSPAC + 3*N )
This causes the lwork query to ignore WRKBL, and return the minimum
workspace size, BDSPAC + 3*N, instead of the optimal workspace size.
However, it only affects the result for small sizes where min(M,N) < NB.
Path 5t has the correct maxwrk formula.
Complex is correct for both Path 5 and 5t.
Test case:
Compare lwork query with
M = 30, N = 20, jobz='A', lwork query is 1340
M = 20, N = 30, jobz='A', lwork query is 3260
These should be the same.
Solution: fix code as above.
================================================================================
f) Not a bug, just a suggestion.
The lwork minimum sizes are not actually minimums, and can be larger than
the queried lwork size.
Solution: add a comment:
These are not tight minimums in all cases; see comments inside code.
================================================================================
g) [sd]bdsdc segfaults due to too small workspace size. Its documentation claims:
If COMPQ = 'N' then LWORK >= (4 * N).
Based on this, in zgesdd, the rwork size >= 5*min(M,N).
However, LAPACK bug 111 found that rwork size >= 7*min(M,N) was required.
In dbdsdc, if uplo='L', then it rotates lower bidiagonal to upper bidiagonal,
and saves 2 vectors of Givens rotations in work. It shifts WSTART from
1 to 2*N-1. Then it calls dlasdq( ..., work( wstart ), info ).
As dlasdq requires 4*N, dbdsdc would now require 6*N in this case.
This caused zgesdd to require rwork size >= 7*min(M,N) when N > M and jobz='N'.
My preferred solution is to change WSTART to 1 in the DLASDQ call inside dbdsdc:
IF( ICOMPQ.EQ.0 ) THEN
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK( WSTART ), INFO )
GO TO 40
END IF
to:
IF( ICOMPQ.EQ.0 ) THEN
* Ignores WSTART, which is needed only for ICOMPQ = 1 or 2;
* using WSTART would change required workspace to 6*N for uplo='L'.
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
$ LDU, WORK( 1 ), INFO )
GO TO 40
END IF
The [cz]gesdd documentation, which was changed to 7*min(M,N) in LAPACK 3.6,
may be reverted to 5*min(M,N), if desired.
================================================================================
h) [sd]gesdd for jobz='N' requires bdspac = 7*n for the dbdsdc workspace.
However, dbdsdc requires only 4*n, or 6*n before fixing bug (g).
For backwards compatability, I did not change the code, but added a comment
for clarification.
================================================================================
i) [cz]gesdd returns info = -13 instead of info = -12 for lwork errors.
================================================================================
j) In zgesdd, for computing maxwrk, these paths:
Path 6, jobz=A
Path 6t, jobz=S
Path 6t, jobz=A
query ilaenv( 1, zungbr, ... )
when the code actually calls zunmbr (twice). I corrected it.
================================================================================
k) In zgesdd documentation, currently
lrwork >= max( 5*mn*mn + 7*mn, 2*mx*mn + 2*mn*mn + mn )
It doesn't need that much, particularly for (mx >> mn) case.
If (mx >> mn), lrwork >= 5*mn*mn + 5*mn;
else, lrwork >= max( 5*mn*mn + 5*mn,
2*mx*mn + 2*mn*mn + mn ).
I changed this in the documentation. Feel free to revert if you prefer.
================================================================================
m) [cz]gesvd, Path 10 and 10t, have minwrk inside the wrong conditional:
IF( .NOT.WNTVN ) THEN
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_P )
MINWRK = 2*N + M
END IF
So Path 10 with jobvt='N', and Path 10t with jobu='N', have minwrk = 1,
so an invalid lwork is not correctly rejected.
================================================================================
mt) transposed case
broken: with old routine, Path 10t with jobu='N' doesn't enforce minwrk
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
-*> The leading dimension of the array U. LDU >= 1; if
-*> JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
+*> The leading dimension of the array U. LDU >= 1;
+*> if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
-*> The leading dimension of the array VT. LDVT >= 1; if
-*> JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*> The leading dimension of the array VT. LDVT >= 1;
+*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
-*> if JOBZ = 'N', LWORK >= 2*min(M,N)+max(M,N).
-*> if JOBZ = 'O',
-*> LWORK >= 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-*> if JOBZ = 'S' or 'A',
-*> LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-*> For good performance, LWORK should generally be larger.
-*>
*> If LWORK = -1, a workspace query is assumed. The optimal
*> size for the WORK array is calculated and stored in WORK(1),
*> and no other work except argument checking is performed.
+*>
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LWORK >= 2*mn + mx.
+*> If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
+*> If JOBZ = 'S', LWORK >= mn*mn + 3*mn.
+*> If JOBZ = 'A', LWORK >= mn*mn + 2*mn + mx.
+*> These are not tight minimums in all cases; see comments inside code.
+*> For good performance, LWORK should generally be larger;
+*> a query is recommended.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (MAX(1,LRWORK))
-*> If JOBZ = 'N', LRWORK >= 7*min(M,N).
-*> Otherwise,
-*> LRWORK >= min(M,N)*max(5*min(M,N)+7,2*max(M,N)+2*min(M,N)+1)
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
+*> else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
+*> else LRWORK >= max( 5*mn*mn + 5*mn,
+*> 2*mx*mn + 2*mn*mn + mn ).
*> \endverbatim
*>
*> \param[out] IWORK
* =====================================================================
SUBROUTINE CGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
$ WORK, LWORK, RWORK, IWORK, INFO )
+ implicit none
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* =====================================================================
*
* .. Parameters ..
- INTEGER LQUERV
- PARAMETER ( LQUERV = -1 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
$ CONE = ( 1.0E+0, 0.0E+0 ) )
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
- LOGICAL WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
+ LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BLK, CHUNK, I, IE, IERR, IL, IR, IRU, IRVT,
$ ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR1, MNTHR2, NRWORK, NWORK, WRKBL
- REAL ANRM, BIGNUM, EPS, SMLNUM
+ INTEGER LWORK_CGEBRD_MN, LWORK_CGEBRD_MM,
+ $ LWORK_CGEBRD_NN, LWORK_CGELQF_MN,
+ $ LWORK_CGEQRF_MN,
+ $ LWORK_CUNGBR_P_MN, LWORK_CUNGBR_P_NN,
+ $ LWORK_CUNGBR_Q_MN, LWORK_CUNGBR_Q_MM,
+ $ LWORK_CUNGLQ_MN, LWORK_CUNGLQ_NN,
+ $ LWORK_CUNGQR_MM, LWORK_CUNGQR_MN,
+ $ LWORK_CUNMBR_PRC_MM, LWORK_CUNMBR_QLN_MM,
+ $ LWORK_CUNMBR_PRC_MN, LWORK_CUNMBR_QLN_MN,
+ $ LWORK_CUNMBR_PRC_NN, LWORK_CUNMBR_QLN_NN
+ REAL ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
REAL DUM( 1 )
+ COMPLEX CDUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEBRD, CGELQF, CGEMM, CGEQRF, CLACP2, CLACPY,
* ..
* .. External Functions ..
LOGICAL LSAME
- INTEGER ILAENV
- REAL CLANGE, SLAMCH
- EXTERNAL CLANGE, SLAMCH, ILAENV, LSAME
+ REAL SLAMCH, CLANGE
+ EXTERNAL LSAME, SLAMCH, CLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
*
* Test the input arguments
*
- INFO = 0
- MINMN = MIN( M, N )
- MNTHR1 = INT( MINMN*17.0 / 9.0 )
- MNTHR2 = INT( MINMN*5.0 / 3.0 )
- WNTQA = LSAME( JOBZ, 'A' )
- WNTQS = LSAME( JOBZ, 'S' )
+ INFO = 0
+ MINMN = MIN( M, N )
+ MNTHR1 = INT( MINMN*17.0E0 / 9.0E0 )
+ MNTHR2 = INT( MINMN*5.0E0 / 3.0E0 )
+ WNTQA = LSAME( JOBZ, 'A' )
+ WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
- WNTQO = LSAME( JOBZ, 'O' )
- WNTQN = LSAME( JOBZ, 'N' )
+ WNTQO = LSAME( JOBZ, 'O' )
+ WNTQN = LSAME( JOBZ, 'N' )
+ LQUERY = ( LWORK.EQ.-1 )
MINWRK = 1
MAXWRK = 1
*
END IF
*
* Compute workspace
-* (Note: Comments in the code beginning "Workspace:" describe the
-* minimal amount of workspace needed at that point in the code,
+* Note: Comments in the code beginning "Workspace:" describe the
+* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* CWorkspace refers to complex workspace, and RWorkspace to
* real workspace. NB refers to the optimal block size for the
IF( M.GE.N ) THEN
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*N*N + 7*N
-* BDSPAN = MAX(7*N+4, 3*N+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (sbdsdc) is
+* BDSPAC = 3*N*N + 4*N for singular values and vectors;
+* BDSPAC = 4*N for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL CGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGEBRD_MN = INT( CDUM(1) )
+*
+ CALL CGEBRD( N, N, CDUM(1), N, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGEBRD_NN = INT( CDUM(1) )
+*
+ CALL CGEQRF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGEQRF_MN = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'P', N, N, N, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'Q', M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_Q_MN = INT( CDUM(1) )
+*
+ CALL CUNGQR( M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGQR_MM = INT( CDUM(1) )
+*
+ CALL CUNGQR( M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGQR_MN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'P', 'R', 'C', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'Q', 'L', 'N', M, M, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_QLN_MM = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'Q', 'L', 'N', M, N, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_QLN_MN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'Q', 'L', 'N', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_QLN_NN = INT( CDUM(1) )
*
IF( M.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
*
- MAXWRK = N + N*ILAENV( 1, 'CGEQRF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*N+2*N*
- $ ILAENV( 1, 'CGEBRD', ' ', N, N, -1, -1 ) )
+ MAXWRK = N + LWORK_CGEQRF_MN
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CGEBRD_NN )
MINWRK = 3*N
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
-*
- WRKBL = N + N*ILAENV( 1, 'CGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'CUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'CGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 2 (M >> N, JOBZ='O')
+*
+ WRKBL = N + LWORK_CGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = M*N + N*N + WRKBL
MINWRK = 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
-*
- WRKBL = N + N*ILAENV( 1, 'CGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'CUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'CGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 3 (M >> N, JOBZ='S')
+*
+ WRKBL = N + LWORK_CGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
MINWRK = N*N + 3*N
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
-*
- WRKBL = N + N*ILAENV( 1, 'CGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'CUNGQR', ' ', M,
- $ M, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'CGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 4 (M >> N, JOBZ='A')
+*
+ WRKBL = N + LWORK_CGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_CUNGQR_MM )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_CUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
- MINWRK = N*N + 2*N + M
+ MINWRK = N*N + MAX( 3*N, N + M )
END IF
ELSE IF( M.GE.MNTHR2 ) THEN
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'CGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_CGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5o (M >> N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5s (M >> N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5a (M >> N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNGBR_Q_MM )
END IF
ELSE
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'CGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_CGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6o (M >= N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6s (M >= N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'CUNGBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'CUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6a (M >= N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_CUNMBR_PRC_NN )
END IF
END IF
ELSE
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*M*M + 7*M
-* BDSPAN = MAX(7*M+4, 3*M+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (sbdsdc) is
+* BDSPAC = 3*M*M + 4*M for singular values and vectors;
+* BDSPAC = 4*M for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL CGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGEBRD_MN = INT( CDUM(1) )
+*
+ CALL CGEBRD( M, M, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGEBRD_MM = INT( CDUM(1) )
+*
+ CALL CGELQF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_CGELQF_MN = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'P', M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_P_MN = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'P', N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL CUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL CUNGLQ( M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGLQ_MN = INT( CDUM(1) )
+*
+ CALL CUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_CUNGLQ_NN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'P', 'R', 'C', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_PRC_MM = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'P', 'R', 'C', M, N, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_PRC_MN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'P', 'R', 'C', N, N, M, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL CUNMBR( 'Q', 'L', 'N', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_CUNMBR_QLN_MM = INT( CDUM(1) )
*
IF( N.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
*
- MAXWRK = M + M*ILAENV( 1, 'CGELQF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*M+2*M*
- $ ILAENV( 1, 'CGEBRD', ' ', M, M, -1, -1 ) )
+ MAXWRK = M + LWORK_CGELQF_MN
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CGEBRD_MM )
MINWRK = 3*M
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
-*
- WRKBL = M + M*ILAENV( 1, 'CGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'CUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'CGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 2t (N >> M, JOBZ='O')
+*
+ WRKBL = M + LWORK_CGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*N + M*M + WRKBL
MINWRK = 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-*
- WRKBL = M + M*ILAENV( 1, 'CGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'CUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'CGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 3t (N >> M, JOBZ='S')
+*
+ WRKBL = M + LWORK_CGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
MINWRK = M*M + 3*M
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
-*
- WRKBL = M + M*ILAENV( 1, 'CGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'CUNGLQ', ' ', N,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'CGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 4t (N >> M, JOBZ='A')
+*
+ WRKBL = M + LWORK_CGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_CUNGLQ_NN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_CUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
- MINWRK = M*M + 2*M + N
+ MINWRK = M*M + MAX( 3*M, M + N )
END IF
ELSE IF( N.GE.MNTHR2 ) THEN
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'CGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_CGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5to (N >> M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ts (N >> M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'CUNGBR', 'P', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ta (N >> M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNGBR_P_NN )
END IF
ELSE
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'CGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_CGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNMBR', 'QLN', M, M, N, -1 ) )
+* Path 6to (N > M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ts (N > M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'CUNGBR', 'PRC', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'CUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ta (N > M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_CUNMBR_PRC_NN )
END IF
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = MAXWRK
- IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV )
- $ INFO = -13
+ IF( LWORK.LT.MINWRK .AND. .NOT. LQUERY ) THEN
+ INFO = -12
+ END IF
END IF
-*
-* Quick returns
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGESDD', -INFO )
RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
END IF
- IF( LWORK.EQ.LQUERV )
- $ RETURN
+*
+* Quick return if possible
+*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: need 0)
+* CWorkspace: need N [tau] + N [work]
+* CWorkspace: prefer N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NRWORK = IE + N
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
+* Path 2 (M >> N, JOBZ='O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
*
LDWRKU = N
IR = IU + LDWRKU*N
- IF( LWORK.GE.M*N+N*N+3*N ) THEN
+ IF( LWORK .GE. M*N + N*N + 3*N ) THEN
*
* WORK(IR) is M by N
*
LDWRKR = M
ELSE
- LDWRKR = ( LWORK-N*N-3*N ) / N
+ LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer M*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of R in WORK(IRU) and computing right singular vectors
* of R in WORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of R
-* (CWorkspace: need 2*N*N+3*N, prefer M*N+N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by the right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
-* (CWorkspace: need 2*N*N, prefer N*N+M*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R]
+* CWorkspace: prefer N*N [U] + M*N [R]
+* RWorkspace: need 0
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M-I+1, LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R]
+* RWorkspace: need 0
*
CALL CLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL CGEMM( 'N', 'N', M, N, N, CONE, A, LDA, WORK( IR ),
*
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL CLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (CWorkspace: need N+M, prefer N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + M [work]
+* CWorkspace: prefer N*N [U] + N [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CUNGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U]
+* RWorkspace: need 0
*
CALL CGEMM( 'N', 'N', M, N, N, CONE, U, LDU, WORK( IU ),
$ LDWRKU, CZERO, A, LDA )
*
* MNTHR2 <= M < MNTHR1
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* CUNGBR and matrix multiplication to compute singular vectors
*
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5n (M >> N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
IRVT = IRU + N*N
NRWORK = IRVT + N*N
*
+* Path 5o (M >> N, JOBZ='O')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL CUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
*
* WORK(IU) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in WORK(IU), copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL CLARCM( N, N, RWORK( IRVT ), N, VT, LDVT,
$ WORK( IU ), LDWRKU, RWORK( NRWORK ) )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 20 I = 1, M, LDWRKU
*
ELSE IF( WNTQS ) THEN
*
+* Path 5s (M >> N, JOBZ='S')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL CUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'L', M, N, A, LDA, U, LDU )
CALL CUNGBR( 'Q', M, N, N, U, LDU, WORK( ITAUQ ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL CLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need N*N+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL CLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
CALL CLACPY( 'F', M, N, A, LDA, U, LDU )
ELSE
*
+* Path 5a (M >> N, JOBZ='A')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL CUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'L', M, N, A, LDA, U, LDU )
CALL CUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL CLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL CLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
*
* M .LT. MNTHR2
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
* Use CUNMBR to compute singular vectors
*
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL CGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6n (M >= N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
IRU = NRWORK
IRVT = IRU + N*N
NRWORK = IRVT + N*N
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
*
* WORK( IU ) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
+* Path 6o (M >= N, JOBZ='O')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
-* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
-* Overwrite WORK(IU) by left singular vectors of A, copying
-* to A
-* (Cworkspace: need M*N+2*N, prefer M*N+N+N*NB)
-* (Rworkspace: need 0)
+* Path 6o-fast
+* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
+* Overwrite WORK(IU) by left singular vectors of A, copying
+* to A
+* CWorkspace: need 2*N [tauq, taup] + M*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU]
*
CALL CLASET( 'F', M, N, CZERO, CZERO, WORK( IU ),
$ LDWRKU )
CALL CLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
+* Path 6o-slow
* Generate Q in A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need 0
*
CALL CUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 30 I = 1, M, LDWRKU
*
ELSE IF( WNTQS ) THEN
*
+* Path 6s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL CLASET( 'F', M, N, CZERO, CZERO, U, LDU )
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL CLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL CLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
*
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M [tau] + M [work]
+* CWorkspace: prefer M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NRWORK = IE + M
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL SBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
* WORK(IVT) is M by M
*
IL = IVT + LDWKVT*M
- IF( LWORK.GE.M*N+M*M+3*M ) THEN
+ IF( LWORK .GE. M*N + M*M + 3*M ) THEN
*
* WORK(IL) M by N
*
* WORK(IL) is M by CHUNK
*
LDWRKL = M
- CHUNK = ( LWORK-M*M-3*M ) / M
+ CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
ITAU = IL + LDWRKL*CHUNK
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by the right singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
*
* Multiply right singular vectors of L in WORK(IL) by Q
* in A, storing result in WORK(IL) and copying to A
-* (CWorkspace: need 2*M*M, prefer M*M+M*N))
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L]
+* CWorkspace: prefer M*M [VT] + M*N [L]
+* RWorkspace: need 0
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-* M right singular vectors to be computed in VT and
-* M left singular vectors to be computed in U
+* Path 3t (N >> M, JOBZ='S')
+* M right singular vectors to be computed in VT and
+* M left singular vectors to be computed in U
*
IL = 1
*
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', M, M, M, WORK( IL ), LDWRKL,
*
* Copy VT to WORK(IL), multiply right singular vectors of L
* in WORK(IL) by Q in A, storing result in VT
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L]
+* RWorkspace: need 0
*
CALL CLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL CGEMM( 'N', 'N', M, N, M, CONE, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQA ) THEN
*
-* Path 9t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL CGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL CLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (CWorkspace: need M+N, prefer M+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + N [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL CUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by right singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT]
+* RWorkspace: need 0
*
- CALL CGEMM( 'N', 'N', M, N, M, CONE, WORK( IVT ),
- $ LDWKVT, VT, LDVT, CZERO, A, LDA )
+ CALL CGEMM( 'N', 'N', M, N, M, CONE, WORK( IVT ), LDWKVT,
+ $ VT, LDVT, CZERO, A, LDA )
*
* Copy right singular vectors of A from A to VT
*
*
* MNTHR2 <= N < MNTHR1
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* CUNGBR and matrix multiplication to compute singular vectors
-*
*
IE = 1
NRWORK = IE + M
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
*
IF( WNTQN ) THEN
*
+* Path 5tn (N >> M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL SBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IRVT = NRWORK
NRWORK = IRU + M*M
IVT = NWORK
*
+* Path 5to (N >> M, JOBZ='O')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'L', M, M, A, LDA, U, LDU )
CALL CUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
LDWKVT = M
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
CALL SBDSDC( 'L', 'I', M, S, RWORK( IE ), RWORK( IRU ),
$ M, RWORK( IRVT ), M, DUM, IDUM,
*
* Multiply Q in U by real matrix RWORK(IRVT)
* storing the result in WORK(IVT), copying to U
-* (Cworkspace: need 0)
-* (Rworkspace: need 2*M*M)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL CLACRM( M, M, U, LDU, RWORK( IRU ), M, WORK( IVT ),
$ LDWKVT, RWORK( NRWORK ) )
*
* Multiply RWORK(IRVT) by P**H in A, storing the
* result in WORK(IVT), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 2*M*M, prefer 2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 50 I = 1, N, CHUNK
50 CONTINUE
ELSE IF( WNTQS ) THEN
*
+* Path 5ts (N >> M, JOBZ='S')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'L', M, M, A, LDA, U, LDU )
CALL CUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL CUNGBR( 'P', M, N, M, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL CLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
CALL CLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
CALL CLACPY( 'F', M, N, A, LDA, VT, LDVT )
ELSE
*
+* Path 5ta (N >> M, JOBZ='A')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'L', M, M, A, LDA, U, LDU )
CALL CUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL CLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL CUNGBR( 'P', N, N, M, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL CLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
+ NRWORK = IRU
CALL CLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
CALL CLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
* N .LT. MNTHR2
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
* Use CUNMBR to compute singular vectors
*
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL CGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6tn (N > M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL SBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL SBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 6to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
-* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
-* Overwrite WORK(IVT) by right singular vectors of A,
-* copying to A
-* (Cworkspace: need M*N+2*M, prefer M*N+M+M*NB)
-* (Rworkspace: need 0)
+* Path 6to-fast
+* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
+* Overwrite WORK(IVT) by right singular vectors of A,
+* copying to A
+* CWorkspace: need 2*M [tauq, taup] + M*N [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
CALL CLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
+* Path 6to-slow
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need 0
*
CALL CUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 3*M*M, prefer M*M+2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 60 I = 1, N, CHUNK
END IF
ELSE IF( WNTQS ) THEN
*
+* Path 6ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL CLASET( 'F', M, N, CZERO, CZERO, VT, LDVT )
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL CLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL CUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL CLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL CUNMBR( 'P', 'R', 'C', N, N, M, A, LDA,
MNTHR = ILAENV( 6, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for CGEQRF
CALL CGEQRF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGEQRF=CDUM(1)
+ LWORK_CGEQRF = INT( CDUM(1) )
* Compute space needed for CUNGQR
CALL CUNGQR( M, N, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_CUNGQR_N=CDUM(1)
+ LWORK_CUNGQR_N = INT( CDUM(1) )
CALL CUNGQR( M, M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_CUNGQR_M=CDUM(1)
+ LWORK_CUNGQR_M = INT( CDUM(1) )
* Compute space needed for CGEBRD
CALL CGEBRD( N, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGEBRD=CDUM(1)
+ LWORK_CGEBRD = INT( CDUM(1) )
* Compute space needed for CUNGBR
CALL CUNGBR( 'P', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_P=CDUM(1)
+ LWORK_CUNGBR_P = INT( CDUM(1) )
CALL CUNGBR( 'Q', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_Q=CDUM(1)
+ LWORK_CUNGBR_Q = INT( CDUM(1) )
*
MNTHR = ILAENV( 6, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
IF( M.GE.MNTHR ) THEN
*
CALL CGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGEBRD=CDUM(1)
+ LWORK_CGEBRD = INT( CDUM(1) )
MAXWRK = 2*N + LWORK_CGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL CUNGBR( 'Q', M, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_Q=CDUM(1)
+ LWORK_CUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_Q )
END IF
IF( WNTUA ) THEN
CALL CUNGBR( 'Q', M, M, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_Q=CDUM(1)
+ LWORK_CUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
MAXWRK = MAX( MAXWRK, 2*N+LWORK_CUNGBR_P )
- MINWRK = 2*N + M
END IF
+ MINWRK = 2*N + M
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
MNTHR = ILAENV( 6, 'CGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for CGELQF
CALL CGELQF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGELQF=CDUM(1)
+ LWORK_CGELQF = INT( CDUM(1) )
* Compute space needed for CUNGLQ
CALL CUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1), -1,
$ IERR )
- LWORK_CUNGLQ_N=CDUM(1)
+ LWORK_CUNGLQ_N = INT( CDUM(1) )
CALL CUNGLQ( M, N, M, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_CUNGLQ_M=CDUM(1)
+ LWORK_CUNGLQ_M = INT( CDUM(1) )
* Compute space needed for CGEBRD
CALL CGEBRD( M, M, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGEBRD=CDUM(1)
+ LWORK_CGEBRD = INT( CDUM(1) )
* Compute space needed for CUNGBR P
CALL CUNGBR( 'P', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_P=CDUM(1)
+ LWORK_CUNGBR_P = INT( CDUM(1) )
* Compute space needed for CUNGBR Q
CALL CUNGBR( 'Q', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_Q=CDUM(1)
+ LWORK_CUNGBR_Q = INT( CDUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
*
CALL CGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_CGEBRD=CDUM(1)
+ LWORK_CGEBRD = INT( CDUM(1) )
MAXWRK = 2*M + LWORK_CGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for CUNGBR P
CALL CUNGBR( 'P', M, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_P=CDUM(1)
+ LWORK_CUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_P )
END IF
IF( WNTVA ) THEN
CALL CUNGBR( 'P', N, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_CUNGBR_P=CDUM(1)
+ LWORK_CUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
MAXWRK = MAX( MAXWRK, 2*M+LWORK_CUNGBR_Q )
- MINWRK = 2*M + N
END IF
+ MINWRK = 2*M + N
END IF
END IF
MAXWRK = MAX( MINWRK, MAXWRK )
WSTART = 1
QSTART = 3
IF( ICOMPQ.EQ.1 ) THEN
- CALL DCOPY( N, D, 1, Q( 1 ), 1 )
+ CALL DCOPY( N, D, 1, Q( 1 ), 1 )
CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 )
END IF
IF( IUPLO.EQ.2 ) THEN
* If ICOMPQ = 0, use DLASDQ to compute the singular values.
*
IF( ICOMPQ.EQ.0 ) THEN
+* Ignore WSTART, instead using WORK( 1 ), since the two vectors
+* for CS and -SN above are added only if ICOMPQ == 2,
+* and adding them exceeds documented WORK size of 4*n.
CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
- $ LDU, WORK( WSTART ), INFO )
+ $ LDU, WORK( 1 ), INFO )
GO TO 40
END IF
*
DO 30 I = 1, NM1
IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
*
-* Subproblem found. First determine its size and then
-* apply divide and conquer on it.
+* Subproblem found. First determine its size and then
+* apply divide and conquer on it.
*
IF( I.LT.NM1 ) THEN
*
-* A subproblem with E(I) small for I < NM1.
+* A subproblem with E(I) small for I < NM1.
*
NSIZE = I - START + 1
ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
-* A subproblem with E(NM1) not too small but I = NM1.
+* A subproblem with E(NM1) not too small but I = NM1.
*
NSIZE = N - START + 1
ELSE
*
-* A subproblem with E(NM1) small. This implies an
-* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem
-* first.
+* A subproblem with E(NM1) small. This implies an
+* 1-by-1 subproblem at D(N). Solve this 1-by-1 problem
+* first.
*
NSIZE = I - START + 1
IF( ICOMPQ.EQ.2 ) THEN
* Definition:
* ===========
*
-* SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
-* LWORK, IWORK, INFO )
+* SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
-*> The leading dimension of the array VT. LDVT >= 1; if
-*> JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*> The leading dimension of the array VT. LDVT >= 1;
+*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
-*> If JOBZ = 'N',
-*> LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)).
-*> If JOBZ = 'O',
-*> LWORK >= 3*min(M,N) +
-*> max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
-*> If JOBZ = 'S' or 'A'
-*> LWORK >= min(M,N)*(7+4*min(M,N))
-*> For good performance, LWORK should generally be larger.
-*> If LWORK = -1 but other input arguments are legal, WORK(1)
-*> returns the optimal LWORK.
+*> If LWORK = -1, a workspace query is assumed. The optimal
+*> size for the WORK array is calculated and stored in WORK(1),
+*> and no other work except argument checking is performed.
+*>
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
+*> If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
+*> If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
+*> If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
+*> These are not tight minimums in all cases; see comments inside code.
+*> For good performance, LWORK should generally be larger;
+*> a query is recommended.
*> \endverbatim
*>
*> \param[out] IWORK
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
+*> @precisions fortran d -> s
* =====================================================================
- SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
- $ LWORK, IWORK, INFO )
+ SUBROUTINE DGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+ $ WORK, LWORK, IWORK, INFO )
+ implicit none
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
$ IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR, NWORK, WRKBL
+ INTEGER LWORK_DGEBRD_MN, LWORK_DGEBRD_MM,
+ $ LWORK_DGEBRD_NN, LWORK_DGELQF_MN,
+ $ LWORK_DGEQRF_MN,
+ $ LWORK_DORGBR_P_MM, LWORK_DORGBR_Q_NN,
+ $ LWORK_DORGLQ_MN, LWORK_DORGLQ_NN,
+ $ LWORK_DORGQR_MM, LWORK_DORGQR_MN,
+ $ LWORK_DORMBR_PRT_MM, LWORK_DORMBR_QLN_MM,
+ $ LWORK_DORMBR_PRT_MN, LWORK_DORMBR_QLN_MN,
+ $ LWORK_DORMBR_PRT_NN, LWORK_DORMBR_QLN_NN
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
* ..
* .. External Functions ..
LOGICAL LSAME
- INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
- EXTERNAL DLAMCH, DLANGE, ILAENV, LSAME
+ EXTERNAL DLAMCH, DLANGE, LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
*
* Test the input arguments
*
- INFO = 0
- MINMN = MIN( M, N )
- WNTQA = LSAME( JOBZ, 'A' )
- WNTQS = LSAME( JOBZ, 'S' )
+ INFO = 0
+ MINMN = MIN( M, N )
+ WNTQA = LSAME( JOBZ, 'A' )
+ WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
- WNTQO = LSAME( JOBZ, 'O' )
- WNTQN = LSAME( JOBZ, 'N' )
+ WNTQO = LSAME( JOBZ, 'O' )
+ WNTQN = LSAME( JOBZ, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
END IF
*
* Compute workspace
-* (Note: Comments in the code beginning "Workspace:" describe the
-* minimal amount of workspace needed at that point in the code,
+* Note: Comments in the code beginning "Workspace:" describe the
+* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
-* following subroutine, as returned by ILAENV.)
+* following subroutine, as returned by ILAENV.
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
+ BDSPAC = 0
+ MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for DBDSDC
*
- MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
IF( WNTQN ) THEN
+* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+* keep 7*N for backwards compatability.
BDSPAC = 7*N
ELSE
BDSPAC = 3*N*N + 4*N
END IF
+*
+* Compute space preferred for each routine
+ CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_DGEBRD_MN = INT( DUM(1) )
+*
+ CALL DGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_DGEBRD_NN = INT( DUM(1) )
+*
+ CALL DGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_DGEQRF_MN = INT( DUM(1) )
+*
+ CALL DORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
+ $ IERR )
+ LWORK_DORGBR_Q_NN = INT( DUM(1) )
+*
+ CALL DORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_DORGQR_MM = INT( DUM(1) )
+*
+ CALL DORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_DORGQR_MN = INT( DUM(1) )
+*
+ CALL DORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_DORMBR_PRT_NN = INT( DUM(1) )
+*
+ CALL DORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_DORMBR_QLN_NN = INT( DUM(1) )
+*
+ CALL DORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_DORMBR_QLN_MN = INT( DUM(1) )
+*
+ CALL DORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_DORMBR_QLN_MM = INT( DUM(1) )
+*
IF( M.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
*
- WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1,
- $ -1 )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+N )
+ WRKBL = N + LWORK_DGEQRF_MN
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+ MAXWRK = MAX( WRKBL, BDSPAC + N )
MINWRK = BDSPAC + N
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
-*
- WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 2 (M >> N, JOBZ='O')
+*
+ WRKBL = N + LWORK_DGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + 2*N*N
MINWRK = BDSPAC + 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
-*
- WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'DORGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 3 (M >> N, JOBZ='S')
+*
+ WRKBL = N + LWORK_DGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = BDSPAC + N*N + 3*N
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
-*
- WRKBL = N + N*ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'DORGQR', ' ', M,
- $ M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'DGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 4 (M >> N, JOBZ='A')
+*
+ WRKBL = N + LWORK_DGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_MM )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
- MINWRK = BDSPAC + N*N + 2*N + M
+ MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
END IF
ELSE
*
-* Path 5 (M at least N, but not much larger)
+* Path 5 (M >= N, but not much larger)
*
- WRKBL = 3*N + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
- $ -1 )
+ WRKBL = 3*N + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
- MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+* Path 5n (M >= N, jobz='N')
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQO ) THEN
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 5o (M >= N, jobz='O')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + M*N
- MINWRK = 3*N + MAX( M, N*N+BDSPAC )
+ MINWRK = 3*N + MAX( M, N*N + BDSPAC )
ELSE IF( WNTQS ) THEN
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+* Path 5s (M >= N, jobz='S')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQA ) THEN
- WRKBL = MAX( WRKBL, 3*N+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
+* Path 5a (M >= N, jobz='A')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
END IF
END IF
*
* Compute space needed for DBDSDC
*
- MNTHR = INT( MINMN*11.0D0 / 6.0D0 )
IF( WNTQN ) THEN
+* dbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+* keep 7*N for backwards compatability.
BDSPAC = 7*M
ELSE
BDSPAC = 3*M*M + 4*M
END IF
+*
+* Compute space preferred for each routine
+ CALL DGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_DGEBRD_MN = INT( DUM(1) )
+*
+ CALL DGEBRD( M, M, A, M, S, DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_DGEBRD_MM = INT( DUM(1) )
+*
+ CALL DGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
+ LWORK_DGELQF_MN = INT( DUM(1) )
+*
+ CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
+ LWORK_DORGLQ_NN = INT( DUM(1) )
+*
+ CALL DORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
+ LWORK_DORGLQ_MN = INT( DUM(1) )
+*
+ CALL DORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1, IERR )
+ LWORK_DORGBR_P_MM = INT( DUM(1) )
+*
+ CALL DORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_DORMBR_PRT_MM = INT( DUM(1) )
+*
+ CALL DORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_DORMBR_PRT_MN = INT( DUM(1) )
+*
+ CALL DORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_DORMBR_PRT_NN = INT( DUM(1) )
+*
+ CALL DORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_DORMBR_QLN_MM = INT( DUM(1) )
+*
IF( N.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
*
- WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1,
- $ -1 )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+M )
+ WRKBL = M + LWORK_DGELQF_MN
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+ MAXWRK = MAX( WRKBL, BDSPAC + M )
MINWRK = BDSPAC + M
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
-*
- WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 2t (N >> M, JOBZ='O')
+*
+ WRKBL = M + LWORK_DGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + 2*M*M
MINWRK = BDSPAC + 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-*
- WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'DORGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 3t (N >> M, JOBZ='S')
+*
+ WRKBL = M + LWORK_DGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_MN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = BDSPAC + M*M + 3*M
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
-*
- WRKBL = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'DORGLQ', ' ', N,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 4t (N >> M, JOBZ='A')
+*
+ WRKBL = M + LWORK_DGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_NN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
- MINWRK = BDSPAC + M*M + 3*M
+ MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
END IF
ELSE
*
-* Path 5t (N greater than M, but not much larger)
+* Path 5t (N > M, but not much larger)
*
- WRKBL = 3*M + ( M+N )*ILAENV( 1, 'DGEBRD', ' ', M, N, -1,
- $ -1 )
+ WRKBL = 3*M + LWORK_DGEBRD_MN
IF( WNTQN ) THEN
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5tn (N > M, jobz='N')
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQO ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 5to (N > M, jobz='O')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*N
- MINWRK = 3*M + MAX( N, M*M+BDSPAC )
+ MINWRK = 3*M + MAX( N, M*M + BDSPAC )
ELSE IF( WNTQS ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', M, N, M, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5ts (N > M, jobz='S')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_MN )
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQA ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'DORMBR', 'PRT', N, N, M, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5ta (N > M, jobz='A')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
END IF
END IF
END IF
+
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = MAXWRK
*
*
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need 2*N, prefer N+N*NB)
+* Workspace: need N [tau] + N [work]
+* Workspace: prefer N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Zero out below R
*
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NWORK = IE + N
*
* Perform bidiagonal SVD, computing singular values only
-* (Workspace: need N+BDSPAC)
+* Workspace: need N [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ = 'O')
+* Path 2 (M >> N, JOBZ = 'O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
*
* WORK(IR) is LDWRKR by N
*
- IF( LWORK.GE.LDA*N+N*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
LDWRKR = LDA
ELSE
- LDWRKR = ( LWORK-N*N-3*N-BDSPAC ) / N
+ LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
- CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+ CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
-* Bidiagonalize R in VT, copying result to WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Bidiagonalize R in WORK(IR)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* WORK(IU) is N by N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite WORK(IU) by left singular vectors of R
* and VT by right singular vectors of R
-* (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
-* (Workspace: need 2*N*N, prefer N*N+M*N)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U]
+* Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
DO 10 I = 1, M, LDWRKR
- CHUNK = MIN( M-I+1, LDWRKR )
+ CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), N, ZERO, WORK( IR ),
$ LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL DLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
- CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+ CALL DLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagoal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite U by left singular vectors of R and VT
* by right singular vectors of R
-* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
-* (Workspace: need N*N)
+* Workspace: need N*N [R]
*
CALL DLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL DGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ),
*
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* Workspace: need N*N [U] + N [tau] + N [work]
+* Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* Workspace: need N*N [U] + N [tau] + M [work]
+* Workspace: prefer N*N [U] + N [tau] + M*NB [work]
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce R in A, zeroing out other entries
*
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite WORK(IU) by left singular vectors of R and VT
* by right singular vectors of R
-* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
-* (Workspace: need N*N)
+* Workspace: need N*N [U]
*
CALL DGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ),
$ LDWRKU, ZERO, A, LDA )
*
* M .LT. MNTHR
*
-* Path 5 (M at least N, but not much larger)
+* Path 5 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
+* Workspace: need 3*N [e, tauq, taup] + M [work]
+* Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5n (M >= N, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 5o (M >= N, JOBZ='O')
IU = NWORK
- IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* WORK( IU ) is M by N
*
NWORK = IU + LDWRKU*N
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
$ LDWRKU )
+* IR is unused; silence compile warnings
+ IR = -1
ELSE
*
* WORK( IU ) is N by N
* WORK(IR) is LDWRKR by N
*
IR = NWORK
- LDWRKR = ( LWORK-N*N-3*N ) / N
+ LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite VT by right singular vectors of A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
- IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
+* Path 5o-fast
* Overwrite WORK(IU) by left singular vectors of A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + M*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy left singular vectors of A from WORK(IU) to A
*
CALL DLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
+* Path 5o-slow
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of
* bidiagonal matrix in WORK(IU), storing result in
* WORK(IR) and copying to A
-* (Workspace: need 2*N*N, prefer N*N+M*N)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + NB*N [R]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N [R]
*
DO 20 I = 1, M, LDWRKR
- CHUNK = MIN( M-I+1, LDWRKR )
+ CHUNK = MIN( M - I + 1, LDWRKR )
CALL DGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), LDWRKU, ZERO,
$ WORK( IR ), LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
+* Path 5s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 3*N, prefer 2*N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
+* Path 5a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, M, ZERO, ZERO, U, LDU )
CALL DBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
* Set the right corner of U to identity matrix
*
IF( M.GT.N ) THEN
- CALL DLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ),
+ CALL DLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,N+1),
$ LDU )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB)
+* Workspace: need 3*N [e, tauq, taup] + M [work]
+* Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M, prefer M+M*NB)
+* Workspace: need M [tau] + M [work]
+* Workspace: prefer M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Zero out above L
*
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NWORK = IE + M
*
* Perform bidiagonal SVD, computing singular values only
-* (Workspace: need M+BDSPAC)
+* Workspace: need M [e] + BDSPAC
*
CALL DBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IVT = 1
*
-* IVT is M by M
+* WORK(IVT) is M by M
+* WORK(IL) is M by M; it is later resized to M by chunk for gemm
*
IL = IVT + M*M
- IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN
-*
-* WORK(IL) is M by N
-*
+ IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
LDWRKL = M
CHUNK = N
ELSE
LDWRKL = M
- CHUNK = ( LWORK-M*M ) / M
+ CHUNK = ( LWORK - M*M ) / M
END IF
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing about above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
- CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
- $ WORK( IL+LDWRKL ), LDWRKL )
+ CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+ $ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U, and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M+M*M+BDSPAC)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
-* (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), WORK( IVT ), M,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by Q
* in A, storing result in WORK(IL) and copying to A
-* (Workspace: need 2*M*M, prefer M*M+M*N)
+* Workspace: need M*M [VT] + M*M [L]
+* Workspace: prefer M*M [VT] + M*N [L]
+* At this point, L is resized as M by chunk.
*
DO 30 I = 1, N, CHUNK
- BLK = MIN( N-I+1, CHUNK )
+ BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M,
$ A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
CALL DLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
+* Path 3t (N >> M, JOBZ='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing out above it
*
CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
- CALL DLASET( 'U', M-1, M-1, ZERO, ZERO,
- $ WORK( IL+LDWRKL ), LDWRKL )
+ CALL DLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+ $ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
-* Bidiagonalize L in WORK(IU), copying result to U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Bidiagonalize L in WORK(IU).
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite U by left singular vectors of L and VT
* by right singular vectors of L
-* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IL) by
* Q in A, storing result in VT
-* (Workspace: need M*M)
+* Workspace: need M*M [L]
*
CALL DLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M [tau] + N [work]
+* Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce L in A, zeroing out other entries
*
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M+M*M+BDSPAC)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
-* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup]+ M [work]
+* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
-* (Workspace: need M*M)
+* Workspace: need M*M [VT]
*
CALL DGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT,
$ VT, LDVT, ZERO, A, LDA )
*
* N .LT. MNTHR
*
-* Path 5t (N greater than M, but not much larger)
+* Path 5t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
+* Workspace: need 3*M [e, tauq, taup] + N [work]
+* Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5tn (N > M, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 5to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
- IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* WORK( IVT ) is M by N
*
CALL DLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
$ LDWKVT )
NWORK = IVT + LDWKVT*N
+* IL is unused; silence compile warnings
+ IL = -1
ELSE
*
* WORK( IVT ) is M by M
*
* WORK(IL) is M by CHUNK
*
- CHUNK = ( LWORK-M*M-3*M ) / M
+ CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M*M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
- IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
+* Path 5to-fast
* Overwrite WORK(IVT) by left singular vectors of A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*N [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy right singular vectors of A from WORK(IVT) to A
*
CALL DLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
+* Path 5to-slow
* Generate P**T in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by right singular vectors of
* bidiagonal matrix in WORK(IVT), storing result in
* WORK(IL) and copying to A
-* (Workspace: need 2*M*M, prefer M*M+M*N)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N [L]
*
DO 40 I = 1, N, CHUNK
- BLK = MIN( N-I+1, CHUNK )
+ BLK = MIN( N - I + 1, CHUNK )
CALL DGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
$ LDWKVT, A( 1, I ), LDA, ZERO,
$ WORK( IL ), M )
END IF
ELSE IF( WNTQS ) THEN
*
+* Path 5ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 3*M, prefer 2*M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
+* Path 5ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL DLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
CALL DBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
* Set the right corner of VT to identity matrix
*
IF( N.GT.M ) THEN
- CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ),
+ CALL DLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
$ LDVT )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 2*M+N, prefer 2*M+N*NB)
+* Workspace: need 3*M [e, tauq, taup] + N [work]
+* Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
CALL DORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL DORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*> LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):
*> - PATH 1 (M much larger than N, JOBU='N')
*> - PATH 1t (N much larger than M, JOBVT='N')
-*> LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(M,N)) for the other paths
+*> LWORK >= MAX(1,3*MIN(M,N) + MAX(M,N),5*MIN(M,N)) for the other paths
*> For good performance, LWORK should generally be larger.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
BDSPAC = 5*N
* Compute space needed for DGEQRF
CALL DGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_DGEQRF=DUM(1)
+ LWORK_DGEQRF = INT( DUM(1) )
* Compute space needed for DORGQR
CALL DORGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_DORGQR_N=DUM(1)
+ LWORK_DORGQR_N = INT( DUM(1) )
CALL DORGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_DORGQR_M=DUM(1)
+ LWORK_DORGQR_M = INT( DUM(1) )
* Compute space needed for DGEBRD
CALL DGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_DGEBRD=DUM(1)
+ LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORGBR P
CALL DORGBR( 'P', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_P=DUM(1)
+ LWORK_DORGBR_P = INT( DUM(1) )
* Compute space needed for DORGBR Q
CALL DORGBR( 'Q', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_Q=DUM(1)
+ LWORK_DORGBR_Q = INT( DUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
* Path 1 (M much larger than N, JOBU='N')
*
MAXWRK = N + LWORK_DGEQRF
- MAXWRK = MAX( MAXWRK, 3*N+LWORK_DGEBRD )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DGEBRD )
IF( WNTVO .OR. WNTVAS )
- $ MAXWRK = MAX( MAXWRK, 3*N+LWORK_DORGBR_P )
+ $ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*N, BDSPAC )
ELSE IF( WNTUO .AND. WNTVN ) THEN
* Path 2 (M much larger than N, JOBU='O', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_N )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
- MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUO .AND. WNTVAS ) THEN
*
* Path 3 (M much larger than N, JOBU='O', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_N )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
- MAXWRK = MAX( N*N+WRKBL, N*N+M*N+N )
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MAXWRK = MAX( N*N + WRKBL, N*N + M*N + N )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVN ) THEN
*
* Path 4 (M much larger than N, JOBU='S', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_N )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVO ) THEN
*
* Path 5 (M much larger than N, JOBU='S', JOBVT='O')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_N )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUS .AND. WNTVAS ) THEN
*
* Path 6 (M much larger than N, JOBU='S', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_N )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_N )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVN ) THEN
*
* Path 7 (M much larger than N, JOBU='A', JOBVT='N')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_M )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVO ) THEN
*
* Path 8 (M much larger than N, JOBU='A', JOBVT='O')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_M )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
ELSE IF( WNTUA .AND. WNTVAS ) THEN
*
* Path 9 (M much larger than N, JOBU='A', JOBVT='S' or
* 'A')
*
WRKBL = N + LWORK_DGEQRF
- WRKBL = MAX( WRKBL, N+LWORK_DORGQR_M )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_Q )
- WRKBL = MAX( WRKBL, 3*N+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, N + LWORK_DORGQR_M )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = N*N + WRKBL
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
END IF
ELSE
*
*
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_DGEBRD=DUM(1)
+ LWORK_DGEBRD = INT( DUM(1) )
MAXWRK = 3*N + LWORK_DGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL DORGBR( 'Q', M, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_Q=DUM(1)
- MAXWRK = MAX( MAXWRK, 3*N+LWORK_DORGBR_Q )
+ LWORK_DORGBR_Q = INT( DUM(1) )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
END IF
IF( WNTUA ) THEN
CALL DORGBR( 'Q', M, M, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_Q=DUM(1)
- MAXWRK = MAX( MAXWRK, 3*N+LWORK_DORGBR_Q )
+ LWORK_DORGBR_Q = INT( DUM(1) )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
- MAXWRK = MAX( MAXWRK, 3*N+LWORK_DORGBR_P )
+ MAXWRK = MAX( MAXWRK, 3*N + LWORK_DORGBR_P )
END IF
MAXWRK = MAX( MAXWRK, BDSPAC )
- MINWRK = MAX( 3*N+M, BDSPAC )
+ MINWRK = MAX( 3*N + M, BDSPAC )
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
BDSPAC = 5*M
* Compute space needed for DGELQF
CALL DGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_DGELQF=DUM(1)
+ LWORK_DGELQF = INT( DUM(1) )
* Compute space needed for DORGLQ
CALL DORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
- LWORK_DORGLQ_N=DUM(1)
+ LWORK_DORGLQ_N = INT( DUM(1) )
CALL DORGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_DORGLQ_M=DUM(1)
+ LWORK_DORGLQ_M = INT( DUM(1) )
* Compute space needed for DGEBRD
CALL DGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_DGEBRD=DUM(1)
+ LWORK_DGEBRD = INT( DUM(1) )
* Compute space needed for DORGBR P
CALL DORGBR( 'P', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_P=DUM(1)
+ LWORK_DORGBR_P = INT( DUM(1) )
* Compute space needed for DORGBR Q
CALL DORGBR( 'Q', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_Q=DUM(1)
+ LWORK_DORGBR_Q = INT( DUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
* Path 1t(N much larger than M, JOBVT='N')
*
MAXWRK = M + LWORK_DGELQF
- MAXWRK = MAX( MAXWRK, 3*M+LWORK_DGEBRD )
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DGEBRD )
IF( WNTUO .OR. WNTUAS )
- $ MAXWRK = MAX( MAXWRK, 3*M+LWORK_DORGBR_Q )
+ $ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
MAXWRK = MAX( MAXWRK, BDSPAC )
MINWRK = MAX( 4*M, BDSPAC )
ELSE IF( WNTVO .AND. WNTUN ) THEN
* Path 2t(N much larger than M, JOBU='N', JOBVT='O')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_M )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
- MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVO .AND. WNTUAS ) THEN
*
* Path 3t(N much larger than M, JOBU='S' or 'A',
* JOBVT='O')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_M )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
- MAXWRK = MAX( M*M+WRKBL, M*M+M*N+M )
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MAXWRK = MAX( M*M + WRKBL, M*M + M*N + M )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUN ) THEN
*
* Path 4t(N much larger than M, JOBU='N', JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_M )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUO ) THEN
*
* Path 5t(N much larger than M, JOBU='O', JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_M )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVS .AND. WNTUAS ) THEN
*
* Path 6t(N much larger than M, JOBU='S' or 'A',
* JOBVT='S')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_M )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_M )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUN ) THEN
*
* Path 7t(N much larger than M, JOBU='N', JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_N )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUO ) THEN
*
* Path 8t(N much larger than M, JOBU='O', JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_N )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = 2*M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
ELSE IF( WNTVA .AND. WNTUAS ) THEN
*
* Path 9t(N much larger than M, JOBU='S' or 'A',
* JOBVT='A')
*
WRKBL = M + LWORK_DGELQF
- WRKBL = MAX( WRKBL, M+LWORK_DORGLQ_N )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DGEBRD )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_P )
- WRKBL = MAX( WRKBL, 3*M+LWORK_DORGBR_Q )
+ WRKBL = MAX( WRKBL, M + LWORK_DORGLQ_N )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DGEBRD )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_P )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_DORGBR_Q )
WRKBL = MAX( WRKBL, BDSPAC )
MAXWRK = M*M + WRKBL
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
END IF
ELSE
*
*
CALL DGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_DGEBRD=DUM(1)
+ LWORK_DGEBRD = INT( DUM(1) )
MAXWRK = 3*M + LWORK_DGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for DORGBR P
CALL DORGBR( 'P', M, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_P=DUM(1)
- MAXWRK = MAX( MAXWRK, 3*M+LWORK_DORGBR_P )
+ LWORK_DORGBR_P = INT( DUM(1) )
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
END IF
IF( WNTVA ) THEN
CALL DORGBR( 'P', N, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_DORGBR_P=DUM(1)
- MAXWRK = MAX( MAXWRK, 3*M+LWORK_DORGBR_P )
+ LWORK_DORGBR_P = INT( DUM(1) )
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
- MAXWRK = MAX( MAXWRK, 3*M+LWORK_DORGBR_Q )
+ MAXWRK = MAX( MAXWRK, 3*M + LWORK_DORGBR_Q )
END IF
MAXWRK = MAX( MAXWRK, BDSPAC )
- MINWRK = MAX( 3*M+N, BDSPAC )
+ MINWRK = MAX( 3*M + N, BDSPAC )
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
IF( WNTVO .OR. WNTVAS ) THEN
*
* If right singular vectors desired, generate P'.
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Sufficient workspace for a fast algorithm
*
IR = 1
- IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN
+ IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
- ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN
+ ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
* WORK(IU) is LDA by N, WORK(IR) is N by N
*
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM, 1,
$ WORK( IR ), LDWRKR, DUM, 1,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
-* (Workspace: need N*N+2*N, prefer N*N+M*N+N)
+* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
DO 10 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
IWORK = ITAUP + N
*
* Bidiagonalize A
-* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
+* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing A
-* (Workspace: need 4*N, prefer 3*N+N*NB)
+* (Workspace: need 4*N, prefer 3*N + N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Sufficient workspace for a fast algorithm
*
IR = 1
- IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+LDA*N ) THEN
+ IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + LDA*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by N
*
LDWRKU = LDA
LDWRKR = LDA
- ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+N )+N*N ) THEN
+ ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + N ) + N*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in VT, copying result to WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
CALL DLACPY( 'L', N, N, VT, LDVT, WORK( IR ), LDWRKR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
-* (Workspace: need N*N+4*N-1, prefer N*N+3*N+(N-1)*NB)
+* (Workspace: need N*N + 4*N-1, prefer N*N + 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR) and computing right
* singular vectors of R in VT
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT, LDVT,
$ WORK( IR ), LDWRKR, DUM, 1,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in WORK(IU) and copying to A
-* (Workspace: need N*N+2*N, prefer N*N+M*N+N)
+* (Workspace: need N*N + 2*N, prefer N*N + M*N + N)
*
DO 20 I = 1, M, LDWRKU
CHUNK = MIN( M-I+1, LDWRKU )
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ VT( 2, 1 ), LDVT )
*
* Generate Q in A
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in A by left vectors bidiagonalizing R
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in VT
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
- ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
+ ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
+* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
-* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
+* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
-* (Workspace: need 2*N*N+4*N,
+* (Workspace: need 2*N*N + 4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
-* (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB)
+* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
-* (Workspace: need 2*N*N+4*N-1,
+* (Workspace: need 2*N*N + 4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
-* (Workspace: need 2*N*N+BDSPAC)
+* (Workspace: need 2*N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply Q in U by left vectors bidiagonalizing R
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing R in A
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IU+1 ), LDWRKU )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DORGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
-* (Workspace: need N*N+4*N-1,
+* (Workspace: need N*N + 4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DORGQR( M, N, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IR+1 ), LDWRKR )
*
* Generate Q in U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
*
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IR)
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, 0, N, 0, S, WORK( IE ), DUM,
$ 1, WORK( IR ), LDWRKR, DUM, 1,
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N+M, prefer N+M*NB)
+* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
LDWRKU = LDA
IR = IU + LDWRKU*N
LDWRKR = LDA
- ELSE IF( LWORK.GE.WRKBL+( LDA+N )*N ) THEN
+ ELSE IF( LWORK.GE.WRKBL+( LDA + N )*N ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is N by N
*
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N*N+2*N, prefer 2*N*N+N+N*NB)
+* (Workspace: need 2*N*N + 2*N, prefer 2*N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need 2*N*N+N+M, prefer 2*N*N+N+M*NB)
+* (Workspace: need 2*N*N + N + M, prefer 2*N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Bidiagonalize R in WORK(IU), copying result to
* WORK(IR)
-* (Workspace: need 2*N*N+4*N,
+* (Workspace: need 2*N*N + 4*N,
* prefer 2*N*N+3*N+2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IR ), LDWRKR )
*
* Generate left bidiagonalizing vectors in WORK(IU)
-* (Workspace: need 2*N*N+4*N, prefer 2*N*N+3*N+N*NB)
+* (Workspace: need 2*N*N + 4*N, prefer 2*N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
-* (Workspace: need 2*N*N+4*N-1,
+* (Workspace: need 2*N*N + 4*N-1,
* prefer 2*N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, WORK( IR ), LDWRKR,
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in WORK(IR)
-* (Workspace: need 2*N*N+BDSPAC)
+* (Workspace: need 2*N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, WORK( IU ),
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N+M, prefer N+M*NB)
+* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
END IF
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply Q in U by left bidiagonalizing vectors
* in A
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in A
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* (Workspace: need N*N + 2*N, prefer N*N + N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* (Workspace: need N*N + N + M, prefer N*N + N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IU), copying result to VT
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LDVT )
*
* Generate left bidiagonalizing vectors in WORK(IU)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+N*NB)
+* (Workspace: need N*N + 4*N, prefer N*N + 3*N + N*NB)
*
CALL DORGBR( 'Q', N, N, N, WORK( IU ), LDWRKU,
$ WORK( ITAUQ ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
-* (Workspace: need N*N+4*N-1,
+* (Workspace: need N*N + 4*N-1,
* prefer N*N+3*N+(N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal QR iteration, computing left
* singular vectors of R in WORK(IU) and computing
* right singular vectors of R in VT
-* (Workspace: need N*N+BDSPAC)
+* (Workspace: need N*N + BDSPAC)
*
CALL DBDSQR( 'U', N, N, N, 0, S, WORK( IE ), VT,
$ LDVT, WORK( IU ), LDWRKU, DUM, 1,
IWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need 2*N, prefer N+N*NB)
+* (Workspace: need 2*N, prefer N + N*NB)
*
CALL DGEQRF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N+M, prefer N+M*NB)
+* (Workspace: need N + M, prefer N + M*NB)
*
CALL DORGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize R in VT
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* (Workspace: need 4*N, prefer 3*N + 2*N*NB)
*
CALL DGEBRD( N, N, VT, LDVT, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply Q in U by left bidiagonalizing vectors
* in VT
-* (Workspace: need 3*N+M, prefer 3*N+M*NB)
+* (Workspace: need 3*N + M, prefer 3*N + M*NB)
*
CALL DORMBR( 'Q', 'R', 'N', M, N, N, VT, LDVT,
$ WORK( ITAUQ ), U, LDU, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in VT
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + N
*
* Bidiagonalize A
-* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
+* (Workspace: need 3*N + M, prefer 3*N + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
-* (Workspace: need 3*N+NCU, prefer 3*N+NCU*NB)
+* (Workspace: need 3*N + NCU, prefer 3*N + NCU*NB)
*
CALL DLACPY( 'L', M, N, A, LDA, U, LDU )
IF( WNTUS )
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL DORGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
-* (Workspace: need 4*N, prefer 3*N+N*NB)
+* (Workspace: need 4*N, prefer 3*N + N*NB)
*
CALL DORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
-* (Workspace: need 4*N-1, prefer 3*N+(N-1)*NB)
+* (Workspace: need 4*N-1, prefer 3*N + (N-1)*NB)
*
CALL DORGBR( 'P', N, N, N, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
IF( WNTUO .OR. WNTUAS ) THEN
*
* If left singular vectors desired, generate Q
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Sufficient workspace for a fast algorithm
*
IR = 1
- IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN
+ IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
- ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN
+ ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing L
-* (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB)
+* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
-* (Workspace: need M*M+2*M, prefer M*M+M*N+M)
+* (Workspace: need M*M + 2*M, prefer M*M + M*N + M)
*
DO 30 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
IWORK = ITAUP + M
*
* Bidiagonalize A
-* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
+* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate right vectors bidiagonalizing A
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Sufficient workspace for a fast algorithm
*
IR = 1
- IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+LDA*M ) THEN
+ IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + LDA*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is LDA by M
*
LDWRKU = LDA
CHUNK = N
LDWRKR = LDA
- ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N+M )+M*M ) THEN
+ ELSE IF( LWORK.GE.MAX( WRKBL, LDA*N + M ) + M*M ) THEN
*
* WORK(IU) is LDA by N and WORK(IR) is M by M
*
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDU )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in U, copying result to WORK(IR)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
CALL DLACPY( 'U', M, M, U, LDU, WORK( IR ), LDWRKR )
*
* Generate right vectors bidiagonalizing L in WORK(IR)
-* (Workspace: need M*M+4*M-1, prefer M*M+3*M+(M-1)*NB)
+* (Workspace: need M*M + 4*M-1, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U, and computing right
* singular vectors of L in WORK(IR)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, U, LDU, DUM, 1,
*
* Multiply right singular vectors of L in WORK(IR) by Q
* in A, storing result in WORK(IU) and copying to A
-* (Workspace: need M*M+2*M, prefer M*M+M*N+M))
+* (Workspace: need M*M + 2*M, prefer M*M + M*N + M))
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDU )
*
* Generate Q in A
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in U
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in A
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), A, LDA, WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left vectors bidiagonalizing L in U
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
*
* Generate right vectors bidiagonalizing L in
* WORK(IR)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), WORK( IWORK ),
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDA )
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
- ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
+ ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
+* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
-* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
+* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
-* (Workspace: need 2*M*M+4*M,
+* (Workspace: need 2*M*M + 4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
-* (Workspace: need 2*M*M+4*M-1,
+* (Workspace: need 2*M*M + 4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
-* (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB)
+* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
-* (Workspace: need 2*M*M+BDSPAC)
+* (Workspace: need 2*M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDA )
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Multiply right vectors bidiagonalizing L by Q in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors of L in A
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IU+LDWRKU ), LDWRKU )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DORGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
-* (Workspace: need M*M+4*M-1,
+* (Workspace: need M*M + 4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DORGLQ( M, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in U
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* N right singular vectors to be computed in VT and
* no left singular vectors to be computed
*
- IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
+ IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ WORK( IR+LDWRKR ), LDWRKR )
*
* Generate Q in VT
-* (Workspace: need M*M+M+N, prefer M*M+M+N*NB)
+* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IR)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IR ), LDWRKR, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LWORK-IWORK+1, IERR )
*
* Generate right bidiagonalizing vectors in WORK(IR)
-* (Workspace: need M*M+4*M-1,
+* (Workspace: need M*M + 4*M-1,
* prefer M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IR ), LDWRKR,
*
* Perform bidiagonal QR iteration, computing right
* singular vectors of L in WORK(IR)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, 0, 0, S, WORK( IE ),
$ WORK( IR ), LDWRKR, DUM, 1, DUM, 1,
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M+N, prefer M+N*NB)
+* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDA )
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
* N right singular vectors to be computed in VT and
* M left singular vectors to be overwritten on A
*
- IF( LWORK.GE.2*M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
+ IF( LWORK.GE.2*M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
LDWRKU = LDA
IR = IU + LDWRKU*M
LDWRKR = LDA
- ELSE IF( LWORK.GE.WRKBL+( LDA+M )*M ) THEN
+ ELSE IF( LWORK.GE.WRKBL+( LDA + M )*M ) THEN
*
* WORK(IU) is LDA by M and WORK(IR) is M by M
*
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M*M+2*M, prefer 2*M*M+M+M*NB)
+* (Workspace: need 2*M*M + 2*M, prefer 2*M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need 2*M*M+M+N, prefer 2*M*M+M+N*NB)
+* (Workspace: need 2*M*M + M + N, prefer 2*M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Bidiagonalize L in WORK(IU), copying result to
* WORK(IR)
-* (Workspace: need 2*M*M+4*M,
+* (Workspace: need 2*M*M + 4*M,
* prefer 2*M*M+3*M+2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IR ), LDWRKR )
*
* Generate right bidiagonalizing vectors in WORK(IU)
-* (Workspace: need 2*M*M+4*M-1,
+* (Workspace: need 2*M*M + 4*M-1,
* prefer 2*M*M+3*M+(M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in WORK(IR)
-* (Workspace: need 2*M*M+4*M, prefer 2*M*M+3*M+M*NB)
+* (Workspace: need 2*M*M + 4*M, prefer 2*M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IWORK ),
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in WORK(IR) and computing
* right singular vectors of L in WORK(IU)
-* (Workspace: need 2*M*M+BDSPAC)
+* (Workspace: need 2*M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, WORK( IR ),
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M+N, prefer M+N*NB)
+* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
$ LDA )
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, A, LDA, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply right bidiagonalizing vectors in A by Q
* in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in A
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
- IF( LWORK.GE.M*M+MAX( N+M, 4*M, BDSPAC ) ) THEN
+ IF( LWORK.GE.M*M+MAX( N + M, 4*M, BDSPAC ) ) THEN
*
* Sufficient workspace for a fast algorithm
*
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* (Workspace: need M*M + 2*M, prefer M*M + M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M*M+M+N, prefer M*M+M+N*NB)
+* (Workspace: need M*M + M + N, prefer M*M + M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IU), copying result to U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, WORK( IU ), LDWRKU, S,
$ WORK( IE ), WORK( ITAUQ ),
$ LDU )
*
* Generate right bidiagonalizing vectors in WORK(IU)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+(M-1)*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + (M-1)*NB)
*
CALL DORGBR( 'P', M, M, M, WORK( IU ), LDWRKU,
$ WORK( ITAUP ), WORK( IWORK ),
$ LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+M*NB)
+* (Workspace: need M*M + 4*M, prefer M*M + 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
* Perform bidiagonal QR iteration, computing left
* singular vectors of L in U and computing right
* singular vectors of L in WORK(IU)
-* (Workspace: need M*M+BDSPAC)
+* (Workspace: need M*M + BDSPAC)
*
CALL DBDSQR( 'U', M, M, M, 0, S, WORK( IE ),
$ WORK( IU ), LDWRKU, U, LDU, DUM, 1,
IWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need 2*M, prefer M+M*NB)
+* (Workspace: need 2*M, prefer M + M*NB)
*
CALL DGELQF( M, N, A, LDA, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M+N, prefer M+N*NB)
+* (Workspace: need M + N, prefer M + N*NB)
*
CALL DORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize L in U
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* (Workspace: need 4*M, prefer 3*M + 2*M*NB)
*
CALL DGEBRD( M, M, U, LDU, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ),
*
* Multiply right bidiagonalizing vectors in U by Q
* in VT
-* (Workspace: need 3*M+N, prefer 3*M+N*NB)
+* (Workspace: need 3*M + N, prefer 3*M + N*NB)
*
CALL DORMBR( 'P', 'L', 'T', M, N, M, U, LDU,
$ WORK( ITAUP ), VT, LDVT,
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* Generate left bidiagonalizing vectors in U
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'Q', M, M, M, U, LDU, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
IWORK = ITAUP + M
*
* Bidiagonalize A
-* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
+* (Workspace: need 3*M + N, prefer 3*M + (M + N)*NB)
*
CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( IWORK ), LWORK-IWORK+1,
*
* If left singular vectors desired in U, copy result to U
* and generate left bidiagonalizing vectors in U
-* (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB)
+* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
CALL DLACPY( 'L', M, M, A, LDA, U, LDU )
CALL DORGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
*
* If right singular vectors desired in VT, copy result to
* VT and generate right bidiagonalizing vectors in VT
-* (Workspace: need 3*M+NRVT, prefer 3*M+NRVT*NB)
+* (Workspace: need 3*M + NRVT, prefer 3*M + NRVT*NB)
*
CALL DLACPY( 'U', M, N, A, LDA, VT, LDVT )
IF( WNTVA )
*
* If left singular vectors desired in A, generate left
* bidiagonalizing vectors in A
-* (Workspace: need 4*M-1, prefer 3*M+(M-1)*NB)
+* (Workspace: need 4*M-1, prefer 3*M + (M-1)*NB)
*
CALL DORGBR( 'Q', M, M, N, A, LDA, WORK( ITAUQ ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
*
* If right singular vectors desired in A, generate right
* bidiagonalizing vectors in A
-* (Workspace: need 4*M, prefer 3*M+M*NB)
+* (Workspace: need 4*M, prefer 3*M + M*NB)
*
CALL DORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( IWORK ), LWORK-IWORK+1, IERR )
WSTART = 1
QSTART = 3
IF( ICOMPQ.EQ.1 ) THEN
- CALL SCOPY( N, D, 1, Q( 1 ), 1 )
+ CALL SCOPY( N, D, 1, Q( 1 ), 1 )
CALL SCOPY( N-1, E, 1, Q( N+1 ), 1 )
END IF
IF( IUPLO.EQ.2 ) THEN
* If ICOMPQ = 0, use SLASDQ to compute the singular values.
*
IF( ICOMPQ.EQ.0 ) THEN
+* Ignore WSTART, instead using WORK( 1 ), since the two vectors
+* for CS and -SN above are added only if ICOMPQ == 2,
+* and adding them exceeds documented WORK size of 4*n.
CALL SLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
- $ LDU, WORK( WSTART ), INFO )
+ $ LDU, WORK( 1 ), INFO )
GO TO 40
END IF
*
* Definition:
* ===========
*
-* SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
-* LWORK, IWORK, INFO )
+* SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
* ..
* .. Array Arguments ..
* INTEGER IWORK( * )
-* REAL A( LDA, * ), S( * ), U( LDU, * ),
+* REAL A( LDA, * ), S( * ), U( LDU, * ),
* $ VT( LDVT, * ), WORK( * )
* ..
*
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
-*> The leading dimension of the array VT. LDVT >= 1; if
-*> JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*> The leading dimension of the array VT. LDVT >= 1;
+*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
-*> If JOBZ = 'N',
-*> LWORK >= 3*min(M,N) + max(max(M,N),6*min(M,N)).
-*> If JOBZ = 'O',
-*> LWORK >= 3*min(M,N) +
-*> max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
-*> If JOBZ = 'S' or 'A'
-*> LWORK >= min(M,N)*(7+4*min(M,N))
-*> For good performance, LWORK should generally be larger.
-*> If LWORK = -1 but other input arguments are legal, WORK(1)
-*> returns the optimal LWORK.
+*> If LWORK = -1, a workspace query is assumed. The optimal
+*> size for the WORK array is calculated and stored in WORK(1),
+*> and no other work except argument checking is performed.
+*>
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).
+*> If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).
+*> If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.
+*> If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.
+*> These are not tight minimums in all cases; see comments inside code.
+*> For good performance, LWORK should generally be larger;
+*> a query is recommended.
*> \endverbatim
*>
*> \param[out] IWORK
*> California at Berkeley, USA
*>
* =====================================================================
- SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
- $ LWORK, IWORK, INFO )
+ SUBROUTINE SGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+ $ WORK, LWORK, IWORK, INFO )
+ implicit none
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* ..
* .. Array Arguments ..
INTEGER IWORK( * )
- REAL A( LDA, * ), S( * ), U( LDU, * ),
+ REAL A( LDA, * ), S( * ), U( LDU, * ),
$ VT( LDVT, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
- REAL ZERO, ONE
+ REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
$ IR, ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR, NWORK, WRKBL
- REAL ANRM, BIGNUM, EPS, SMLNUM
+ INTEGER LWORK_SGEBRD_MN, LWORK_SGEBRD_MM,
+ $ LWORK_SGEBRD_NN, LWORK_SGELQF_MN,
+ $ LWORK_SGEQRF_MN,
+ $ LWORK_SORGBR_P_MM, LWORK_SORGBR_Q_NN,
+ $ LWORK_SORGLQ_MN, LWORK_SORGLQ_NN,
+ $ LWORK_SORGQR_MM, LWORK_SORGQR_MN,
+ $ LWORK_SORMBR_PRT_MM, LWORK_SORMBR_QLN_MM,
+ $ LWORK_SORMBR_PRT_MN, LWORK_SORMBR_QLN_MN,
+ $ LWORK_SORMBR_PRT_NN, LWORK_SORMBR_QLN_NN
+ REAL ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
- INTEGER ILAENV
REAL SLAMCH, SLANGE
- EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
+ EXTERNAL SLAMCH, SLANGE, LSAME
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
*
* Test the input arguments
*
- INFO = 0
- MINMN = MIN( M, N )
- WNTQA = LSAME( JOBZ, 'A' )
- WNTQS = LSAME( JOBZ, 'S' )
+ INFO = 0
+ MINMN = MIN( M, N )
+ WNTQA = LSAME( JOBZ, 'A' )
+ WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
- WNTQO = LSAME( JOBZ, 'O' )
- WNTQN = LSAME( JOBZ, 'N' )
+ WNTQO = LSAME( JOBZ, 'O' )
+ WNTQN = LSAME( JOBZ, 'N' )
LQUERY = ( LWORK.EQ.-1 )
*
IF( .NOT.( WNTQA .OR. WNTQS .OR. WNTQO .OR. WNTQN ) ) THEN
END IF
*
* Compute workspace
-* (Note: Comments in the code beginning "Workspace:" describe the
-* minimal amount of workspace needed at that point in the code,
+* Note: Comments in the code beginning "Workspace:" describe the
+* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* NB refers to the optimal block size for the immediately
-* following subroutine, as returned by ILAENV.)
+* following subroutine, as returned by ILAENV.
*
IF( INFO.EQ.0 ) THEN
MINWRK = 1
MAXWRK = 1
+ BDSPAC = 0
+ MNTHR = INT( MINMN*11.0E0 / 6.0E0 )
IF( M.GE.N .AND. MINMN.GT.0 ) THEN
*
* Compute space needed for SBDSDC
*
- MNTHR = INT( MINMN*11.0E0 / 6.0E0 )
IF( WNTQN ) THEN
+* sbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+* keep 7*N for backwards compatability.
BDSPAC = 7*N
ELSE
BDSPAC = 3*N*N + 4*N
END IF
+*
+* Compute space preferred for each routine
+ CALL SGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_SGEBRD_MN = INT( DUM(1) )
+*
+ CALL SGEBRD( N, N, DUM(1), N, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_SGEBRD_NN = INT( DUM(1) )
+*
+ CALL SGEQRF( M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_SGEQRF_MN = INT( DUM(1) )
+*
+ CALL SORGBR( 'Q', N, N, N, DUM(1), N, DUM(1), DUM(1), -1,
+ $ IERR )
+ LWORK_SORGBR_Q_NN = INT( DUM(1) )
+*
+ CALL SORGQR( M, M, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_SORGQR_MM = INT( DUM(1) )
+*
+ CALL SORGQR( M, N, N, DUM(1), M, DUM(1), DUM(1), -1, IERR )
+ LWORK_SORGQR_MN = INT( DUM(1) )
+*
+ CALL SORMBR( 'P', 'R', 'T', N, N, N, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_SORMBR_PRT_NN = INT( DUM(1) )
+*
+ CALL SORMBR( 'Q', 'L', 'N', N, N, N, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_SORMBR_QLN_NN = INT( DUM(1) )
+*
+ CALL SORMBR( 'Q', 'L', 'N', M, N, N, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_SORMBR_QLN_MN = INT( DUM(1) )
+*
+ CALL SORMBR( 'Q', 'L', 'N', M, M, N, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_SORMBR_QLN_MM = INT( DUM(1) )
+*
IF( M.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
*
- WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1,
- $ -1 )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+N )
+ WRKBL = N + LWORK_SGEQRF_MN
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SGEBRD_NN )
+ MAXWRK = MAX( WRKBL, BDSPAC + N )
MINWRK = BDSPAC + N
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
-*
- WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'SORGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 2 (M >> N, JOBZ='O')
+*
+ WRKBL = N + LWORK_SGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_SORGQR_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + 2*N*N
MINWRK = BDSPAC + 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
-*
- WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'SORGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 3 (M >> N, JOBZ='S')
+*
+ WRKBL = N + LWORK_SGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_SORGQR_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
MINWRK = BDSPAC + N*N + 3*N
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
-*
- WRKBL = N + N*ILAENV( 1, 'SGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'SORGQR', ' ', M,
- $ M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+2*N*
- $ ILAENV( 1, 'SGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 4 (M >> N, JOBZ='A')
+*
+ WRKBL = N + LWORK_SGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_SORGQR_MM )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SGEBRD_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + N*N
- MINWRK = BDSPAC + N*N + 2*N + M
+ MINWRK = N*N + MAX( 3*N + BDSPAC, N + M )
END IF
ELSE
*
-* Path 5 (M at least N, but not much larger)
+* Path 5 (M >= N, but not much larger)
*
- WRKBL = 3*N + ( M+N )*ILAENV( 1, 'SGEBRD', ' ', M, N, -1,
- $ -1 )
+ WRKBL = 3*N + LWORK_SGEBRD_MN
IF( WNTQN ) THEN
- MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+* Path 5n (M >= N, jobz='N')
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQO ) THEN
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*N )
+* Path 5o (M >= N, jobz='O')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_MN )
+ WRKBL = MAX( WRKBL, 3*N + BDSPAC )
MAXWRK = WRKBL + M*N
- MINWRK = 3*N + MAX( M, N*N+BDSPAC )
+ MINWRK = 3*N + MAX( M, N*N + BDSPAC )
ELSE IF( WNTQS ) THEN
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*N )
+* Path 5s (M >= N, jobz='S')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_MN )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
ELSE IF( WNTQA ) THEN
- WRKBL = MAX( WRKBL, 3*N+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*N+N*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, BDSPAC+3*N )
+* Path 5a (M >= N, jobz='A')
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*N + LWORK_SORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*N + BDSPAC )
MINWRK = 3*N + MAX( M, BDSPAC )
END IF
END IF
- ELSE IF ( MINMN.GT.0 ) THEN
+ ELSE IF( MINMN.GT.0 ) THEN
*
* Compute space needed for SBDSDC
*
- MNTHR = INT( MINMN*11.0E0 / 6.0E0 )
IF( WNTQN ) THEN
+* sbdsdc needs only 4*N (or 6*N for uplo=L for LAPACK <= 3.6)
+* keep 7*N for backwards compatability.
BDSPAC = 7*M
ELSE
BDSPAC = 3*M*M + 4*M
END IF
+*
+* Compute space preferred for each routine
+ CALL SGEBRD( M, N, DUM(1), M, DUM(1), DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_SGEBRD_MN = INT( DUM(1) )
+*
+ CALL SGEBRD( M, M, A, M, S, DUM(1), DUM(1),
+ $ DUM(1), DUM(1), -1, IERR )
+ LWORK_SGEBRD_MM = INT( DUM(1) )
+*
+ CALL SGELQF( M, N, A, M, DUM(1), DUM(1), -1, IERR )
+ LWORK_SGELQF_MN = INT( DUM(1) )
+*
+ CALL SORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
+ LWORK_SORGLQ_NN = INT( DUM(1) )
+*
+ CALL SORGLQ( M, N, M, A, M, DUM(1), DUM(1), -1, IERR )
+ LWORK_SORGLQ_MN = INT( DUM(1) )
+*
+ CALL SORGBR( 'P', M, M, M, A, N, DUM(1), DUM(1), -1, IERR )
+ LWORK_SORGBR_P_MM = INT( DUM(1) )
+*
+ CALL SORMBR( 'P', 'R', 'T', M, M, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_SORMBR_PRT_MM = INT( DUM(1) )
+*
+ CALL SORMBR( 'P', 'R', 'T', M, N, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_SORMBR_PRT_MN = INT( DUM(1) )
+*
+ CALL SORMBR( 'P', 'R', 'T', N, N, M, DUM(1), N,
+ $ DUM(1), DUM(1), N, DUM(1), -1, IERR )
+ LWORK_SORMBR_PRT_NN = INT( DUM(1) )
+*
+ CALL SORMBR( 'Q', 'L', 'N', M, M, M, DUM(1), M,
+ $ DUM(1), DUM(1), M, DUM(1), -1, IERR )
+ LWORK_SORMBR_QLN_MM = INT( DUM(1) )
+*
IF( N.GE.MNTHR ) THEN
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
*
- WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1,
- $ -1 )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+M )
+ WRKBL = M + LWORK_SGELQF_MN
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SGEBRD_MM )
+ MAXWRK = MAX( WRKBL, BDSPAC + M )
MINWRK = BDSPAC + M
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
-*
- WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'SORGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 2t (N >> M, JOBZ='O')
+*
+ WRKBL = M + LWORK_SGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_SORGLQ_MN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + 2*M*M
MINWRK = BDSPAC + 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-*
- WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'SORGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 3t (N >> M, JOBZ='S')
+*
+ WRKBL = M + LWORK_SGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_SORGLQ_MN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
MINWRK = BDSPAC + M*M + 3*M
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
-*
- WRKBL = M + M*ILAENV( 1, 'SGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'SORGLQ', ' ', N,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+2*M*
- $ ILAENV( 1, 'SGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 4t (N >> M, JOBZ='A')
+*
+ WRKBL = M + LWORK_SGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_SORGLQ_NN )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SGEBRD_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_MM )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*M
- MINWRK = BDSPAC + M*M + 3*M
+ MINWRK = M*M + MAX( 3*M + BDSPAC, M + N )
END IF
ELSE
*
-* Path 5t (N greater than M, but not much larger)
+* Path 5t (N > M, but not much larger)
*
- WRKBL = 3*M + ( M+N )*ILAENV( 1, 'SGEBRD', ' ', M, N, -1,
- $ -1 )
+ WRKBL = 3*M + LWORK_SGEBRD_MN
IF( WNTQN ) THEN
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5tn (N > M, jobz='N')
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQO ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', M, N, M, -1 ) )
- WRKBL = MAX( WRKBL, BDSPAC+3*M )
+* Path 5to (N > M, jobz='O')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_MN )
+ WRKBL = MAX( WRKBL, 3*M + BDSPAC )
MAXWRK = WRKBL + M*N
- MINWRK = 3*M + MAX( N, M*M+BDSPAC )
+ MINWRK = 3*M + MAX( N, M*M + BDSPAC )
ELSE IF( WNTQS ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', M, N, M, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5ts (N > M, jobz='S')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_MN )
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
ELSE IF( WNTQA ) THEN
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'QLN', M, M, N, -1 ) )
- WRKBL = MAX( WRKBL, 3*M+M*
- $ ILAENV( 1, 'SORMBR', 'PRT', N, N, M, -1 ) )
- MAXWRK = MAX( WRKBL, BDSPAC+3*M )
+* Path 5ta (N > M, jobz='A')
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 3*M + LWORK_SORMBR_PRT_NN )
+ MAXWRK = MAX( WRKBL, 3*M + BDSPAC )
MINWRK = 3*M + MAX( N, BDSPAC )
END IF
END IF
END IF
+
MAXWRK = MAX( MAXWRK, MINWRK )
WORK( 1 ) = MAXWRK
*
*
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need 2*N, prefer N+N*NB)
+* Workspace: need N [tau] + N [work]
+* Workspace: prefer N [tau] + N*NB [work]
*
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Zero out below R
*
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (Workspace: need 4*N, prefer 3*N+2*N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL SGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NWORK = IE + N
*
* Perform bidiagonal SVD, computing singular values only
-* (Workspace: need N+BDSPAC)
+* Workspace: need N [e] + BDSPAC
*
CALL SBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ = 'O')
+* Path 2 (M >> N, JOBZ = 'O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
*
* WORK(IR) is LDWRKR by N
*
- IF( LWORK.GE.LDA*N+N*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. LDA*N + N*N + 3*N + BDSPAC ) THEN
LDWRKR = LDA
ELSE
- LDWRKR = ( LWORK-N*N-3*N-BDSPAC ) / N
+ LDWRKR = ( LWORK - N*N - 3*N - BDSPAC ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL SLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
- CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+ CALL SLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL SORGQR( M, N, N, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
-* Bidiagonalize R in VT, copying result to WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Bidiagonalize R in WORK(IR)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL SGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* WORK(IU) is N by N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite WORK(IU) by left singular vectors of R
* and VT by right singular vectors of R
-* (Workspace: need 2*N*N+3*N, prefer 2*N*N+2*N+N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), WORK( IU ), N, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
-* (Workspace: need 2*N*N, prefer N*N+M*N)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N*N [U]
+* Workspace: prefer M*N [R] + 3*N [e, tauq, taup] + N*N [U]
*
DO 10 I = 1, M, LDWRKR
- CHUNK = MIN( M-I+1, LDWRKR )
+ CHUNK = MIN( M - I + 1, LDWRKR )
CALL SGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), N, ZERO, WORK( IR ),
$ LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy R to WORK(IR), zeroing out below it
*
CALL SLACPY( 'U', N, N, A, LDA, WORK( IR ), LDWRKR )
- CALL SLASET( 'L', N-1, N-1, ZERO, ZERO, WORK( IR+1 ),
+ CALL SLASET( 'L', N - 1, N - 1, ZERO, ZERO, WORK(IR+1),
$ LDWRKR )
*
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need N*N [R] + N [tau] + N [work]
+* Workspace: prefer N*N [R] + N [tau] + N*NB [work]
*
CALL SORGQR( M, N, N, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + N
ITAUP = ITAUQ + N
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL SGEBRD( N, N, WORK( IR ), LDWRKR, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagoal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite U by left singular vectors of R and VT
* by right singular vectors of R
-* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+* Workspace: need N*N [R] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [R] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
CALL SORMBR( 'P', 'R', 'T', N, N, N, WORK( IR ), LDWRKR,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
-* (Workspace: need N*N)
+* Workspace: need N*N [R]
*
CALL SLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL SGEMM( 'N', 'N', M, N, N, ONE, A, LDA, WORK( IR ),
*
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* Workspace: need N*N [U] + N [tau] + N [work]
+* Workspace: prefer N*N [U] + N [tau] + N*NB [work]
*
CALL SGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (Workspace: need N*N+N+M, prefer N*N+N+M*NB)
+* Workspace: need N*N [U] + N [tau] + M [work]
+* Workspace: prefer N*N [U] + N [tau] + M*NB [work]
CALL SORGQR( M, M, N, U, LDU, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce R in A, zeroing out other entries
*
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (Workspace: need N*N+4*N, prefer N*N+3*N+2*N*NB)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + 2*N*NB [work]
*
CALL SGEBRD( N, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ), N,
$ VT, LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite WORK(IU) by left singular vectors of R and VT
* by right singular vectors of R
-* (Workspace: need N*N+3*N, prefer N*N+2*N+N*NB)
+* Workspace: need N*N [U] + 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer N*N [U] + 3*N [e, tauq, taup] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', N, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
-* (Workspace: need N*N)
+* Workspace: need N*N [U]
*
CALL SGEMM( 'N', 'N', M, N, N, ONE, U, LDU, WORK( IU ),
$ LDWRKU, ZERO, A, LDA )
*
* M .LT. MNTHR
*
-* Path 5 (M at least N, but not much larger)
+* Path 5 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
*
IE = 1
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (Workspace: need 3*N+M, prefer 3*N+(M+N)*NB)
+* Workspace: need 3*N [e, tauq, taup] + M [work]
+* Workspace: prefer 3*N [e, tauq, taup] + (M+N)*NB [work]
*
CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5n (M >= N, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'N', N, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 5o (M >= N, JOBZ='O')
IU = NWORK
- IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
* WORK( IU ) is M by N
*
NWORK = IU + LDWRKU*N
CALL SLASET( 'F', M, N, ZERO, ZERO, WORK( IU ),
$ LDWRKU )
+* IR is unused; silence compile warnings
+ IR = -1
ELSE
*
* WORK( IU ) is N by N
* WORK(IR) is LDWRKR by N
*
IR = NWORK
- LDWRKR = ( LWORK-N*N-3*N ) / N
+ LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in WORK(IU) and computing right
* singular vectors of bidiagonal matrix in VT
-* (Workspace: need N+N*N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + BDSPAC
*
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), WORK( IU ),
$ LDWRKU, VT, LDVT, DUM, IDUM, WORK( NWORK ),
$ IWORK, INFO )
*
* Overwrite VT by right singular vectors of A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
- IF( LWORK.GE.M*N+3*N+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*N + BDSPAC ) THEN
*
+* Path 5o-fast
* Overwrite WORK(IU) by left singular vectors of A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + M*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + M*N [U] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), WORK( IU ), LDWRKU,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy left singular vectors of A from WORK(IU) to A
*
CALL SLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
+* Path 5o-slow
* Generate Q in A
-* (Workspace: need N*N+2*N, prefer N*N+N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + N*NB [work]
*
CALL SORGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by left singular vectors of
* bidiagonal matrix in WORK(IU), storing result in
* WORK(IR) and copying to A
-* (Workspace: need 2*N*N, prefer N*N+M*N)
+* Workspace: need 3*N [e, tauq, taup] + N*N [U] + NB*N [R]
+* Workspace: prefer 3*N [e, tauq, taup] + N*N [U] + M*N [R]
*
DO 20 I = 1, M, LDWRKR
- CHUNK = MIN( M-I+1, LDWRKR )
+ CHUNK = MIN( M - I + 1, LDWRKR )
CALL SGEMM( 'N', 'N', CHUNK, N, N, ONE, A( I, 1 ),
$ LDA, WORK( IU ), LDWRKU, ZERO,
$ WORK( IR ), LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
+* Path 5s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL SLASET( 'F', M, N, ZERO, ZERO, U, LDU )
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 3*N, prefer 2*N+N*NB)
+* Workspace: need 3*N [e, tauq, taup] + N [work]
+* Workspace: prefer 3*N [e, tauq, taup] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, N, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
+* Path 5a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need N+BDSPAC)
+* Workspace: need 3*N [e, tauq, taup] + BDSPAC
*
CALL SLASET( 'F', M, M, ZERO, ZERO, U, LDU )
CALL SBDSDC( 'U', 'I', N, S, WORK( IE ), U, LDU, VT,
* Set the right corner of U to identity matrix
*
IF( M.GT.N ) THEN
- CALL SLASET( 'F', M-N, M-N, ZERO, ONE, U( N+1, N+1 ),
+ CALL SLASET( 'F', M - N, M - N, ZERO, ONE, U(N+1,N+1),
$ LDU )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need N*N+2*N+M, prefer N*N+2*N+M*NB)
+* Workspace: need 3*N [e, tauq, taup] + M [work]
+* Workspace: prefer 3*N [e, tauq, taup] + M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
END IF
*
END IF
*
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need 2*M, prefer M+M*NB)
+* Workspace: need M [tau] + M [work]
+* Workspace: prefer M [tau] + M*NB [work]
*
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Zero out above L
*
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (Workspace: need 4*M, prefer 3*M+2*M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL SGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NWORK = IE + M
*
* Perform bidiagonal SVD, computing singular values only
-* (Workspace: need M+BDSPAC)
+* Workspace: need M [e] + BDSPAC
*
CALL SBDSDC( 'U', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
IVT = 1
*
-* IVT is M by M
+* WORK(IVT) is M by M
+* WORK(IL) is M by M; it is later resized to M by chunk for gemm
*
IL = IVT + M*M
- IF( LWORK.GE.M*N+M*M+3*M+BDSPAC ) THEN
-*
-* WORK(IL) is M by N
-*
+ IF( LWORK .GE. M*N + M*M + 3*M + BDSPAC ) THEN
LDWRKL = M
CHUNK = N
ELSE
LDWRKL = M
- CHUNK = ( LWORK-M*M ) / M
+ CHUNK = ( LWORK - M*M ) / M
END IF
ITAU = IL + LDWRKL*M
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing about above it
*
CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
- CALL SLASET( 'U', M-1, M-1, ZERO, ZERO,
- $ WORK( IL+LDWRKL ), LDWRKL )
+ CALL SLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+ $ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
*
CALL SORGLQ( M, N, M, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL SGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U, and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M+M*M+BDSPAC)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), M, DUM, IDUM, WORK( NWORK ),
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
-* (Workspace: need 2*M*M+3*M, prefer 2*M*M+2*M+M*NB)
+* Workspace: need M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), WORK( IVT ), M,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by Q
* in A, storing result in WORK(IL) and copying to A
-* (Workspace: need 2*M*M, prefer M*M+M*N)
+* Workspace: need M*M [VT] + M*M [L]
+* Workspace: prefer M*M [VT] + M*N [L]
+* At this point, L is resized as M by chunk.
*
DO 30 I = 1, N, CHUNK
- BLK = MIN( N-I+1, CHUNK )
+ BLK = MIN( N - I + 1, CHUNK )
CALL SGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ), M,
$ A( 1, I ), LDA, ZERO, WORK( IL ), LDWRKL )
CALL SLACPY( 'F', M, BLK, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
+* Path 3t (N >> M, JOBZ='S')
* M right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Copy L to WORK(IL), zeroing out above it
*
CALL SLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWRKL )
- CALL SLASET( 'U', M-1, M-1, ZERO, ZERO,
- $ WORK( IL+LDWRKL ), LDWRKL )
+ CALL SLASET( 'U', M - 1, M - 1, ZERO, ZERO,
+ $ WORK( IL + LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [L] + M [tau] + M [work]
+* Workspace: prefer M*M [L] + M [tau] + M*NB [work]
*
CALL SORGLQ( M, N, M, A, LDA, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
IE = ITAU
ITAUQ = IE + M
ITAUP = ITAUQ + M
NWORK = ITAUP + M
*
-* Bidiagonalize L in WORK(IU), copying result to U
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Bidiagonalize L in WORK(IU).
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL SGEBRD( M, M, WORK( IL ), LDWRKL, S, WORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU, VT,
$ LDVT, DUM, IDUM, WORK( NWORK ), IWORK,
*
* Overwrite U by left singular vectors of L and VT
* by right singular vectors of L
-* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+* Workspace: need M*M [L] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [L] + 3*M [e, tauq, taup] + M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', M, M, M, WORK( IL ), LDWRKL,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IL) by
* Q in A, storing result in VT
-* (Workspace: need M*M)
+* Workspace: need M*M [L]
*
CALL SLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL SGEMM( 'N', 'N', M, N, M, ONE, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M [tau] + M [work]
+* Workspace: prefer M*M [VT] + M [tau] + M*NB [work]
*
CALL SGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need M*M [VT] + M [tau] + N [work]
+* Workspace: prefer M*M [VT] + M [tau] + N*NB [work]
*
CALL SORGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Produce L in A, zeroing out other entries
*
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (Workspace: need M*M+4*M, prefer M*M+3*M+2*M*NB)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup] + 2*M*NB [work]
*
CALL SGEBRD( M, M, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M+M*M+BDSPAC)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'U', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
*
* Overwrite U by left singular vectors of L and WORK(IVT)
* by right singular vectors of L
-* (Workspace: need M*M+3*M, prefer M*M+2*M+M*NB)
+* Workspace: need M*M [VT] + 3*M [e, tauq, taup]+ M [work]
+* Workspace: prefer M*M [VT] + 3*M [e, tauq, taup]+ M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', M, M, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
-* (Workspace: need M*M)
+* Workspace: need M*M [VT]
*
CALL SGEMM( 'N', 'N', M, N, M, ONE, WORK( IVT ), LDWKVT,
$ VT, LDVT, ZERO, A, LDA )
*
* N .LT. MNTHR
*
-* Path 5t (N greater than M, but not much larger)
+* Path 5t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
*
IE = 1
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
+* Workspace: need 3*M [e, tauq, taup] + N [work]
+* Workspace: prefer 3*M [e, tauq, taup] + (M+N)*NB [work]
*
CALL SGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5tn (N > M, JOBZ='N')
* Perform bidiagonal SVD, only computing singular values
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL SBDSDC( 'L', 'N', M, S, WORK( IE ), DUM, 1, DUM, 1,
$ DUM, IDUM, WORK( NWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 5to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
- IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
* WORK( IVT ) is M by N
*
CALL SLASET( 'F', M, N, ZERO, ZERO, WORK( IVT ),
$ LDWKVT )
NWORK = IVT + LDWKVT*N
+* IL is unused; silence compile warnings
+ IL = -1
ELSE
*
* WORK( IVT ) is M by M
*
* WORK(IL) is M by CHUNK
*
- CHUNK = ( LWORK-M*M-3*M ) / M
+ CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in WORK(IVT)
-* (Workspace: need M*M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + BDSPAC
*
CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU,
$ WORK( IVT ), LDWKVT, DUM, IDUM,
$ WORK( NWORK ), IWORK, INFO )
*
* Overwrite U by left singular vectors of A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
*
- IF( LWORK.GE.M*N+3*M+BDSPAC ) THEN
+ IF( LWORK .GE. M*N + 3*M + BDSPAC ) THEN
*
+* Path 5to-fast
* Overwrite WORK(IVT) by left singular vectors of A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*N [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*N [VT] + M*NB [work]
*
CALL SORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), WORK( IVT ), LDWKVT,
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Copy right singular vectors of A from WORK(IVT) to A
*
CALL SLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
+* Path 5to-slow
* Generate P**T in A
-* (Workspace: need M*M+2*M, prefer M*M+M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*NB [work]
*
CALL SORGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
- $ WORK( NWORK ), LWORK-NWORK+1, IERR )
+ $ WORK( NWORK ), LWORK - NWORK + 1, IERR )
*
* Multiply Q in A by right singular vectors of
* bidiagonal matrix in WORK(IVT), storing result in
* WORK(IL) and copying to A
-* (Workspace: need 2*M*M, prefer M*M+M*N)
+* Workspace: need 3*M [e, tauq, taup] + M*M [VT] + M*NB [L]
+* Workspace: prefer 3*M [e, tauq, taup] + M*M [VT] + M*N [L]
*
DO 40 I = 1, N, CHUNK
- BLK = MIN( N-I+1, CHUNK )
+ BLK = MIN( N - I + 1, CHUNK )
CALL SGEMM( 'N', 'N', M, BLK, M, ONE, WORK( IVT ),
$ LDWKVT, A( 1, I ), LDA, ZERO,
$ WORK( IL ), M )
END IF
ELSE IF( WNTQS ) THEN
*
+* Path 5ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL SLASET( 'F', M, N, ZERO, ZERO, VT, LDVT )
CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 3*M, prefer 2*M+M*NB)
+* Workspace: need 3*M [e, tauq, taup] + M [work]
+* Workspace: prefer 3*M [e, tauq, taup] + M*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', M, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
ELSE IF( WNTQA ) THEN
*
+* Path 5ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in U and computing right singular
* vectors of bidiagonal matrix in VT
-* (Workspace: need M+BDSPAC)
+* Workspace: need 3*M [e, tauq, taup] + BDSPAC
*
CALL SLASET( 'F', N, N, ZERO, ZERO, VT, LDVT )
CALL SBDSDC( 'L', 'I', M, S, WORK( IE ), U, LDU, VT,
* Set the right corner of VT to identity matrix
*
IF( N.GT.M ) THEN
- CALL SLASET( 'F', N-M, N-M, ZERO, ONE, VT( M+1, M+1 ),
+ CALL SLASET( 'F', N-M, N-M, ZERO, ONE, VT(M+1,M+1),
$ LDVT )
END IF
*
* Overwrite U by left singular vectors of A and VT
* by right singular vectors of A
-* (Workspace: need 2*M+N, prefer 2*M+N*NB)
+* Workspace: need 3*M [e, tauq, taup] + N [work]
+* Workspace: prefer 3*M [e, tauq, taup] + N*NB [work]
*
CALL SORMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
CALL SORMBR( 'P', 'R', 'T', N, N, M, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
- $ LWORK-NWORK+1, IERR )
+ $ LWORK - NWORK + 1, IERR )
END IF
*
END IF
BDSPAC = 5*N
* Compute space needed for SGEQRF
CALL SGEQRF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_SGEQRF=DUM(1)
+ LWORK_SGEQRF = INT( DUM(1) )
* Compute space needed for SORGQR
CALL SORGQR( M, N, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_SORGQR_N=DUM(1)
+ LWORK_SORGQR_N = INT( DUM(1) )
CALL SORGQR( M, M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_SORGQR_M=DUM(1)
+ LWORK_SORGQR_M = INT( DUM(1) )
* Compute space needed for SGEBRD
CALL SGEBRD( N, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_SGEBRD=DUM(1)
+ LWORK_SGEBRD = INT( DUM(1) )
* Compute space needed for SORGBR P
CALL SORGBR( 'P', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_P=DUM(1)
+ LWORK_SORGBR_P = INT( DUM(1) )
* Compute space needed for SORGBR Q
CALL SORGBR( 'Q', N, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_Q=DUM(1)
+ LWORK_SORGBR_Q = INT( DUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
*
CALL SGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_SGEBRD=DUM(1)
+ LWORK_SGEBRD = INT( DUM(1) )
MAXWRK = 3*N + LWORK_SGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL SORGBR( 'Q', M, N, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_Q=DUM(1)
+ LWORK_SORGBR_Q = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*N+LWORK_SORGBR_Q )
END IF
IF( WNTUA ) THEN
CALL SORGBR( 'Q', M, M, N, A, LDA, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_Q=DUM(1)
+ LWORK_SORGBR_Q = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*N+LWORK_SORGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
BDSPAC = 5*M
* Compute space needed for SGELQF
CALL SGELQF( M, N, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_SGELQF=DUM(1)
+ LWORK_SGELQF = INT( DUM(1) )
* Compute space needed for SORGLQ
CALL SORGLQ( N, N, M, DUM(1), N, DUM(1), DUM(1), -1, IERR )
- LWORK_SORGLQ_N=DUM(1)
+ LWORK_SORGLQ_N = INT( DUM(1) )
CALL SORGLQ( M, N, M, A, LDA, DUM(1), DUM(1), -1, IERR )
- LWORK_SORGLQ_M=DUM(1)
+ LWORK_SORGLQ_M = INT( DUM(1) )
* Compute space needed for SGEBRD
CALL SGEBRD( M, M, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_SGEBRD=DUM(1)
+ LWORK_SGEBRD = INT( DUM(1) )
* Compute space needed for SORGBR P
CALL SORGBR( 'P', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_P=DUM(1)
+ LWORK_SORGBR_P = INT( DUM(1) )
* Compute space needed for SORGBR Q
CALL SORGBR( 'Q', M, M, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_Q=DUM(1)
+ LWORK_SORGBR_Q = INT( DUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
*
CALL SGEBRD( M, N, A, LDA, S, DUM(1), DUM(1),
$ DUM(1), DUM(1), -1, IERR )
- LWORK_SGEBRD=DUM(1)
+ LWORK_SGEBRD = INT( DUM(1) )
MAXWRK = 3*M + LWORK_SGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for SORGBR P
CALL SORGBR( 'P', M, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_P=DUM(1)
+ LWORK_SORGBR_P = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*M+LWORK_SORGBR_P )
END IF
IF( WNTVA ) THEN
CALL SORGBR( 'P', N, N, M, A, N, DUM(1),
$ DUM(1), -1, IERR )
- LWORK_SORGBR_P=DUM(1)
+ LWORK_SORGBR_P = INT( DUM(1) )
MAXWRK = MAX( MAXWRK, 3*M+LWORK_SORGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
* Definition:
* ===========
*
-* SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
-* LWORK, RWORK, IWORK, INFO )
+* SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+* WORK, LWORK, RWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBZ
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
-*> The leading dimension of the array U. LDU >= 1; if
-*> JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
+*> The leading dimension of the array U. LDU >= 1;
+*> if JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.
*> \endverbatim
*>
*> \param[out] VT
*> \param[in] LDVT
*> \verbatim
*> LDVT is INTEGER
-*> The leading dimension of the array VT. LDVT >= 1; if
-*> JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
+*> The leading dimension of the array VT. LDVT >= 1;
+*> if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;
*> if JOBZ = 'S', LDVT >= min(M,N).
*> \endverbatim
*>
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
-*> if JOBZ = 'N', LWORK >= 2*min(M,N)+max(M,N).
-*> if JOBZ = 'O',
-*> LWORK >= 2*min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-*> if JOBZ = 'S' or 'A',
-*> LWORK >= min(M,N)*min(M,N)+2*min(M,N)+max(M,N).
-*> For good performance, LWORK should generally be larger.
-*>
*> If LWORK = -1, a workspace query is assumed. The optimal
*> size for the WORK array is calculated and stored in WORK(1),
*> and no other work except argument checking is performed.
+*>
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LWORK >= 2*mn + mx.
+*> If JOBZ = 'O', LWORK >= 2*mn*mn + 2*mn + mx.
+*> If JOBZ = 'S', LWORK >= mn*mn + 3*mn.
+*> If JOBZ = 'A', LWORK >= mn*mn + 2*mn + mx.
+*> These are not tight minimums in all cases; see comments inside code.
+*> For good performance, LWORK should generally be larger;
+*> a query is recommended.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
-*> If JOBZ = 'N', LRWORK >= 7*min(M,N).
-*> Otherwise,
-*> LRWORK >= min(M,N)*max(5*min(M,N)+7,2*max(M,N)+2*min(M,N)+1)
+*> Let mx = max(M,N) and mn = min(M,N).
+*> If JOBZ = 'N', LRWORK >= 5*mn (LAPACK <= 3.6 needs 7*mn);
+*> else if mx >> mn, LRWORK >= 5*mn*mn + 5*mn;
+*> else LRWORK >= max( 5*mn*mn + 5*mn,
+*> 2*mx*mn + 2*mn*mn + mn ).
*> \endverbatim
*>
*> \param[out] IWORK
*> Ming Gu and Huan Ren, Computer Science Division, University of
*> California at Berkeley, USA
*>
+*> @precisions fortran z -> c
* =====================================================================
- SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
- $ LWORK, RWORK, IWORK, INFO )
+ SUBROUTINE ZGESDD( JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT,
+ $ WORK, LWORK, RWORK, IWORK, INFO )
+ implicit none
*
* -- LAPACK driver routine (version 3.6.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* =====================================================================
*
* .. Parameters ..
- INTEGER LQUERV
- PARAMETER ( LQUERV = -1 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
- LOGICAL WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
+ LOGICAL LQUERY, WNTQA, WNTQAS, WNTQN, WNTQO, WNTQS
INTEGER BLK, CHUNK, I, IE, IERR, IL, IR, IRU, IRVT,
$ ISCL, ITAU, ITAUP, ITAUQ, IU, IVT, LDWKVT,
$ LDWRKL, LDWRKR, LDWRKU, MAXWRK, MINMN, MINWRK,
$ MNTHR1, MNTHR2, NRWORK, NWORK, WRKBL
+ INTEGER LWORK_ZGEBRD_MN, LWORK_ZGEBRD_MM,
+ $ LWORK_ZGEBRD_NN, LWORK_ZGELQF_MN,
+ $ LWORK_ZGEQRF_MN,
+ $ LWORK_ZUNGBR_P_MN, LWORK_ZUNGBR_P_NN,
+ $ LWORK_ZUNGBR_Q_MN, LWORK_ZUNGBR_Q_MM,
+ $ LWORK_ZUNGLQ_MN, LWORK_ZUNGLQ_NN,
+ $ LWORK_ZUNGQR_MM, LWORK_ZUNGQR_MN,
+ $ LWORK_ZUNMBR_PRC_MM, LWORK_ZUNMBR_QLN_MM,
+ $ LWORK_ZUNMBR_PRC_MN, LWORK_ZUNMBR_QLN_MN,
+ $ LWORK_ZUNMBR_PRC_NN, LWORK_ZUNMBR_QLN_NN
DOUBLE PRECISION ANRM, BIGNUM, EPS, SMLNUM
* ..
* .. Local Arrays ..
INTEGER IDUM( 1 )
DOUBLE PRECISION DUM( 1 )
+ COMPLEX*16 CDUM( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DBDSDC, DLASCL, XERBLA, ZGEBRD, ZGELQF, ZGEMM,
* ..
* .. External Functions ..
LOGICAL LSAME
- INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
- EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
+ EXTERNAL LSAME, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX, MIN, SQRT
*
* Test the input arguments
*
- INFO = 0
- MINMN = MIN( M, N )
+ INFO = 0
+ MINMN = MIN( M, N )
MNTHR1 = INT( MINMN*17.0D0 / 9.0D0 )
MNTHR2 = INT( MINMN*5.0D0 / 3.0D0 )
- WNTQA = LSAME( JOBZ, 'A' )
- WNTQS = LSAME( JOBZ, 'S' )
+ WNTQA = LSAME( JOBZ, 'A' )
+ WNTQS = LSAME( JOBZ, 'S' )
WNTQAS = WNTQA .OR. WNTQS
- WNTQO = LSAME( JOBZ, 'O' )
- WNTQN = LSAME( JOBZ, 'N' )
+ WNTQO = LSAME( JOBZ, 'O' )
+ WNTQN = LSAME( JOBZ, 'N' )
+ LQUERY = ( LWORK.EQ.-1 )
MINWRK = 1
MAXWRK = 1
*
END IF
*
* Compute workspace
-* (Note: Comments in the code beginning "Workspace:" describe the
-* minimal amount of workspace needed at that point in the code,
+* Note: Comments in the code beginning "Workspace:" describe the
+* minimal amount of workspace allocated at that point in the code,
* as well as the preferred amount for good performance.
* CWorkspace refers to complex workspace, and RWorkspace to
* real workspace. NB refers to the optimal block size for the
IF( M.GE.N ) THEN
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*N*N + 7*N
-* BDSPAN = MAX(7*N+4, 3*N+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (dbdsdc) is
+* BDSPAC = 3*N*N + 4*N for singular values and vectors;
+* BDSPAC = 4*N for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL ZGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MN = INT( CDUM(1) )
+*
+ CALL ZGEBRD( N, N, CDUM(1), N, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_NN = INT( CDUM(1) )
+*
+ CALL ZGEQRF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEQRF_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', N, N, N, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MN = INT( CDUM(1) )
+*
+ CALL ZUNGQR( M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGQR_MM = INT( CDUM(1) )
+*
+ CALL ZUNGQR( M, N, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGQR_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MM = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, N, N, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', N, N, N, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_NN = INT( CDUM(1) )
*
IF( M.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
*
- MAXWRK = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
+ MAXWRK = N + LWORK_ZGEQRF_MN
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZGEBRD_NN )
MINWRK = 3*N
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
-*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 2 (M >> N, JOBZ='O')
+*
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = M*N + N*N + WRKBL
MINWRK = 2*N*N + 3*N
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
-*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+N*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 3 (M >> N, JOBZ='S')
+*
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
MINWRK = N*N + 3*N
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
-*
- WRKBL = N + N*ILAENV( 1, 'ZGEQRF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, N+M*ILAENV( 1, 'ZUNGQR', ' ', M,
- $ M, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+2*N*
- $ ILAENV( 1, 'ZGEBRD', ' ', N, N, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', N, N, N, -1 ) )
- WRKBL = MAX( WRKBL, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
+* Path 4 (M >> N, JOBZ='A')
+*
+ WRKBL = N + LWORK_ZGEQRF_MN
+ WRKBL = MAX( WRKBL, N + LWORK_ZUNGQR_MM )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZGEBRD_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_QLN_NN )
+ WRKBL = MAX( WRKBL, 2*N + LWORK_ZUNMBR_PRC_NN )
MAXWRK = N*N + WRKBL
- MINWRK = N*N + 2*N + M
+ MINWRK = N*N + MAX( 3*N, N + M )
END IF
ELSE IF( M.GE.MNTHR2 ) THEN
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_ZGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5o (M >> N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, N, N, -1 ) )
+* Path 5s (M >> N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5a (M >> N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_P_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNGBR_Q_MM )
END IF
ELSE
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
*
- MAXWRK = 2*N + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*N + LWORK_ZGEBRD_MN
MINWRK = 2*N + M
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6o (M >= N, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + N*N
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, N, N, -1 ) )
+* Path 6s (M >= N, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MN )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*N+N*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', N, N, N, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*N+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6a (M >= N, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*N + LWORK_ZUNMBR_PRC_NN )
END IF
END IF
ELSE
*
* There is no complex work space needed for bidiagonal SVD
-* The real work space needed for bidiagonal SVD is BDSPAC
-* for computing singular values and singular vectors; BDSPAN
-* for computing singular values only.
-* BDSPAC = 5*M*M + 7*M
-* BDSPAN = MAX(7*M+4, 3*M+2+SMLSIZ*(SMLSIZ+8))
+* The real work space needed for bidiagonal SVD (dbdsdc) is
+* BDSPAC = 3*M*M + 4*M for singular values and vectors;
+* BDSPAC = 4*M for singular values only;
+* not including e, RU, and RVT matrices.
+*
+* Compute space preferred for each routine
+ CALL ZGEBRD( M, N, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MN = INT( CDUM(1) )
+*
+ CALL ZGEBRD( M, M, CDUM(1), M, CDUM(1), DUM(1), CDUM(1),
+ $ CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGEBRD_MM = INT( CDUM(1) )
+*
+ CALL ZGELQF( M, N, CDUM(1), M, CDUM(1), CDUM(1), -1, IERR )
+ LWORK_ZGELQF_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_MN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'P', N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_P_NN = INT( CDUM(1) )
+*
+ CALL ZUNGBR( 'Q', M, M, N, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGBR_Q_MM = INT( CDUM(1) )
+*
+ CALL ZUNGLQ( M, N, M, CDUM(1), M, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGLQ_MN = INT( CDUM(1) )
+*
+ CALL ZUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1),
+ $ -1, IERR )
+ LWORK_ZUNGLQ_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_MM = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', M, N, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_MN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'P', 'R', 'C', N, N, M, CDUM(1), N, CDUM(1),
+ $ CDUM(1), N, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_PRC_NN = INT( CDUM(1) )
+*
+ CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, CDUM(1), M, CDUM(1),
+ $ CDUM(1), M, CDUM(1), -1, IERR )
+ LWORK_ZUNMBR_QLN_MM = INT( CDUM(1) )
*
IF( N.GE.MNTHR1 ) THEN
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
*
- MAXWRK = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1,
- $ -1 )
- MAXWRK = MAX( MAXWRK, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
+ MAXWRK = M + LWORK_ZGELQF_MN
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZGEBRD_MM )
MINWRK = 3*M
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
-*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'ZUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 2t (N >> M, JOBZ='O')
+*
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*N + M*M + WRKBL
MINWRK = 2*M*M + 3*M
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+M*ILAENV( 1, 'ZUNGLQ', ' ', M,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 3t (N >> M, JOBZ='S')
+*
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_MN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
MINWRK = M*M + 3*M
ELSE IF( WNTQA ) THEN
*
-* Path 4t (N much larger than M, JOBZ='A')
-*
- WRKBL = M + M*ILAENV( 1, 'ZGELQF', ' ', M, N, -1, -1 )
- WRKBL = MAX( WRKBL, M+N*ILAENV( 1, 'ZUNGLQ', ' ', N,
- $ N, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+2*M*
- $ ILAENV( 1, 'ZGEBRD', ' ', M, M, -1, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, M, M, -1 ) )
- WRKBL = MAX( WRKBL, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, M, -1 ) )
+* Path 4t (N >> M, JOBZ='A')
+*
+ WRKBL = M + LWORK_ZGELQF_MN
+ WRKBL = MAX( WRKBL, M + LWORK_ZUNGLQ_NN )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZGEBRD_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_QLN_MM )
+ WRKBL = MAX( WRKBL, 2*M + LWORK_ZUNMBR_PRC_MM )
MAXWRK = M*M + WRKBL
- MINWRK = M*M + 2*M + N
+ MINWRK = M*M + MAX( 3*M, M + N )
END IF
ELSE IF( N.GE.MNTHR2 ) THEN
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_ZGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5to (N >> M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'P', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ts (N >> M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'ZUNGBR', 'P', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'Q', M, M, N, -1 ) )
+* Path 5ta (N >> M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_Q_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNGBR_P_NN )
END IF
ELSE
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
*
- MAXWRK = 2*M + ( M+N )*ILAENV( 1, 'ZGEBRD', ' ', M, N,
- $ -1, -1 )
+ MAXWRK = 2*M + LWORK_ZGEBRD_MN
MINWRK = 2*M + N
IF( WNTQO ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNMBR', 'QLN', M, M, N, -1 ) )
+* Path 6to (N > M, JOBZ='O')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_MN )
MAXWRK = MAXWRK + M*N
MINWRK = MINWRK + M*M
ELSE IF( WNTQS ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', M, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ts (N > M, JOBZ='S')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_MN )
ELSE IF( WNTQA ) THEN
- MAXWRK = MAX( MAXWRK, 2*M+N*
- $ ILAENV( 1, 'ZUNGBR', 'PRC', N, N, M, -1 ) )
- MAXWRK = MAX( MAXWRK, 2*M+M*
- $ ILAENV( 1, 'ZUNGBR', 'QLN', M, M, N, -1 ) )
+* Path 6ta (N > M, JOBZ='A')
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_QLN_MM )
+ MAXWRK = MAX( MAXWRK, 2*M + LWORK_ZUNMBR_PRC_NN )
END IF
END IF
END IF
END IF
IF( INFO.EQ.0 ) THEN
WORK( 1 ) = MAXWRK
- IF( LWORK.LT.MINWRK .AND. LWORK.NE.LQUERV )
- $ INFO = -13
+ IF( LWORK.LT.MINWRK .AND. .NOT. LQUERY ) THEN
+ INFO = -12
+ END IF
END IF
-*
-* Quick returns
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGESDD', -INFO )
RETURN
+ ELSE IF( LQUERY ) THEN
+ RETURN
END IF
- IF( LWORK.EQ.LQUERV )
- $ RETURN
+*
+* Quick return if possible
+*
IF( M.EQ.0 .OR. N.EQ.0 ) THEN
RETURN
END IF
*
IF( WNTQN ) THEN
*
-* Path 1 (M much larger than N, JOBZ='N')
+* Path 1 (M >> N, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: need 0)
+* CWorkspace: need N [tau] + N [work]
+* CWorkspace: prefer N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NRWORK = IE + N
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2 (M much larger than N, JOBZ='O')
+* Path 2 (M >> N, JOBZ='O')
* N left singular vectors to be overwritten on A and
* N right singular vectors to be computed in VT
*
*
LDWRKU = N
IR = IU + LDWRKU*N
- IF( LWORK.GE.M*N+N*N+3*N ) THEN
+ IF( LWORK .GE. M*N + N*N + 3*N ) THEN
*
* WORK(IR) is M by N
*
LDWRKR = M
ELSE
- LDWRKR = ( LWORK-N*N-3*N ) / N
+ LDWRKR = ( LWORK - N*N - 3*N ) / N
END IF
ITAU = IR + LDWRKR*N
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer M*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of R in WORK(IRU) and computing right singular vectors
* of R in WORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of R
-* (CWorkspace: need 2*N*N+3*N, prefer M*N+N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by the right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IU), storing result in WORK(IR) and copying to A
-* (CWorkspace: need 2*N*N, prefer N*N+M*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N*N [R]
+* CWorkspace: prefer N*N [U] + M*N [R]
+* RWorkspace: need 0
*
DO 10 I = 1, M, LDWRKR
CHUNK = MIN( M-I+1, LDWRKR )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3 (M much larger than N, JOBZ='S')
+* Path 3 (M >> N, JOBZ='S')
* N left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R
-* (CWorkspace: need N*N+2*N, prefer N*N+N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ LDWRKR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + N [tau] + N [work]
+* CWorkspace: prefer N*N [R] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, N, N, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in WORK(IR)
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, WORK( IR ), LDWRKR, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = IE + N
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', N, N, N, WORK( IR ), LDWRKR,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [R] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, WORK( IR ), LDWRKR,
*
* Multiply Q in A by left singular vectors of R in
* WORK(IR), storing result in U
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [R]
+* RWorkspace: need 0
*
CALL ZLACPY( 'F', N, N, U, LDU, WORK( IR ), LDWRKR )
CALL ZGEMM( 'N', 'N', M, N, N, CONE, A, LDA, WORK( IR ),
*
ELSE IF( WNTQA ) THEN
*
-* Path 4 (M much larger than N, JOBZ='A')
+* Path 4 (M >> N, JOBZ='A')
* M left singular vectors to be computed in U and
* N right singular vectors to be computed in VT
*
NWORK = ITAU + N
*
* Compute A=Q*R, copying result to U
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + N [work]
+* CWorkspace: prefer N*N [U] + N [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
*
* Generate Q in U
-* (CWorkspace: need N+M, prefer N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + N [tau] + M [work]
+* CWorkspace: prefer N*N [U] + N [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGQR( M, M, N, U, LDU, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + N
*
* Bidiagonalize R in A
-* (CWorkspace: need 3*N, prefer 2*N+2*N*NB)
-* (RWorkspace: need N)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + 2*N*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( N, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by left singular vectors of R
-* (CWorkspace: need N*N+3*N, prefer N*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, WORK( IU ),
$ LDWRKU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of R
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U] + 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer N*N [U] + 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
*
* Multiply Q in U by left singular vectors of R in
* WORK(IU), storing result in A
-* (CWorkspace: need N*N)
-* (RWorkspace: 0)
+* CWorkspace: need N*N [U]
+* RWorkspace: need 0
*
CALL ZGEMM( 'N', 'N', M, N, N, CONE, U, LDU, WORK( IU ),
$ LDWRKU, CZERO, A, LDA )
*
* MNTHR2 <= M < MNTHR1
*
-* Path 5 (M much larger than N, but not as much as MNTHR1)
+* Path 5 (M >> N, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* ZUNGBR and matrix multiplication to compute singular vectors
*
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 5n (M >> N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
IRVT = IRU + N*N
NRWORK = IRVT + N*N
*
+* Path 5o (M >> N, JOBZ='O')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate Q in A
-* (CWorkspace: need 2*N, prefer N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
*
* WORK(IU) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in WORK(IU), copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT,
$ WORK( IU ), LDWRKU, RWORK( NRWORK ) )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 20 I = 1, M, LDWRKU
*
ELSE IF( WNTQS ) THEN
*
+* Path 5s (M >> N, JOBZ='S')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, N, N, U, LDU, WORK( ITAUQ ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need N*N+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL ZLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
CALL ZLACPY( 'F', M, N, A, LDA, U, LDU )
ELSE
*
+* Path 5a (M >> N, JOBZ='A')
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', N, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, N, VT, LDVT, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to U, generate Q
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, N, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + 2*N*N [rwork]
*
CALL ZLARCM( N, N, RWORK( IRVT ), N, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: 0)
-* (Rworkspace: need 3*N*N)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
CALL ZLACRM( M, N, U, LDU, RWORK( IRU ), N, A, LDA,
*
* M .LT. MNTHR2
*
-* Path 6 (M at least N, but not much larger)
+* Path 6 (M >= N, but not much larger)
* Reduce to bidiagonal form without QR decomposition
* Use ZUNMBR to compute singular vectors
*
NWORK = ITAUP + N
*
* Bidiagonalize A
-* (CWorkspace: need 2*N+M, prefer 2*N+(M+N)*NB)
-* (RWorkspace: need N)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need N [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6n (M >= N, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', N, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IU = NWORK
IRU = NRWORK
IRVT = IRU + N*N
NRWORK = IRVT + N*N
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
* WORK( IU ) is M by N
*
*
* WORK( IU ) is LDWRKU by N
*
- LDWRKU = ( LWORK-3*N ) / N
+ LDWRKU = ( LWORK - 3*N ) / N
END IF
NWORK = IU + LDWRKU*N
*
+* Path 6o (M >= N, JOBZ='O')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
CALL DBDSDC( 'U', 'I', N, S, RWORK( IE ), RWORK( IRU ),
$ N, RWORK( IRVT ), N, DUM, IDUM,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ WORK( ITAUP ), VT, LDVT, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*N ) THEN
+ IF( LWORK .GE. M*N + 3*N ) THEN
*
-* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
-* Overwrite WORK(IU) by left singular vectors of A, copying
-* to A
-* (Cworkspace: need M*N+2*N, prefer M*N+N+N*NB)
-* (Rworkspace: need 0)
+* Path 6o-fast
+* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
+* Overwrite WORK(IU) by left singular vectors of A, copying
+* to A
+* CWorkspace: need 2*N [tauq, taup] + M*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, WORK( IU ),
$ LDWRKU )
CALL ZLACPY( 'F', M, N, WORK( IU ), LDWRKU, A, LDA )
ELSE
*
+* Path 6o-slow
* Generate Q in A
-* (Cworkspace: need 2*N, prefer N+N*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*N [U] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'Q', M, N, N, A, LDA, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need N*N, prefer M*N)
-* (Rworkspace: need 3*N*N, prefer N*N+2*M*N)
+* CWorkspace: need 2*N [tauq, taup] + N*N [U]
+* CWorkspace: prefer 2*N [tauq, taup] + M*N [U]
+* RWorkspace: need N [e] + N*N [RU] + 2*N*N [rwork]
+* RWorkspace: prefer N [e] + N*N [RU] + 2*M*N [rwork] < N + 5*N*N since M < 2*N here
*
NRWORK = IRVT
DO 30 I = 1, M, LDWRKU
*
ELSE IF( WNTQS ) THEN
*
+* Path 6s (M >= N, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, U, LDU )
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6a (M >= N, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT] + BDSPAC
*
IRU = NRWORK
IRVT = IRU + N*N
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 2*N+M, prefer 2*N+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + M [work]
+* CWorkspace: prefer 2*N [tauq, taup] + M*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRU ), N, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*N, prefer 2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need 2*N [tauq, taup] + N [work]
+* CWorkspace: prefer 2*N [tauq, taup] + N*NB [work]
+* RWorkspace: need N [e] + N*N [RU] + N*N [RVT]
*
CALL ZLACP2( 'F', N, N, RWORK( IRVT ), N, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, N, A, LDA,
*
IF( WNTQN ) THEN
*
-* Path 1t (N much larger than M, JOBZ='N')
+* Path 1t (N >> M, JOBZ='N')
* No singular vectors to be computed
*
ITAU = 1
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M [tau] + M [work]
+* CWorkspace: prefer M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need 3*M, prefer 2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
NRWORK = IE + M
*
* Perform bidiagonal SVD, compute singular values only
-* (CWorkspace: 0)
-* (RWorkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'U', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
*
ELSE IF( WNTQO ) THEN
*
-* Path 2t (N much larger than M, JOBZ='O')
+* Path 2t (N >> M, JOBZ='O')
* M right singular vectors to be overwritten on A and
* M left singular vectors to be computed in U
*
* WORK(IVT) is M by M
*
IL = IVT + LDWKVT*M
- IF( LWORK.GE.M*N+M*M+3*M ) THEN
+ IF( LWORK .GE. M*N + M*M + 3*M ) THEN
*
* WORK(IL) M by N
*
* WORK(IL) is M by CHUNK
*
LDWRKL = M
- CHUNK = ( LWORK-M*M-3*M ) / M
+ CHUNK = ( LWORK - M*M - 3*M ) / M
END IF
ITAU = IL + LDWRKL*CHUNK
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix WORK(IU)
* Overwrite WORK(IU) by the left singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by the right singular vectors of L
-* (CWorkspace: need N*N+3*N, prefer M*N+2*N+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
*
* Multiply right singular vectors of L in WORK(IL) by Q
* in A, storing result in WORK(IL) and copying to A
-* (CWorkspace: need 2*M*M, prefer M*M+M*N))
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M*M [L]
+* CWorkspace: prefer M*M [VT] + M*N [L]
+* RWorkspace: need 0
*
DO 40 I = 1, N, CHUNK
BLK = MIN( N-I+1, CHUNK )
*
ELSE IF( WNTQS ) THEN
*
-* Path 3t (N much larger than M, JOBZ='S')
-* M right singular vectors to be computed in VT and
-* M left singular vectors to be computed in U
+* Path 3t (N >> M, JOBZ='S')
+* M right singular vectors to be computed in VT and
+* M left singular vectors to be computed in U
*
IL = 1
*
NWORK = ITAU + M
*
* Compute A=L*Q
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
$ WORK( IL+LDWRKL ), LDWRKL )
*
* Generate Q in A
-* (CWorkspace: need M*M+2*M, prefer M*M+M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + M [tau] + M [work]
+* CWorkspace: prefer M*M [L] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( M, N, M, A, LDA, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in WORK(IL)
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, WORK( IL ), LDWRKL, S, RWORK( IE ),
$ WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, WORK( IL ), LDWRKL,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by left singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [L] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', M, M, M, WORK( IL ), LDWRKL,
*
* Copy VT to WORK(IL), multiply right singular vectors of L
* in WORK(IL) by Q in A, storing result in VT
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [L]
+* RWorkspace: need 0
*
CALL ZLACPY( 'F', M, M, VT, LDVT, WORK( IL ), LDWRKL )
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IL ), LDWRKL,
*
ELSE IF( WNTQA ) THEN
*
-* Path 9t (N much larger than M, JOBZ='A')
+* Path 4t (N >> M, JOBZ='A')
* N right singular vectors to be computed in VT and
* M left singular vectors to be computed in U
*
NWORK = ITAU + M
*
* Compute A=L*Q, copying result to VT
-* (CWorkspace: need 2*M, prefer M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + M [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
*
* Generate Q in VT
-* (CWorkspace: need M+N, prefer M+N*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + M [tau] + N [work]
+* CWorkspace: prefer M*M [VT] + M [tau] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGLQ( N, N, M, VT, LDVT, WORK( ITAU ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
NWORK = ITAUP + M
*
* Bidiagonalize L in A
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+2*M*NB)
-* (RWorkspace: need M)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + 2*M*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, M, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RU] + M*M [RVT] + BDSPAC
*
IRU = IE + M
IRVT = IRU + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of L
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, M, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
* Overwrite WORK(IVT) by right singular vectors of L
-* (CWorkspace: need M*M+3*M, prefer M*M+2*M+M*NB)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT] + 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer M*M [VT] + 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
*
* Multiply right singular vectors of L in WORK(IVT) by
* Q in VT, storing result in A
-* (CWorkspace: need M*M)
-* (RWorkspace: 0)
+* CWorkspace: need M*M [VT]
+* RWorkspace: need 0
*
CALL ZGEMM( 'N', 'N', M, N, M, CONE, WORK( IVT ), LDWKVT,
$ VT, LDVT, CZERO, A, LDA )
*
* MNTHR2 <= N < MNTHR1
*
-* Path 5t (N much larger than M, but not as much as MNTHR1)
+* Path 5t (N >> M, but not as much as MNTHR1)
* Reduce to bidiagonal form without QR decomposition, use
* ZUNGBR and matrix multiplication to compute singular vectors
-*
*
IE = 1
NRWORK = IE + M
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
*
IF( WNTQN ) THEN
*
+* Path 5tn (N >> M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
IRVT = NRWORK
NRWORK = IRU + M*M
IVT = NWORK
*
+* Path 5to (N >> M, JOBZ='O')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
LDWKVT = M
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
CALL DBDSDC( 'L', 'I', M, S, RWORK( IE ), RWORK( IRU ),
$ M, RWORK( IRVT ), M, DUM, IDUM,
*
* Multiply Q in U by real matrix RWORK(IRVT)
* storing the result in WORK(IVT), copying to U
-* (Cworkspace: need 0)
-* (Rworkspace: need 2*M*M)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, WORK( IVT ),
$ LDWKVT, RWORK( NRWORK ) )
*
* Multiply RWORK(IRVT) by P**H in A, storing the
* result in WORK(IVT), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 2*M*M, prefer 2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 50 I = 1, N, CHUNK
50 CONTINUE
ELSE IF( WNTQS ) THEN
*
+* Path 5ts (N >> M, JOBZ='S')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', M, N, M, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
CALL ZLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
ELSE
*
+* Path 5ta (N >> M, JOBZ='A')
* Copy A to U, generate Q
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'L', M, M, A, LDA, U, LDU )
CALL ZUNGBR( 'Q', M, M, N, U, LDU, WORK( ITAUQ ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Copy A to VT, generate P**H
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: 0)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need 0
*
CALL ZLACPY( 'U', M, N, A, LDA, VT, LDVT )
CALL ZUNGBR( 'P', N, N, M, VT, LDVT, WORK( ITAUP ),
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Multiply Q in U by real matrix RWORK(IRU), storing the
* result in A, copying to U
-* (CWorkspace: need 0)
-* (Rworkspace: need 3*M*M)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + 2*M*M [rwork]
*
CALL ZLACRM( M, M, U, LDU, RWORK( IRU ), M, A, LDA,
$ RWORK( NRWORK ) )
*
* Multiply real matrix RWORK(IRVT) by P**H in VT,
* storing the result in A, copying to VT
-* (Cworkspace: need 0)
-* (Rworkspace: need M*M+2*M*N)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
+ NRWORK = IRU
CALL ZLARCM( M, N, RWORK( IRVT ), M, VT, LDVT, A, LDA,
$ RWORK( NRWORK ) )
CALL ZLACPY( 'F', M, N, A, LDA, VT, LDVT )
*
* N .LT. MNTHR2
*
-* Path 6t (N greater than M, but not much larger)
+* Path 6t (N > M, but not much larger)
* Reduce to bidiagonal form without LQ decomposition
* Use ZUNMBR to compute singular vectors
*
NWORK = ITAUP + M
*
* Bidiagonalize A
-* (CWorkspace: need 2*M+N, prefer 2*M+(M+N)*NB)
-* (RWorkspace: M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + (M+N)*NB [work]
+* RWorkspace: need M [e]
*
CALL ZGEBRD( M, N, A, LDA, S, RWORK( IE ), WORK( ITAUQ ),
$ WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
$ IERR )
IF( WNTQN ) THEN
*
+* Path 6tn (N > M, JOBZ='N')
* Compute singular values only
-* (Cworkspace: 0)
-* (Rworkspace: need BDSPAN)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + BDSPAC
*
- CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM, 1, DUM, 1,
+ CALL DBDSDC( 'L', 'N', M, S, RWORK( IE ), DUM,1,DUM,1,
$ DUM, IDUM, RWORK( NRWORK ), IWORK, INFO )
ELSE IF( WNTQO ) THEN
+* Path 6to (N > M, JOBZ='O')
LDWKVT = M
IVT = NWORK
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
* WORK( IVT ) is M by N
*
*
* WORK( IVT ) is M by CHUNK
*
- CHUNK = ( LWORK-3*M ) / M
+ CHUNK = ( LWORK - 3*M ) / M
NWORK = IVT + LDWKVT*CHUNK
END IF
*
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
$ WORK( ITAUQ ), U, LDU, WORK( NWORK ),
$ LWORK-NWORK+1, IERR )
*
- IF( LWORK.GE.M*N+3*M ) THEN
+ IF( LWORK .GE. M*N + 3*M ) THEN
*
-* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
-* Overwrite WORK(IVT) by right singular vectors of A,
-* copying to A
-* (Cworkspace: need M*N+2*M, prefer M*N+M+M*NB)
-* (Rworkspace: need 0)
+* Path 6to-fast
+* Copy real matrix RWORK(IRVT) to complex matrix WORK(IVT)
+* Overwrite WORK(IVT) by right singular vectors of A,
+* copying to A
+* CWorkspace: need 2*M [tauq, taup] + M*N [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, WORK( IVT ),
$ LDWKVT )
CALL ZLACPY( 'F', M, N, WORK( IVT ), LDWKVT, A, LDA )
ELSE
*
+* Path 6to-slow
* Generate P**H in A
-* (Cworkspace: need 2*M, prefer M+M*NB)
-* (Rworkspace: need 0)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*M [VT] + M*NB [work]
+* RWorkspace: need 0
*
CALL ZUNGBR( 'P', M, N, M, A, LDA, WORK( ITAUP ),
$ WORK( NWORK ), LWORK-NWORK+1, IERR )
*
* Multiply Q in A by real matrix RWORK(IRU), storing the
* result in WORK(IU), copying to A
-* (CWorkspace: need M*M, prefer M*N)
-* (Rworkspace: need 3*M*M, prefer M*M+2*M*N)
+* CWorkspace: need 2*M [tauq, taup] + M*M [VT]
+* CWorkspace: prefer 2*M [tauq, taup] + M*N [VT]
+* RWorkspace: need M [e] + M*M [RVT] + 2*M*M [rwork]
+* RWorkspace: prefer M [e] + M*M [RVT] + 2*M*N [rwork] < M + 5*M*M since N < 2*M here
*
NRWORK = IRU
DO 60 I = 1, N, CHUNK
END IF
ELSE IF( WNTQS ) THEN
*
+* Path 6ts (N > M, JOBZ='S')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLASET( 'F', M, N, CZERO, CZERO, VT, LDVT )
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
$ LWORK-NWORK+1, IERR )
ELSE
*
+* Path 6ta (N > M, JOBZ='A')
* Perform bidiagonal SVD, computing left singular vectors
* of bidiagonal matrix in RWORK(IRU) and computing right
* singular vectors of bidiagonal matrix in RWORK(IRVT)
-* (CWorkspace: need 0)
-* (RWorkspace: need BDSPAC)
+* CWorkspace: need 0
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU] + BDSPAC
*
IRVT = NRWORK
IRU = IRVT + M*M
*
* Copy real matrix RWORK(IRU) to complex matrix U
* Overwrite U by left singular vectors of A
-* (CWorkspace: need 3*M, prefer 2*M+M*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + M [work]
+* CWorkspace: prefer 2*M [tauq, taup] + M*NB [work]
+* RWorkspace: need M [e] + M*M [RVT] + M*M [RU]
*
CALL ZLACP2( 'F', M, M, RWORK( IRU ), M, U, LDU )
CALL ZUNMBR( 'Q', 'L', 'N', M, M, N, A, LDA,
*
* Copy real matrix RWORK(IRVT) to complex matrix VT
* Overwrite VT by right singular vectors of A
-* (CWorkspace: need 2*M+N, prefer 2*M+N*NB)
-* (RWorkspace: M*M)
+* CWorkspace: need 2*M [tauq, taup] + N [work]
+* CWorkspace: prefer 2*M [tauq, taup] + N*NB [work]
+* RWorkspace: need M [e] + M*M [RVT]
*
CALL ZLACP2( 'F', M, M, RWORK( IRVT ), M, VT, LDVT )
CALL ZUNMBR( 'P', 'R', 'C', N, N, M, A, LDA,
MNTHR = ILAENV( 6, 'ZGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for ZGEQRF
CALL ZGEQRF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGEQRF=CDUM(1)
+ LWORK_ZGEQRF = INT( CDUM(1) )
* Compute space needed for ZUNGQR
CALL ZUNGQR( M, N, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZUNGQR_N=CDUM(1)
+ LWORK_ZUNGQR_N = INT( CDUM(1) )
CALL ZUNGQR( M, M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZUNGQR_M=CDUM(1)
+ LWORK_ZUNGQR_M = INT( CDUM(1) )
* Compute space needed for ZGEBRD
CALL ZGEBRD( N, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGEBRD=CDUM(1)
+ LWORK_ZGEBRD = INT( CDUM(1) )
* Compute space needed for ZUNGBR
CALL ZUNGBR( 'P', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_P=CDUM(1)
+ LWORK_ZUNGBR_P = INT( CDUM(1) )
CALL ZUNGBR( 'Q', N, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_Q=CDUM(1)
+ LWORK_ZUNGBR_Q = INT( CDUM(1) )
*
IF( M.GE.MNTHR ) THEN
IF( WNTUN ) THEN
*
CALL ZGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGEBRD=CDUM(1)
+ LWORK_ZGEBRD = INT( CDUM(1) )
MAXWRK = 2*N + LWORK_ZGEBRD
IF( WNTUS .OR. WNTUO ) THEN
CALL ZUNGBR( 'Q', M, N, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_Q=CDUM(1)
+ LWORK_ZUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_Q )
END IF
IF( WNTUA ) THEN
CALL ZUNGBR( 'Q', M, M, N, A, LDA, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_Q=CDUM(1)
+ LWORK_ZUNGBR_Q = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_Q )
END IF
IF( .NOT.WNTVN ) THEN
MAXWRK = MAX( MAXWRK, 2*N+LWORK_ZUNGBR_P )
- MINWRK = 2*N + M
END IF
+ MINWRK = 2*N + M
END IF
ELSE IF( MINMN.GT.0 ) THEN
*
MNTHR = ILAENV( 6, 'ZGESVD', JOBU // JOBVT, M, N, 0, 0 )
* Compute space needed for ZGELQF
CALL ZGELQF( M, N, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGELQF=CDUM(1)
+ LWORK_ZGELQF = INT( CDUM(1) )
* Compute space needed for ZUNGLQ
CALL ZUNGLQ( N, N, M, CDUM(1), N, CDUM(1), CDUM(1), -1,
$ IERR )
- LWORK_ZUNGLQ_N=CDUM(1)
+ LWORK_ZUNGLQ_N = INT( CDUM(1) )
CALL ZUNGLQ( M, N, M, A, LDA, CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZUNGLQ_M=CDUM(1)
+ LWORK_ZUNGLQ_M = INT( CDUM(1) )
* Compute space needed for ZGEBRD
CALL ZGEBRD( M, M, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGEBRD=CDUM(1)
+ LWORK_ZGEBRD = INT( CDUM(1) )
* Compute space needed for ZUNGBR P
CALL ZUNGBR( 'P', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_P=CDUM(1)
+ LWORK_ZUNGBR_P = INT( CDUM(1) )
* Compute space needed for ZUNGBR Q
CALL ZUNGBR( 'Q', M, M, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_Q=CDUM(1)
+ LWORK_ZUNGBR_Q = INT( CDUM(1) )
IF( N.GE.MNTHR ) THEN
IF( WNTVN ) THEN
*
*
CALL ZGEBRD( M, N, A, LDA, S, DUM(1), CDUM(1),
$ CDUM(1), CDUM(1), -1, IERR )
- LWORK_ZGEBRD=CDUM(1)
+ LWORK_ZGEBRD = INT( CDUM(1) )
MAXWRK = 2*M + LWORK_ZGEBRD
IF( WNTVS .OR. WNTVO ) THEN
* Compute space needed for ZUNGBR P
CALL ZUNGBR( 'P', M, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_P=CDUM(1)
+ LWORK_ZUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_P )
END IF
IF( WNTVA ) THEN
CALL ZUNGBR( 'P', N, N, M, A, N, CDUM(1),
$ CDUM(1), -1, IERR )
- LWORK_ZUNGBR_P=CDUM(1)
+ LWORK_ZUNGBR_P = INT( CDUM(1) )
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_P )
END IF
IF( .NOT.WNTUN ) THEN
MAXWRK = MAX( MAXWRK, 2*M+LWORK_ZUNGBR_Q )
- MINWRK = 2*M + N
END IF
+ MINWRK = 2*M + N
END IF
END IF
MAXWRK = MAX( MAXWRK, MINWRK )
projects / platform / upstream / lapack.git / commitdiff