* Definition: * =========== * * SUBROUTINE ZGETSLS( TRANS, M, N, NRHS, A, LDA, B, LDB * $ , WORK, LWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER TRANS * INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS * .. * .. Array Arguments .. * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGETSLS solves overdetermined or underdetermined real linear systems *> involving an M-by-N matrix A, or its transpose, using a tall skinny *> QR or short wide LQfactorization of A. It is assumed that A has *> full rank. *> *> The following options are provided: *> *> 1. If TRANS = 'N' and m >= n: find the least squares solution of *> an overdetermined system, i.e., solve the least squares problem *> minimize || B - A*X ||. *> *> 2. If TRANS = 'N' and m < n: find the minimum norm solution of *> an underdetermined system A * X = B. *> *> 3. If TRANS = 'C' and m >= n: find the minimum norm solution of *> an undetermined system A**T * X = B. *> *> 4. If TRANS = 'C' and m < n: find the least squares solution of *> an overdetermined system, i.e., solve the least squares problem *> minimize || B - A**T * X ||. *> *> Several right hand side vectors b and solution vectors x can be *> handled in a single call; they are stored as the columns of the *> M-by-NRHS right hand side matrix B and the N-by-NRHS solution *> matrix X. *> \endverbatim * * Arguments: * ========== * *> \param[in] TRANS *> \verbatim *> TRANS is CHARACTER*1 *> = 'N': the linear system involves A; *> = 'C': the linear system involves A**C. *> \endverbatim *> *> \param[in] M *> \verbatim *> M is INTEGER *> The number of rows of the matrix A. M >= 0. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The number of columns of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] NRHS *> \verbatim *> NRHS is INTEGER *> The number of right hand sides, i.e., the number of *> columns of the matrices B and X. NRHS >=0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA,N) *> On entry, the M-by-N matrix A. *> On exit, *> if M >= N, A is overwritten by details of its QR *> factorization as returned by DGEQRF; *> if M < N, A is overwritten by details of its LQ *> factorization as returned by DGELQF. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,M). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX*16 array, dimension (LDB,NRHS) *> On entry, the matrix B of right hand side vectors, stored *> columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS *> if TRANS = 'T'. *> On exit, if INFO = 0, B is overwritten by the solution *> vectors, stored columnwise: *> if TRANS = 'N' and m >= n, rows 1 to n of B contain the least *> squares solution vectors; the residual sum of squares for the *> solution in each column is given by the sum of squares of *> elements N+1 to M in that column; *> if TRANS = 'N' and m < n, rows 1 to N of B contain the *> minimum norm solution vectors; *> if TRANS = 'T' and m >= n, rows 1 to M of B contain the *> minimum norm solution vectors; *> if TRANS = 'T' and m < n, rows 1 to M of B contain the *> least squares solution vectors; the residual sum of squares *> for the solution in each column is given by the sum of *> squares of elements M+1 to N in that column. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= MAX(1,M,N). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. *> LWORK >= max( 1, MN + max( MN, NRHS ) ). *> For optimal performance, *> LWORK >= max( 1, MN + max( MN, NRHS )*NB ). *> where MN = min(M,N) and NB is the optimum block size. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates the optimal size of the WORK array, returns *> this value as the first entry of the WORK array, and no error *> message related to LWORK is issued by XERBLA. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> > 0: if INFO = i, the i-th diagonal element of the *> triangular factor of A is zero, so that A does not have *> full rank; the least squares solution could not be *> computed. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * * ===================================================================== SUBROUTINE ZGETSLS( TRANS, M, N, NRHS, A, LDA, B, LDB $ , WORK, LWORK, INFO ) * * -- LAPACK driver routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. CHARACTER TRANS INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, MB, NB * .. * .. Array Arguments .. COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * ) * * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CZERO PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, TRAN INTEGER I, IASCL, IBSCL, J, MINMN, MAXMN, BROW, LW, $ SCLLEN, MNK, WSIZEO, WSIZEM, LW1, LW2, INFO2 DOUBLE PRECISION ANRM, BIGNUM, BNRM, SMLNUM * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE * .. * .. External Subroutines .. EXTERNAL ZGEQR, ZGEMQR, ZLASCL, ZLASET, $ ZTRTRS, XERBLA, ZGELQ, ZGEMLQ * .. * .. Intrinsic Functions .. INTRINSIC DBLE, MAX, MIN * .. * .. Executable Statements .. * * Test the input arguments. * INFO=0 MINMN = MIN( M, N ) MAXMN = MAX( M, N ) MNK = MAX(MINMN,NRHS) TRAN = LSAME( TRANS, 'C' ) * LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.( LSAME( TRANS, 'N' ) .OR. $ LSAME( TRANS, 'C' ) ) ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( NRHS.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN INFO = -8 END IF * IF( INFO.EQ.0) THEN * * Determine the block size and minimum LWORK * IF ( M.GE.N ) THEN CALL ZGEQR( M, N, A, LDA, WORK(1), -1, WORK(6), -1, $ INFO2) MB = INT(WORK(4)) NB = INT(WORK(5)) LW = INT(WORK(6)) CALL ZGEMQR( 'L', TRANS, M, NRHS, N, A, LDA, WORK(1), $ INT(WORK(2)), B, LDB, WORK(6), -1 , INFO2 ) WSIZEO = INT(WORK(2))+MAX(LW,INT(WORK(6))) WSIZEM = INT(WORK(3))+MAX(LW,INT(WORK(6))) ELSE CALL ZGELQ( M, N, A, LDA, WORK(1), -1, WORK(6), -1, $ INFO2) MB = INT(WORK(4)) NB = INT(WORK(5)) LW = INT(WORK(6)) CALL ZGEMLQ( 'L', TRANS, N, NRHS, M, A, LDA, WORK(1), $ INT(WORK(2)), B, LDB, WORK(6), -1 , INFO2 ) WSIZEO = INT(WORK(2))+MAX(LW,INT(WORK(6))) WSIZEM = INT(WORK(3))+MAX(LW,INT(WORK(6))) END IF * IF((LWORK.LT.WSIZEO).AND.(.NOT.LQUERY)) THEN INFO=-10 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGETSLS', -INFO ) WORK( 1 ) = DBLE( WSIZEO ) WORK( 2 ) = DBLE( WSIZEM ) RETURN ELSE IF (LQUERY) THEN WORK( 1 ) = DBLE( WSIZEO ) WORK( 2 ) = DBLE( WSIZEM ) RETURN END IF IF(LWORK.LT.WSIZEO) THEN LW1=INT(WORK(3)) LW2=MAX(LW,INT(WORK(6))) ELSE LW1=INT(WORK(2)) LW2=MAX(LW,INT(WORK(6))) END IF * * Quick return if possible * IF( MIN( M, N, NRHS ).EQ.0 ) THEN CALL ZLASET( 'FULL', MAX( M, N ), NRHS, CZERO, CZERO, $ B, LDB ) RETURN END IF * * Get machine parameters * SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) * * Scale A, B if max element outside range [SMLNUM,BIGNUM] * ANRM = ZLANGE( 'M', M, N, A, LDA, WORK ) IASCL = 0 IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO ) IASCL = 1 ELSE IF( ANRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO ) IASCL = 2 ELSE IF( ANRM.EQ.ZERO ) THEN * * Matrix all zero. Return zero solution. * CALL ZLASET( 'F', MAXMN, NRHS, CZERO, CZERO, B, LDB ) GO TO 50 END IF * BROW = M IF ( TRAN ) THEN BROW = N END IF BNRM = ZLANGE( 'M', BROW, NRHS, B, LDB, WORK ) IBSCL = 0 IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN * * Scale matrix norm up to SMLNUM * CALL ZLASCL( 'G', 0, 0, BNRM, SMLNUM, BROW, NRHS, B, LDB, $ INFO ) IBSCL = 1 ELSE IF( BNRM.GT.BIGNUM ) THEN * * Scale matrix norm down to BIGNUM * CALL ZLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB, $ INFO ) IBSCL = 2 END IF * IF ( M.GE.N) THEN * * compute QR factorization of A * CALL ZGEQR( M, N, A, LDA, WORK(LW2+1), LW1 $ , WORK(1), LW2, INFO ) IF (.NOT.TRAN) THEN * * Least-Squares Problem min || A * X - B || * * B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS) * CALL ZGEMQR( 'L' , 'C', M, NRHS, N, A, LDA, $ WORK(LW2+1), LW1, B, LDB, WORK(1), LW2, INFO ) * * B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS) * CALL ZTRTRS( 'U', 'N', 'N', N, NRHS, $ A, LDA, B, LDB, INFO ) IF(INFO.GT.0) THEN RETURN END IF SCLLEN = N ELSE * * Overdetermined system of equations A**T * X = B * * B(1:N,1:NRHS) := inv(R**T) * B(1:N,1:NRHS) * CALL ZTRTRS( 'U', 'C', 'N', N, NRHS, $ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * * B(N+1:M,1:NRHS) = CZERO * DO 20 J = 1, NRHS DO 10 I = N + 1, M B( I, J ) = CZERO 10 CONTINUE 20 CONTINUE * * B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS) * CALL ZGEMQR( 'L', 'N', M, NRHS, N, A, LDA, $ WORK( LW2+1), LW1, B, LDB, WORK( 1 ), LW2, $ INFO ) * SCLLEN = M * END IF * ELSE * * Compute LQ factorization of A * CALL ZGELQ( M, N, A, LDA, WORK(LW2+1), LW1 $ , WORK(1), LW2, INFO ) * * workspace at least M, optimally M*NB. * IF( .NOT.TRAN ) THEN * * underdetermined system of equations A * X = B * * B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS) * CALL ZTRTRS( 'L', 'N', 'N', M, NRHS, $ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * * B(M+1:N,1:NRHS) = 0 * DO 40 J = 1, NRHS DO 30 I = M + 1, N B( I, J ) = CZERO 30 CONTINUE 40 CONTINUE * * B(1:N,1:NRHS) := Q(1:N,:)**T * B(1:M,1:NRHS) * CALL ZGEMLQ( 'L', 'C', N, NRHS, M, A, LDA, $ WORK( LW2+1), LW1, B, LDB, WORK( 1 ), LW2, $ INFO ) * * workspace at least NRHS, optimally NRHS*NB * SCLLEN = N * ELSE * * overdetermined system min || A**T * X - B || * * B(1:N,1:NRHS) := Q * B(1:N,1:NRHS) * CALL ZGEMLQ( 'L', 'N', N, NRHS, M, A, LDA, $ WORK( LW2+1), LW1, B, LDB, WORK( 1 ), LW2, $ INFO ) * * workspace at least NRHS, optimally NRHS*NB * * B(1:M,1:NRHS) := inv(L**T) * B(1:M,1:NRHS) * CALL ZTRTRS( 'L', 'C', 'N', M, NRHS, $ A, LDA, B, LDB, INFO ) * IF( INFO.GT.0 ) THEN RETURN END IF * SCLLEN = M * END IF * END IF * * Undo scaling * IF( IASCL.EQ.1 ) THEN CALL ZLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB, $ INFO ) ELSE IF( IASCL.EQ.2 ) THEN CALL ZLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB, $ INFO ) END IF IF( IBSCL.EQ.1 ) THEN CALL ZLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB, $ INFO ) ELSE IF( IBSCL.EQ.2 ) THEN CALL ZLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB, $ INFO ) END IF * 50 CONTINUE WORK( 1 ) = DBLE( WSIZEO ) WORK( 2 ) = DBLE( WSIZEM ) RETURN * * End of ZGETSLS * END