*> \brief \b ZTGSEN * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZTGSEN + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, * ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, * WORK, LWORK, IWORK, LIWORK, INFO ) * * .. Scalar Arguments .. * LOGICAL WANTQ, WANTZ * INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, * $ M, N * DOUBLE PRECISION PL, PR * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * INTEGER IWORK( * ) * DOUBLE PRECISION DIF( * ) * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), * $ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZTGSEN reorders the generalized Schur decomposition of a complex *> matrix pair (A, B) (in terms of an unitary equivalence trans- *> formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues *> appears in the leading diagonal blocks of the pair (A,B). The leading *> columns of Q and Z form unitary bases of the corresponding left and *> right eigenspaces (deflating subspaces). (A, B) must be in *> generalized Schur canonical form, that is, A and B are both upper *> triangular. *> *> ZTGSEN also computes the generalized eigenvalues *> *> w(j)= ALPHA(j) / BETA(j) *> *> of the reordered matrix pair (A, B). *> *> Optionally, the routine computes estimates of reciprocal condition *> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), *> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) *> between the matrix pairs (A11, B11) and (A22,B22) that correspond to *> the selected cluster and the eigenvalues outside the cluster, resp., *> and norms of "projections" onto left and right eigenspaces w.r.t. *> the selected cluster in the (1,1)-block. *> *> \endverbatim * * Arguments: * ========== * *> \param[in] IJOB *> \verbatim *> IJOB is integer *> Specifies whether condition numbers are required for the *> cluster of eigenvalues (PL and PR) or the deflating subspaces *> (Difu and Difl): *> =0: Only reorder w.r.t. SELECT. No extras. *> =1: Reciprocal of norms of "projections" onto left and right *> eigenspaces w.r.t. the selected cluster (PL and PR). *> =2: Upper bounds on Difu and Difl. F-norm-based estimate *> (DIF(1:2)). *> =3: Estimate of Difu and Difl. 1-norm-based estimate *> (DIF(1:2)). *> About 5 times as expensive as IJOB = 2. *> =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic *> version to get it all. *> =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above) *> \endverbatim *> *> \param[in] WANTQ *> \verbatim *> WANTQ is LOGICAL *> .TRUE. : update the left transformation matrix Q; *> .FALSE.: do not update Q. *> \endverbatim *> *> \param[in] WANTZ *> \verbatim *> WANTZ is LOGICAL *> .TRUE. : update the right transformation matrix Z; *> .FALSE.: do not update Z. *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> SELECT is LOGICAL array, dimension (N) *> SELECT specifies the eigenvalues in the selected cluster. To *> select an eigenvalue w(j), SELECT(j) must be set to *> .TRUE.. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A and B. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension(LDA,N) *> On entry, the upper triangular matrix A, in generalized *> Schur canonical form. *> On exit, A is overwritten by the reordered matrix A. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX*16 array, dimension(LDB,N) *> On entry, the upper triangular matrix B, in generalized *> Schur canonical form. *> On exit, B is overwritten by the reordered matrix B. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is COMPLEX*16 array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is COMPLEX*16 array, dimension (N) *> *> The diagonal elements of A and B, respectively, *> when the pair (A,B) has been reduced to generalized Schur *> form. ALPHA(i)/BETA(i) i=1,...,N are the generalized *> eigenvalues. *> \endverbatim *> *> \param[in,out] Q *> \verbatim *> Q is COMPLEX*16 array, dimension (LDQ,N) *> On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. *> On exit, Q has been postmultiplied by the left unitary *> transformation matrix which reorder (A, B); The leading M *> columns of Q form orthonormal bases for the specified pair of *> left eigenspaces (deflating subspaces). *> If WANTQ = .FALSE., Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. LDQ >= 1. *> If WANTQ = .TRUE., LDQ >= N. *> \endverbatim *> *> \param[in,out] Z *> \verbatim *> Z is COMPLEX*16 array, dimension (LDZ,N) *> On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. *> On exit, Z has been postmultiplied by the left unitary *> transformation matrix which reorder (A, B); The leading M *> columns of Z form orthonormal bases for the specified pair of *> left eigenspaces (deflating subspaces). *> If WANTZ = .FALSE., Z is not referenced. *> \endverbatim *> *> \param[in] LDZ *> \verbatim *> LDZ is INTEGER *> The leading dimension of the array Z. LDZ >= 1. *> If WANTZ = .TRUE., LDZ >= N. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The dimension of the specified pair of left and right *> eigenspaces, (deflating subspaces) 0 <= M <= N. *> \endverbatim *> *> \param[out] PL *> \verbatim *> PL is DOUBLE PRECISION *> \endverbatim *> *> \param[out] PR *> \verbatim *> PR is DOUBLE PRECISION *> *> If IJOB = 1, 4 or 5, PL, PR are lower bounds on the *> reciprocal of the norm of "projections" onto left and right *> eigenspace with respect to the selected cluster. *> 0 < PL, PR <= 1. *> If M = 0 or M = N, PL = PR = 1. *> If IJOB = 0, 2 or 3 PL, PR are not referenced. *> \endverbatim *> *> \param[out] DIF *> \verbatim *> DIF is DOUBLE PRECISION array, dimension (2). *> If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl. *> If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on *> Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based *> estimates of Difu and Difl, computed using reversed *> communication with ZLACN2. *> If M = 0 or N, DIF(1:2) = F-norm([A, B]). *> If IJOB = 0 or 1, DIF is not referenced. *> \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 >= 1 *> If IJOB = 1, 2 or 4, LWORK >= 2*M*(N-M) *> If IJOB = 3 or 5, LWORK >= 4*M*(N-M) *> *> 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] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) *> On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. *> \endverbatim *> *> \param[in] LIWORK *> \verbatim *> LIWORK is INTEGER *> The dimension of the array IWORK. LIWORK >= 1. *> If IJOB = 1, 2 or 4, LIWORK >= N+2; *> If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M)); *> *> If LIWORK = -1, then a workspace query is assumed; the *> routine only calculates the optimal size of the IWORK array, *> returns this value as the first entry of the IWORK array, and *> no error message related to LIWORK 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. *> =1: Reordering of (A, B) failed because the transformed *> matrix pair (A, B) would be too far from generalized *> Schur form; the problem is very ill-conditioned. *> (A, B) may have been partially reordered. *> If requested, 0 is returned in DIF(*), PL and PR. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup complex16OTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> ZTGSEN first collects the selected eigenvalues by computing unitary *> U and W that move them to the top left corner of (A, B). In other *> words, the selected eigenvalues are the eigenvalues of (A11, B11) in *> *> U**H*(A, B)*W = (A11 A12) (B11 B12) n1 *> ( 0 A22),( 0 B22) n2 *> n1 n2 n1 n2 *> *> where N = n1+n2 and U**H means the conjugate transpose of U. The first *> n1 columns of U and W span the specified pair of left and right *> eigenspaces (deflating subspaces) of (A, B). *> *> If (A, B) has been obtained from the generalized real Schur *> decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the *> reordered generalized Schur form of (C, D) is given by *> *> (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H, *> *> and the first n1 columns of Q*U and Z*W span the corresponding *> deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.). *> *> Note that if the selected eigenvalue is sufficiently ill-conditioned, *> then its value may differ significantly from its value before *> reordering. *> *> The reciprocal condition numbers of the left and right eigenspaces *> spanned by the first n1 columns of U and W (or Q*U and Z*W) may *> be returned in DIF(1:2), corresponding to Difu and Difl, resp. *> *> The Difu and Difl are defined as: *> *> Difu[(A11, B11), (A22, B22)] = sigma-min( Zu ) *> and *> Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)], *> *> where sigma-min(Zu) is the smallest singular value of the *> (2*n1*n2)-by-(2*n1*n2) matrix *> *> Zu = [ kron(In2, A11) -kron(A22**H, In1) ] *> [ kron(In2, B11) -kron(B22**H, In1) ]. *> *> Here, Inx is the identity matrix of size nx and A22**H is the *> conjugate transpose of A22. kron(X, Y) is the Kronecker product between *> the matrices X and Y. *> *> When DIF(2) is small, small changes in (A, B) can cause large changes *> in the deflating subspace. An approximate (asymptotic) bound on the *> maximum angular error in the computed deflating subspaces is *> *> EPS * norm((A, B)) / DIF(2), *> *> where EPS is the machine precision. *> *> The reciprocal norm of the projectors on the left and right *> eigenspaces associated with (A11, B11) may be returned in PL and PR. *> They are computed as follows. First we compute L and R so that *> P*(A, B)*Q is block diagonal, where *> *> P = ( I -L ) n1 Q = ( I R ) n1 *> ( 0 I ) n2 and ( 0 I ) n2 *> n1 n2 n1 n2 *> *> and (L, R) is the solution to the generalized Sylvester equation *> *> A11*R - L*A22 = -A12 *> B11*R - L*B22 = -B12 *> *> Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2). *> An approximate (asymptotic) bound on the average absolute error of *> the selected eigenvalues is *> *> EPS * norm((A, B)) / PL. *> *> There are also global error bounds which valid for perturbations up *> to a certain restriction: A lower bound (x) on the smallest *> F-norm(E,F) for which an eigenvalue of (A11, B11) may move and *> coalesce with an eigenvalue of (A22, B22) under perturbation (E,F), *> (i.e. (A + E, B + F), is *> *> x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)). *> *> An approximate bound on x can be computed from DIF(1:2), PL and PR. *> *> If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed *> (L', R') and unperturbed (L, R) left and right deflating subspaces *> associated with the selected cluster in the (1,1)-blocks can be *> bounded as *> *> max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2)) *> max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2)) *> *> See LAPACK User's Guide section 4.11 or the following references *> for more information. *> *> Note that if the default method for computing the Frobenius-norm- *> based estimate DIF is not wanted (see ZLATDF), then the parameter *> IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF *> (IJOB = 2 will be used)). See ZTGSYL for more details. *> \endverbatim * *> \par Contributors: * ================== *> *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, *> Umea University, S-901 87 Umea, Sweden. * *> \par References: * ================ *> *> [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the *> Generalized Real Schur Form of a Regular Matrix Pair (A, B), in *> M.S. Moonen et al (eds), Linear Algebra for Large Scale and *> Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. *> \n *> [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified *> Eigenvalues of a Regular Matrix Pair (A, B) and Condition *> Estimation: Theory, Algorithms and Software, Report *> UMINF - 94.04, Department of Computing Science, Umea University, *> S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. *> To appear in Numerical Algorithms, 1996. *> \n *> [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software *> for Solving the Generalized Sylvester Equation and Estimating the *> Separation between Regular Matrix Pairs, Report UMINF - 93.23, *> Department of Computing Science, Umea University, S-901 87 Umea, *> Sweden, December 1993, Revised April 1994, Also as LAPACK working *> Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, *> 1996. *> * ===================================================================== SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, $ ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, $ WORK, LWORK, IWORK, LIWORK, INFO ) * * -- LAPACK computational routine (version 3.6.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * June 2016 * * .. Scalar Arguments .. LOGICAL WANTQ, WANTZ INTEGER IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK, $ M, N DOUBLE PRECISION PL, PR * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) DOUBLE PRECISION DIF( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), $ BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * ) * .. * * ===================================================================== * * .. Parameters .. INTEGER IDIFJB PARAMETER ( IDIFJB = 3 ) DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP INTEGER I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2, $ N1, N2 DOUBLE PRECISION DSCALE, DSUM, RDSCAL, SAFMIN COMPLEX*16 TEMP1, TEMP2 * .. * .. Local Arrays .. INTEGER ISAVE( 3 ) * .. * .. External Subroutines .. EXTERNAL XERBLA, ZLACN2, ZLACPY, ZLASSQ, ZSCAL, ZTGEXC, $ ZTGSYL * .. * .. Intrinsic Functions .. INTRINSIC ABS, DCMPLX, DCONJG, MAX, SQRT * .. * .. External Functions .. DOUBLE PRECISION DLAMCH EXTERNAL DLAMCH * .. * .. Executable Statements .. * * Decode and test the input parameters * INFO = 0 LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) * IF( IJOB.LT.0 .OR. IJOB.GT.5 ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -9 ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN INFO = -13 ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN INFO = -15 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTGSEN', -INFO ) RETURN END IF * IERR = 0 * WANTP = IJOB.EQ.1 .OR. IJOB.GE.4 WANTD1 = IJOB.EQ.2 .OR. IJOB.EQ.4 WANTD2 = IJOB.EQ.3 .OR. IJOB.EQ.5 WANTD = WANTD1 .OR. WANTD2 * * Set M to the dimension of the specified pair of deflating * subspaces. * M = 0 IF( .NOT.LQUERY .OR. IJOB.NE.0 ) THEN DO 10 K = 1, N ALPHA( K ) = A( K, K ) BETA( K ) = B( K, K ) IF( K.LT.N ) THEN IF( SELECT( K ) ) $ M = M + 1 ELSE IF( SELECT( N ) ) $ M = M + 1 END IF 10 CONTINUE END IF * IF( IJOB.EQ.1 .OR. IJOB.EQ.2 .OR. IJOB.EQ.4 ) THEN LWMIN = MAX( 1, 2*M*( N-M ) ) LIWMIN = MAX( 1, N+2 ) ELSE IF( IJOB.EQ.3 .OR. IJOB.EQ.5 ) THEN LWMIN = MAX( 1, 4*M*( N-M ) ) LIWMIN = MAX( 1, 2*M*( N-M ), N+2 ) ELSE LWMIN = 1 LIWMIN = 1 END IF * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -21 ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN INFO = -23 END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZTGSEN', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible. * IF( M.EQ.N .OR. M.EQ.0 ) THEN IF( WANTP ) THEN PL = ONE PR = ONE END IF IF( WANTD ) THEN DSCALE = ZERO DSUM = ONE DO 20 I = 1, N CALL ZLASSQ( N, A( 1, I ), 1, DSCALE, DSUM ) CALL ZLASSQ( N, B( 1, I ), 1, DSCALE, DSUM ) 20 CONTINUE DIF( 1 ) = DSCALE*SQRT( DSUM ) DIF( 2 ) = DIF( 1 ) END IF GO TO 70 END IF * * Get machine constant * SAFMIN = DLAMCH( 'S' ) * * Collect the selected blocks at the top-left corner of (A, B). * KS = 0 DO 30 K = 1, N SWAP = SELECT( K ) IF( SWAP ) THEN KS = KS + 1 * * Swap the K-th block to position KS. Compute unitary Q * and Z that will swap adjacent diagonal blocks in (A, B). * IF( K.NE.KS ) $ CALL ZTGEXC( WANTQ, WANTZ, N, A, LDA, B, LDB, Q, LDQ, Z, $ LDZ, K, KS, IERR ) * IF( IERR.GT.0 ) THEN * * Swap is rejected: exit. * INFO = 1 IF( WANTP ) THEN PL = ZERO PR = ZERO END IF IF( WANTD ) THEN DIF( 1 ) = ZERO DIF( 2 ) = ZERO END IF GO TO 70 END IF END IF 30 CONTINUE IF( WANTP ) THEN * * Solve generalized Sylvester equation for R and L: * A11 * R - L * A22 = A12 * B11 * R - L * B22 = B12 * N1 = M N2 = N - M I = N1 + 1 CALL ZLACPY( 'Full', N1, N2, A( 1, I ), LDA, WORK, N1 ) CALL ZLACPY( 'Full', N1, N2, B( 1, I ), LDB, WORK( N1*N2+1 ), $ N1 ) IJB = 0 CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), N1, $ DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ), $ LWORK-2*N1*N2, IWORK, IERR ) * * Estimate the reciprocal of norms of "projections" onto * left and right eigenspaces * RDSCAL = ZERO DSUM = ONE CALL ZLASSQ( N1*N2, WORK, 1, RDSCAL, DSUM ) PL = RDSCAL*SQRT( DSUM ) IF( PL.EQ.ZERO ) THEN PL = ONE ELSE PL = DSCALE / ( SQRT( DSCALE*DSCALE / PL+PL )*SQRT( PL ) ) END IF RDSCAL = ZERO DSUM = ONE CALL ZLASSQ( N1*N2, WORK( N1*N2+1 ), 1, RDSCAL, DSUM ) PR = RDSCAL*SQRT( DSUM ) IF( PR.EQ.ZERO ) THEN PR = ONE ELSE PR = DSCALE / ( SQRT( DSCALE*DSCALE / PR+PR )*SQRT( PR ) ) END IF END IF IF( WANTD ) THEN * * Compute estimates Difu and Difl. * IF( WANTD1 ) THEN N1 = M N2 = N - M I = N1 + 1 IJB = IDIFJB * * Frobenius norm-based Difu estimate. * CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, WORK, $ N1, B, LDB, B( I, I ), LDB, WORK( N1*N2+1 ), $ N1, DSCALE, DIF( 1 ), WORK( N1*N2*2+1 ), $ LWORK-2*N1*N2, IWORK, IERR ) * * Frobenius norm-based Difl estimate. * CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, WORK, $ N2, B( I, I ), LDB, B, LDB, WORK( N1*N2+1 ), $ N2, DSCALE, DIF( 2 ), WORK( N1*N2*2+1 ), $ LWORK-2*N1*N2, IWORK, IERR ) ELSE * * Compute 1-norm-based estimates of Difu and Difl using * reversed communication with ZLACN2. In each step a * generalized Sylvester equation or a transposed variant * is solved. * KASE = 0 N1 = M N2 = N - M I = N1 + 1 IJB = 0 MN2 = 2*N1*N2 * * 1-norm-based estimate of Difu. * 40 CONTINUE CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 1 ), KASE, $ ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Solve generalized Sylvester equation * CALL ZTGSYL( 'N', IJB, N1, N2, A, LDA, A( I, I ), LDA, $ WORK, N1, B, LDB, B( I, I ), LDB, $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK, $ IERR ) ELSE * * Solve the transposed variant. * CALL ZTGSYL( 'C', IJB, N1, N2, A, LDA, A( I, I ), LDA, $ WORK, N1, B, LDB, B( I, I ), LDB, $ WORK( N1*N2+1 ), N1, DSCALE, DIF( 1 ), $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK, $ IERR ) END IF GO TO 40 END IF DIF( 1 ) = DSCALE / DIF( 1 ) * * 1-norm-based estimate of Difl. * 50 CONTINUE CALL ZLACN2( MN2, WORK( MN2+1 ), WORK, DIF( 2 ), KASE, $ ISAVE ) IF( KASE.NE.0 ) THEN IF( KASE.EQ.1 ) THEN * * Solve generalized Sylvester equation * CALL ZTGSYL( 'N', IJB, N2, N1, A( I, I ), LDA, A, LDA, $ WORK, N2, B( I, I ), LDB, B, LDB, $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK, $ IERR ) ELSE * * Solve the transposed variant. * CALL ZTGSYL( 'C', IJB, N2, N1, A( I, I ), LDA, A, LDA, $ WORK, N2, B, LDB, B( I, I ), LDB, $ WORK( N1*N2+1 ), N2, DSCALE, DIF( 2 ), $ WORK( N1*N2*2+1 ), LWORK-2*N1*N2, IWORK, $ IERR ) END IF GO TO 50 END IF DIF( 2 ) = DSCALE / DIF( 2 ) END IF END IF * * If B(K,K) is complex, make it real and positive (normalization * of the generalized Schur form) and Store the generalized * eigenvalues of reordered pair (A, B) * DO 60 K = 1, N DSCALE = ABS( B( K, K ) ) IF( DSCALE.GT.SAFMIN ) THEN TEMP1 = DCONJG( B( K, K ) / DSCALE ) TEMP2 = B( K, K ) / DSCALE B( K, K ) = DSCALE CALL ZSCAL( N-K, TEMP1, B( K, K+1 ), LDB ) CALL ZSCAL( N-K+1, TEMP1, A( K, K ), LDA ) IF( WANTQ ) $ CALL ZSCAL( N, TEMP2, Q( 1, K ), 1 ) ELSE B( K, K ) = DCMPLX( ZERO, ZERO ) END IF * ALPHA( K ) = A( K, K ) BETA( K ) = B( K, K ) * 60 CONTINUE * 70 CONTINUE * WORK( 1 ) = LWMIN IWORK( 1 ) = LIWMIN * RETURN * * End of ZTGSEN * END