*> \brief ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices (blocked algorithm) * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGGES3 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, * \$ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, * \$ WORK, LWORK, RWORK, BWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVSL, JOBVSR, SORT * INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM * .. * .. Array Arguments .. * LOGICAL BWORK( * ) * DOUBLE PRECISION RWORK( * ) * COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), * \$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), * \$ WORK( * ) * .. * .. Function Arguments .. * LOGICAL SELCTG * EXTERNAL SELCTG * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices *> (A,B), the generalized eigenvalues, the generalized complex Schur *> form (S, T), and optionally left and/or right Schur vectors (VSL *> and VSR). This gives the generalized Schur factorization *> *> (A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H ) *> *> where (VSR)**H is the conjugate-transpose of VSR. *> *> Optionally, it also orders the eigenvalues so that a selected cluster *> of eigenvalues appears in the leading diagonal blocks of the upper *> triangular matrix S and the upper triangular matrix T. The leading *> columns of VSL and VSR then form an unitary basis for the *> corresponding left and right eigenspaces (deflating subspaces). *> *> (If only the generalized eigenvalues are needed, use the driver *> ZGGEV instead, which is faster.) *> *> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w *> or a ratio alpha/beta = w, such that A - w*B is singular. It is *> usually represented as the pair (alpha,beta), as there is a *> reasonable interpretation for beta=0, and even for both being zero. *> *> A pair of matrices (S,T) is in generalized complex Schur form if S *> and T are upper triangular and, in addition, the diagonal elements *> of T are non-negative real numbers. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVSL *> \verbatim *> JOBVSL is CHARACTER*1 *> = 'N': do not compute the left Schur vectors; *> = 'V': compute the left Schur vectors. *> \endverbatim *> *> \param[in] JOBVSR *> \verbatim *> JOBVSR is CHARACTER*1 *> = 'N': do not compute the right Schur vectors; *> = 'V': compute the right Schur vectors. *> \endverbatim *> *> \param[in] SORT *> \verbatim *> SORT is CHARACTER*1 *> Specifies whether or not to order the eigenvalues on the *> diagonal of the generalized Schur form. *> = 'N': Eigenvalues are not ordered; *> = 'S': Eigenvalues are ordered (see SELCTG). *> \endverbatim *> *> \param[in] SELCTG *> \verbatim *> SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments *> SELCTG must be declared EXTERNAL in the calling subroutine. *> If SORT = 'N', SELCTG is not referenced. *> If SORT = 'S', SELCTG is used to select eigenvalues to sort *> to the top left of the Schur form. *> An eigenvalue ALPHA(j)/BETA(j) is selected if *> SELCTG(ALPHA(j),BETA(j)) is true. *> *> Note that a selected complex eigenvalue may no longer satisfy *> SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since *> ordering may change the value of complex eigenvalues *> (especially if the eigenvalue is ill-conditioned), in this *> case INFO is set to N+2 (See INFO below). *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrices A, B, VSL, and VSR. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N) *> On entry, the first of the pair of matrices. *> On exit, A has been overwritten by its generalized Schur *> form S. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of A. LDA >= max(1,N). *> \endverbatim *> *> \param[in,out] B *> \verbatim *> B is COMPLEX*16 array, dimension (LDB, N) *> On entry, the second of the pair of matrices. *> On exit, B has been overwritten by its generalized Schur *> form T. *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of B. LDB >= max(1,N). *> \endverbatim *> *> \param[out] SDIM *> \verbatim *> SDIM is INTEGER *> If SORT = 'N', SDIM = 0. *> If SORT = 'S', SDIM = number of eigenvalues (after sorting) *> for which SELCTG is true. *> \endverbatim *> *> \param[out] ALPHA *> \verbatim *> ALPHA is COMPLEX*16 array, dimension (N) *> \endverbatim *> *> \param[out] BETA *> \verbatim *> BETA is COMPLEX*16 array, dimension (N) *> On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the *> generalized eigenvalues. ALPHA(j), j=1,...,N and BETA(j), *> j=1,...,N are the diagonals of the complex Schur form (A,B) *> output by ZGGES3. The BETA(j) will be non-negative real. *> *> Note: the quotients ALPHA(j)/BETA(j) may easily over- or *> underflow, and BETA(j) may even be zero. Thus, the user *> should avoid naively computing the ratio alpha/beta. *> However, ALPHA will be always less than and usually *> comparable with norm(A) in magnitude, and BETA always less *> than and usually comparable with norm(B). *> \endverbatim *> *> \param[out] VSL *> \verbatim *> VSL is COMPLEX*16 array, dimension (LDVSL,N) *> If JOBVSL = 'V', VSL will contain the left Schur vectors. *> Not referenced if JOBVSL = 'N'. *> \endverbatim *> *> \param[in] LDVSL *> \verbatim *> LDVSL is INTEGER *> The leading dimension of the matrix VSL. LDVSL >= 1, and *> if JOBVSL = 'V', LDVSL >= N. *> \endverbatim *> *> \param[out] VSR *> \verbatim *> VSR is COMPLEX*16 array, dimension (LDVSR,N) *> If JOBVSR = 'V', VSR will contain the right Schur vectors. *> Not referenced if JOBVSR = 'N'. *> \endverbatim *> *> \param[in] LDVSR *> \verbatim *> LDVSR is INTEGER *> The leading dimension of the matrix VSR. LDVSR >= 1, and *> if JOBVSR = 'V', LDVSR >= 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. *> *> 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] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION array, dimension (8*N) *> \endverbatim *> *> \param[out] BWORK *> \verbatim *> BWORK is LOGICAL array, dimension (N) *> Not referenced if SORT = 'N'. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value. *> =1,...,N: *> The QZ iteration failed. (A,B) are not in Schur *> form, but ALPHA(j) and BETA(j) should be correct for *> j=INFO+1,...,N. *> > N: =N+1: other than QZ iteration failed in ZHGEQZ *> =N+2: after reordering, roundoff changed values of *> some complex eigenvalues so that leading *> eigenvalues in the Generalized Schur form no *> longer satisfy SELCTG=.TRUE. This could also *> be caused due to scaling. *> =N+3: reordering failed in ZTGSEN. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date January 2015 * *> \ingroup complex16GEeigen * * ===================================================================== SUBROUTINE ZGGES3( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, \$ LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, \$ WORK, LWORK, RWORK, BWORK, INFO ) * * -- LAPACK driver 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..-- * January 2015 * * .. Scalar Arguments .. CHARACTER JOBVSL, JOBVSR, SORT INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM * .. * .. Array Arguments .. LOGICAL BWORK( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), \$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ), \$ WORK( * ) * .. * .. Function Arguments .. LOGICAL SELCTG EXTERNAL SELCTG * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) COMPLEX*16 CZERO, CONE PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), \$ CONE = ( 1.0D0, 0.0D0 ) ) * .. * .. Local Scalars .. LOGICAL CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL, \$ LQUERY, WANTST INTEGER I, ICOLS, IERR, IHI, IJOBVL, IJOBVR, ILEFT, \$ ILO, IRIGHT, IROWS, IRWRK, ITAU, IWRK, LWKOPT DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PVSL, \$ PVSR, SMLNUM * .. * .. Local Arrays .. INTEGER IDUM( 1 ) DOUBLE PRECISION DIF( 2 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHD3, \$ ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGSEN, ZUNGQR, \$ ZUNMQR * .. * .. External Functions .. LOGICAL LSAME DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Decode the input arguments * IF( LSAME( JOBVSL, 'N' ) ) THEN IJOBVL = 1 ILVSL = .FALSE. ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN IJOBVL = 2 ILVSL = .TRUE. ELSE IJOBVL = -1 ILVSL = .FALSE. END IF * IF( LSAME( JOBVSR, 'N' ) ) THEN IJOBVR = 1 ILVSR = .FALSE. ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN IJOBVR = 2 ILVSR = .TRUE. ELSE IJOBVR = -1 ILVSR = .FALSE. END IF * WANTST = LSAME( SORT, 'S' ) * * Test the input arguments * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) IF( IJOBVL.LE.0 ) THEN INFO = -1 ELSE IF( IJOBVR.LE.0 ) THEN INFO = -2 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -3 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( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN INFO = -14 ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN INFO = -16 ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN INFO = -18 END IF * * Compute workspace * IF( INFO.EQ.0 ) THEN CALL ZGEQRF( N, N, B, LDB, WORK, WORK, -1, IERR ) LWKOPT = MAX( 1, N + INT ( WORK( 1 ) ) ) CALL ZUNMQR( 'L', 'C', N, N, N, B, LDB, WORK, A, LDA, WORK, \$ -1, IERR ) LWKOPT = MAX( LWKOPT, N + INT ( WORK( 1 ) ) ) IF( ILVSL ) THEN CALL ZUNGQR( N, N, N, VSL, LDVSL, WORK, WORK, -1, IERR ) LWKOPT = MAX( LWKOPT, N + INT ( WORK( 1 ) ) ) END IF CALL ZGGHD3( JOBVSL, JOBVSR, N, 1, N, A, LDA, B, LDB, VSL, \$ LDVSL, VSR, LDVSR, WORK, -1, IERR ) LWKOPT = MAX( LWKOPT, N + INT ( WORK( 1 ) ) ) CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, 1, N, A, LDA, B, LDB, \$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, -1, \$ RWORK, IERR ) LWKOPT = MAX( LWKOPT, INT ( WORK( 1 ) ) ) IF( WANTST ) THEN CALL ZTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, \$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, SDIM, \$ PVSL, PVSR, DIF, WORK, -1, IDUM, 1, IERR ) LWKOPT = MAX( LWKOPT, INT ( WORK( 1 ) ) ) END IF WORK( 1 ) = DCMPLX( LWKOPT ) END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGGES3 ', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN SDIM = 0 RETURN END IF * * Get machine constants * EPS = DLAMCH( 'P' ) SMLNUM = DLAMCH( 'S' ) BIGNUM = ONE / SMLNUM CALL DLABAD( SMLNUM, BIGNUM ) SMLNUM = SQRT( SMLNUM ) / EPS BIGNUM = ONE / SMLNUM * * Scale A if max element outside range [SMLNUM,BIGNUM] * ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK ) ILASCL = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN ANRMTO = SMLNUM ILASCL = .TRUE. ELSE IF( ANRM.GT.BIGNUM ) THEN ANRMTO = BIGNUM ILASCL = .TRUE. END IF * IF( ILASCL ) \$ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR ) * * Scale B if max element outside range [SMLNUM,BIGNUM] * BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK ) ILBSCL = .FALSE. IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN BNRMTO = SMLNUM ILBSCL = .TRUE. ELSE IF( BNRM.GT.BIGNUM ) THEN BNRMTO = BIGNUM ILBSCL = .TRUE. END IF * IF( ILBSCL ) \$ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR ) * * Permute the matrix to make it more nearly triangular * ILEFT = 1 IRIGHT = N + 1 IRWRK = IRIGHT + N CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), RWORK( IRWRK ), IERR ) * * Reduce B to triangular form (QR decomposition of B) * IROWS = IHI + 1 - ILO ICOLS = N + 1 - ILO ITAU = 1 IWRK = ITAU + IROWS CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ), \$ WORK( IWRK ), LWORK+1-IWRK, IERR ) * * Apply the orthogonal transformation to matrix A * CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB, \$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ), \$ LWORK+1-IWRK, IERR ) * * Initialize VSL * IF( ILVSL ) THEN CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL ) IF( IROWS.GT.1 ) THEN CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB, \$ VSL( ILO+1, ILO ), LDVSL ) END IF CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL, \$ WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR ) END IF * * Initialize VSR * IF( ILVSR ) \$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR ) * * Reduce to generalized Hessenberg form * CALL ZGGHD3( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL, \$ LDVSL, VSR, LDVSR, WORK( IWRK ), LWORK+1-IWRK, IERR ) * SDIM = 0 * * Perform QZ algorithm, computing Schur vectors if desired * IWRK = ITAU CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, \$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWRK ), \$ LWORK+1-IWRK, RWORK( IRWRK ), IERR ) IF( IERR.NE.0 ) THEN IF( IERR.GT.0 .AND. IERR.LE.N ) THEN INFO = IERR ELSE IF( IERR.GT.N .AND. IERR.LE.2*N ) THEN INFO = IERR - N ELSE INFO = N + 1 END IF GO TO 30 END IF * * Sort eigenvalues ALPHA/BETA if desired * IF( WANTST ) THEN * * Undo scaling on eigenvalues before selecting * IF( ILASCL ) \$ CALL ZLASCL( 'G', 0, 0, ANRM, ANRMTO, N, 1, ALPHA, N, IERR ) IF( ILBSCL ) \$ CALL ZLASCL( 'G', 0, 0, BNRM, BNRMTO, N, 1, BETA, N, IERR ) * * Select eigenvalues * DO 10 I = 1, N BWORK( I ) = SELCTG( ALPHA( I ), BETA( I ) ) 10 CONTINUE * CALL ZTGSEN( 0, ILVSL, ILVSR, BWORK, N, A, LDA, B, LDB, ALPHA, \$ BETA, VSL, LDVSL, VSR, LDVSR, SDIM, PVSL, PVSR, \$ DIF, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1, IERR ) IF( IERR.EQ.1 ) \$ INFO = N + 3 * END IF * * Apply back-permutation to VSL and VSR * IF( ILVSL ) \$ CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), N, VSL, LDVSL, IERR ) IF( ILVSR ) \$ CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ), \$ RWORK( IRIGHT ), N, VSR, LDVSR, IERR ) * * Undo scaling * IF( ILASCL ) THEN CALL ZLASCL( 'U', 0, 0, ANRMTO, ANRM, N, N, A, LDA, IERR ) CALL ZLASCL( 'G', 0, 0, ANRMTO, ANRM, N, 1, ALPHA, N, IERR ) END IF * IF( ILBSCL ) THEN CALL ZLASCL( 'U', 0, 0, BNRMTO, BNRM, N, N, B, LDB, IERR ) CALL ZLASCL( 'G', 0, 0, BNRMTO, BNRM, N, 1, BETA, N, IERR ) END IF * IF( WANTST ) THEN * * Check if reordering is correct * LASTSL = .TRUE. SDIM = 0 DO 20 I = 1, N CURSL = SELCTG( ALPHA( I ), BETA( I ) ) IF( CURSL ) \$ SDIM = SDIM + 1 IF( CURSL .AND. .NOT.LASTSL ) \$ INFO = N + 2 LASTSL = CURSL 20 CONTINUE * END IF * 30 CONTINUE * WORK( 1 ) = DCMPLX( LWKOPT ) * RETURN * * End of ZGGES3 * END