*> \brief ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGEESX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, * VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, * BWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBVS, SENSE, SORT * INTEGER INFO, LDA, LDVS, LWORK, N, SDIM * DOUBLE PRECISION RCONDE, RCONDV * .. * .. Array Arguments .. * LOGICAL BWORK( * ) * DOUBLE PRECISION RWORK( * ) * COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * ) * .. * .. Function Arguments .. * LOGICAL SELECT * EXTERNAL SELECT * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the *> eigenvalues, the Schur form T, and, optionally, the matrix of Schur *> vectors Z. This gives the Schur factorization A = Z*T*(Z**H). *> *> Optionally, it also orders the eigenvalues on the diagonal of the *> Schur form so that selected eigenvalues are at the top left; *> computes a reciprocal condition number for the average of the *> selected eigenvalues (RCONDE); and computes a reciprocal condition *> number for the right invariant subspace corresponding to the *> selected eigenvalues (RCONDV). The leading columns of Z form an *> orthonormal basis for this invariant subspace. *> *> For further explanation of the reciprocal condition numbers RCONDE *> and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where *> these quantities are called s and sep respectively). *> *> A complex matrix is in Schur form if it is upper triangular. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOBVS *> \verbatim *> JOBVS is CHARACTER*1 *> = 'N': Schur vectors are not computed; *> = 'V': Schur vectors are computed. *> \endverbatim *> *> \param[in] SORT *> \verbatim *> SORT is CHARACTER*1 *> Specifies whether or not to order the eigenvalues on the *> diagonal of the Schur form. *> = 'N': Eigenvalues are not ordered; *> = 'S': Eigenvalues are ordered (see SELECT). *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument *> SELECT must be declared EXTERNAL in the calling subroutine. *> If SORT = 'S', SELECT is used to select eigenvalues to order *> to the top left of the Schur form. *> If SORT = 'N', SELECT is not referenced. *> An eigenvalue W(j) is selected if SELECT(W(j)) is true. *> \endverbatim *> *> \param[in] SENSE *> \verbatim *> SENSE is CHARACTER*1 *> Determines which reciprocal condition numbers are computed. *> = 'N': None are computed; *> = 'E': Computed for average of selected eigenvalues only; *> = 'V': Computed for selected right invariant subspace only; *> = 'B': Computed for both. *> If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in,out] A *> \verbatim *> A is COMPLEX*16 array, dimension (LDA, N) *> On entry, the N-by-N matrix A. *> On exit, A is overwritten by its Schur form T. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[out] SDIM *> \verbatim *> SDIM is INTEGER *> If SORT = 'N', SDIM = 0. *> If SORT = 'S', SDIM = number of eigenvalues for which *> SELECT is true. *> \endverbatim *> *> \param[out] W *> \verbatim *> W is COMPLEX*16 array, dimension (N) *> W contains the computed eigenvalues, in the same order *> that they appear on the diagonal of the output Schur form T. *> \endverbatim *> *> \param[out] VS *> \verbatim *> VS is COMPLEX*16 array, dimension (LDVS,N) *> If JOBVS = 'V', VS contains the unitary matrix Z of Schur *> vectors. *> If JOBVS = 'N', VS is not referenced. *> \endverbatim *> *> \param[in] LDVS *> \verbatim *> LDVS is INTEGER *> The leading dimension of the array VS. LDVS >= 1, and if *> JOBVS = 'V', LDVS >= N. *> \endverbatim *> *> \param[out] RCONDE *> \verbatim *> RCONDE is DOUBLE PRECISION *> If SENSE = 'E' or 'B', RCONDE contains the reciprocal *> condition number for the average of the selected eigenvalues. *> Not referenced if SENSE = 'N' or 'V'. *> \endverbatim *> *> \param[out] RCONDV *> \verbatim *> RCONDV is DOUBLE PRECISION *> If SENSE = 'V' or 'B', RCONDV contains the reciprocal *> condition number for the selected right invariant subspace. *> Not referenced if SENSE = 'N' or 'E'. *> \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(1,2*N). *> Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), *> where SDIM is the number of selected eigenvalues computed by *> this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also *> that an error is only returned if LWORK < max(1,2*N), but if *> SENSE = 'E' or 'V' or 'B' this may not be large enough. *> For good performance, LWORK must generally be larger. *> *> If LWORK = -1, then a workspace query is assumed; the routine *> only calculates upper bound on the optimal size of the *> array WORK, 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 (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. *> > 0: if INFO = i, and i is *> <= N: the QR algorithm failed to compute all the *> eigenvalues; elements 1:ILO-1 and i+1:N of W *> contain those eigenvalues which have converged; if *> JOBVS = 'V', VS contains the transformation which *> reduces A to its partially converged Schur form. *> = N+1: the eigenvalues could not be reordered because some *> eigenvalues were too close to separate (the problem *> is very ill-conditioned); *> = N+2: after reordering, roundoff changed values of some *> complex eigenvalues so that leading eigenvalues in *> the Schur form no longer satisfy SELECT=.TRUE. This *> could also be caused by underflow due to scaling. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date June 2016 * *> \ingroup complex16GEeigen * * ===================================================================== SUBROUTINE ZGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, \$ VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, \$ BWORK, INFO ) * * -- LAPACK driver routine (version 3.7.0) -- * -- 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 .. CHARACTER JOBVS, SENSE, SORT INTEGER INFO, LDA, LDVS, LWORK, N, SDIM DOUBLE PRECISION RCONDE, RCONDV * .. * .. Array Arguments .. LOGICAL BWORK( * ) DOUBLE PRECISION RWORK( * ) COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * ) * .. * .. Function Arguments .. LOGICAL SELECT EXTERNAL SELECT * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SCALEA, WANTSB, WANTSE, WANTSN, WANTST, \$ WANTSV, WANTVS INTEGER HSWORK, I, IBAL, ICOND, IERR, IEVAL, IHI, ILO, \$ ITAU, IWRK, LWRK, MAXWRK, MINWRK DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM * .. * .. Local Arrays .. DOUBLE PRECISION DUM( 1 ) * .. * .. External Subroutines .. EXTERNAL DLABAD, DLASCL, XERBLA, ZCOPY, ZGEBAK, ZGEBAL, \$ ZGEHRD, ZHSEQR, ZLACPY, ZLASCL, ZTRSEN, ZUNGHR * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV DOUBLE PRECISION DLAMCH, ZLANGE EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT * .. * .. Executable Statements .. * * Test the input arguments * INFO = 0 WANTVS = LSAME( JOBVS, 'V' ) WANTST = LSAME( SORT, 'S' ) WANTSN = LSAME( SENSE, 'N' ) WANTSE = LSAME( SENSE, 'E' ) WANTSV = LSAME( SENSE, 'V' ) WANTSB = LSAME( SENSE, 'B' ) LQUERY = ( LWORK.EQ.-1 ) * IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN INFO = -1 ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN INFO = -2 ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. \$ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN INFO = -4 ELSE IF( N.LT.0 ) THEN INFO = -5 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -7 ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN INFO = -11 END IF * * Compute workspace * (Note: Comments in the code beginning "Workspace:" describe the * minimal amount of real workspace needed 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 * immediately following subroutine, as returned by ILAENV. * HSWORK refers to the workspace preferred by ZHSEQR, as * calculated below. HSWORK is computed assuming ILO=1 and IHI=N, * the worst case. * If SENSE = 'E', 'V' or 'B', then the amount of workspace needed * depends on SDIM, which is computed by the routine ZTRSEN later * in the code.) * IF( INFO.EQ.0 ) THEN IF( N.EQ.0 ) THEN MINWRK = 1 LWRK = 1 ELSE MAXWRK = N + N*ILAENV( 1, 'ZGEHRD', ' ', N, 1, N, 0 ) MINWRK = 2*N * CALL ZHSEQR( 'S', JOBVS, N, 1, N, A, LDA, W, VS, LDVS, \$ WORK, -1, IEVAL ) HSWORK = WORK( 1 ) * IF( .NOT.WANTVS ) THEN MAXWRK = MAX( MAXWRK, HSWORK ) ELSE MAXWRK = MAX( MAXWRK, N + ( N - 1 )*ILAENV( 1, 'ZUNGHR', \$ ' ', N, 1, N, -1 ) ) MAXWRK = MAX( MAXWRK, HSWORK ) END IF LWRK = MAXWRK IF( .NOT.WANTSN ) \$ LWRK = MAX( LWRK, ( N*N )/2 ) END IF WORK( 1 ) = LWRK * IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN INFO = -15 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGEESX', -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, DUM ) SCALEA = .FALSE. IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN SCALEA = .TRUE. CSCALE = SMLNUM ELSE IF( ANRM.GT.BIGNUM ) THEN SCALEA = .TRUE. CSCALE = BIGNUM END IF IF( SCALEA ) \$ CALL ZLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) * * * Permute the matrix to make it more nearly triangular * (CWorkspace: none) * (RWorkspace: need N) * IBAL = 1 CALL ZGEBAL( 'P', N, A, LDA, ILO, IHI, RWORK( IBAL ), IERR ) * * Reduce to upper Hessenberg form * (CWorkspace: need 2*N, prefer N+N*NB) * (RWorkspace: none) * ITAU = 1 IWRK = N + ITAU CALL ZGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) * IF( WANTVS ) THEN * * Copy Householder vectors to VS * CALL ZLACPY( 'L', N, N, A, LDA, VS, LDVS ) * * Generate unitary matrix in VS * (CWorkspace: need 2*N-1, prefer N+(N-1)*NB) * (RWorkspace: none) * CALL ZUNGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ), \$ LWORK-IWRK+1, IERR ) END IF * SDIM = 0 * * Perform QR iteration, accumulating Schur vectors in VS if desired * (CWorkspace: need 1, prefer HSWORK (see comments) ) * (RWorkspace: none) * IWRK = ITAU CALL ZHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, W, VS, LDVS, \$ WORK( IWRK ), LWORK-IWRK+1, IEVAL ) IF( IEVAL.GT.0 ) \$ INFO = IEVAL * * Sort eigenvalues if desired * IF( WANTST .AND. INFO.EQ.0 ) THEN IF( SCALEA ) \$ CALL ZLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, W, N, IERR ) DO 10 I = 1, N BWORK( I ) = SELECT( W( I ) ) 10 CONTINUE * * Reorder eigenvalues, transform Schur vectors, and compute * reciprocal condition numbers * (CWorkspace: if SENSE is not 'N', need 2*SDIM*(N-SDIM) * otherwise, need none ) * (RWorkspace: none) * CALL ZTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, W, SDIM, \$ RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1, \$ ICOND ) IF( .NOT.WANTSN ) \$ MAXWRK = MAX( MAXWRK, 2*SDIM*( N-SDIM ) ) IF( ICOND.EQ.-14 ) THEN * * Not enough complex workspace * INFO = -15 END IF END IF * IF( WANTVS ) THEN * * Undo balancing * (CWorkspace: none) * (RWorkspace: need N) * CALL ZGEBAK( 'P', 'R', N, ILO, IHI, RWORK( IBAL ), N, VS, LDVS, \$ IERR ) END IF * IF( SCALEA ) THEN * * Undo scaling for the Schur form of A * CALL ZLASCL( 'U', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) CALL ZCOPY( N, A, LDA+1, W, 1 ) IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN DUM( 1 ) = RCONDV CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) RCONDV = DUM( 1 ) END IF END IF * WORK( 1 ) = MAXWRK RETURN * * End of ZGEESX * END