zgges.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 478 行 · 第 1/2 页

      SUBROUTINE ZGGES( 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 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. 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
*     ..
*
*  Purpose
*  =======
*
*  ZGGES 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.
*
*  Arguments
*  =========
*
*  JOBVSL  (input) CHARACTER*1
*          = 'N':  do not compute the left Schur vectors;
*          = 'V':  compute the left Schur vectors.
*
*  JOBVSR  (input) CHARACTER*1
*          = 'N':  do not compute the right Schur vectors;
*          = 'V':  compute the right Schur vectors.
*
*  SORT    (input) 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).
*
*  SELCTG  (external procedure) 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).
*
*  N       (input) INTEGER
*          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*
*  A       (input/output) 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.
*
*  LDA     (input) INTEGER
*          The leading dimension of A.  LDA >= max(1,N).
*
*  B       (input/output) 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.
*
*  LDB     (input) INTEGER
*          The leading dimension of B.  LDB >= max(1,N).
*
*  SDIM    (output) INTEGER
*          If SORT = 'N', SDIM = 0.
*          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*          for which SELCTG is true.
*
*  ALPHA   (output) COMPLEX*16 array, dimension (N)
*  BETA    (output) 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 ZGGES. 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).
*
*  VSL     (output) COMPLEX*16 array, dimension (LDVSL,N)
*          If JOBVSL = 'V', VSL will contain the left Schur vectors.
*          Not referenced if JOBVSL = 'N'.
*
*  LDVSL   (input) INTEGER
*          The leading dimension of the matrix VSL. LDVSL >= 1, and
*          if JOBVSL = 'V', LDVSL >= N.
*
*  VSR     (output) COMPLEX*16 array, dimension (LDVSR,N)
*          If JOBVSR = 'V', VSR will contain the right Schur vectors.
*          Not referenced if JOBVSR = 'N'.
*
*  LDVSR   (input) INTEGER
*          The leading dimension of the matrix VSR. LDVSR >= 1, and
*          if JOBVSR = 'V', LDVSR >= N.
*
*  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.  LWORK >= max(1,2*N).
*          For good performance, LWORK must generally be larger.
*
*          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.
*
*  RWORK   (workspace) DOUBLE PRECISION array, dimension (8*N)
*
*  BWORK   (workspace) LOGICAL array, dimension (N)
*          Not referenced if SORT = 'N'.
*
*  INFO    (output) 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 falied in ZTGSEN.
*
*  =====================================================================
*
*     .. 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, LWKMIN,
     $                   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, ZGGHRD,
     $                   ZHGEQZ, ZLACPY, ZLASCL, ZLASET, ZTGSEN, ZUNGQR,
     $                   ZUNMQR
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, ZLANGE
      EXTERNAL           LSAME, ILAENV, 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 )