NAME

SYNOPSIS

  SUBROUTINE CGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, 
 *      SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, 
 *      BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
  LOGICAL SELCTG
  LOGICAL BWORK(*)
  REAL RWORK(*)
 
  SUBROUTINE CGGES_64( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, 
 *      LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, 
 *      RWORK, BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
  LOGICAL*8 SELCTG
  LOGICAL*8 BWORK(*)
  REAL RWORK(*)
 

F95 INTERFACE

  SUBROUTINE GGES( JOBVSL, JOBVSR, SORT, SELCTG, [N], A, [LDA], B, 
 *       [LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], 
 *       [LWORK], [RWORK], [BWORK], [INFO])
  CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
  INTEGER :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
  LOGICAL :: SELCTG
  LOGICAL, DIMENSION(:) :: BWORK
  REAL, DIMENSION(:) :: RWORK
 
  SUBROUTINE GGES_64( JOBVSL, JOBVSR, SORT, SELCTG, [N], A, [LDA], B, 
 *       [LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], [WORK], 
 *       [LWORK], [RWORK], [BWORK], [INFO])
  CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
  INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, INFO
  LOGICAL(8) :: SELCTG
  LOGICAL(8), DIMENSION(:) :: BWORK
  REAL, DIMENSION(:) :: RWORK
 

C INTERFACE

void cgges(char jobvsl, char jobvsr, char sort, logical(*selctg)(complex,complex), int n, complex *a, int lda, complex *b, int ldb, int *sdim, complex *alpha, complex *beta, complex *vsl, int ldvsl, complex *vsr, int ldvsr, int *info);

void cgges_64(char jobvsl, char jobvsr, char sort, logical(*selctg)(complex,complex), long n, complex *a, long lda, complex *b, long ldb, long *sdim, complex *alpha, complex *beta, complex *vsl, long ldvsl, complex *vsr, long ldvsr, long *info);

PURPOSE

        (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 CGGEV 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)

* JOBVSR (input)

* SORT (input)

Specifies whether or not to order the eigenvalues on the diagonal of the generalized Schur form.

* SELCTG (input)

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)

The order of the matrices A, B, VSL, and VSR. N >= 0.

* A (input/output)

On entry, the first of the pair of matrices. On exit, A has been overwritten by its generalized Schur form S.

* LDA (input)

The leading dimension of A. LDA >= max(1,N).

* B (input/output)

On entry, the second of the pair of matrices. On exit, B has been overwritten by its generalized Schur form T.

* LDB (input)

The leading dimension of B. LDB >= max(1,N).

* SDIM (output)

If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues (after sorting) for which SELCTG is true.

* ALPHA (output)

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 CGGES. 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).

* BETA (output)

See description of ALPHA.

* VSL (input)

If JOBVSL = 'V', VSL will contain the left Schur vectors. Not referenced if JOBVSL = 'N'.

* LDVSL (input)

The leading dimension of the matrix VSL. LDVSL >= 1, and if JOBVSL = 'V', LDVSL >= N.

* VSR (input)

If JOBVSR = 'V', VSR will contain the right Schur vectors. Not referenced if JOBVSR = 'N'.

* LDVSR (input)

The leading dimension of the matrix VSR. LDVSR >= 1, and if JOBVSR = 'V', LDVSR >= N.

* WORK (workspace)

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

* LWORK (input)

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)

dimension(8*N)

* BWORK (workspace)

Not referenced if SORT = 'N'.

* INFO (output)