NAME

cggesx - compute for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T),

SYNOPSIS

  SUBROUTINE CGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, 
 *      B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, 
 *      RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
  INTEGER IWORK(*)
  LOGICAL SELCTG
  LOGICAL BWORK(*)
  REAL RCONDE(*), RCONDV(*), RWORK(*)

  SUBROUTINE CGGESX_64( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, 
 *      LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, 
 *      RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
  COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), WORK(*)
  INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  LOGICAL*8 SELCTG
  LOGICAL*8 BWORK(*)
  REAL RCONDE(*), RCONDV(*), RWORK(*)

F95 INTERFACE

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

  SUBROUTINE GGESX_64( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, [N], A, 
 *       [LDA], B, [LDB], SDIM, ALPHA, BETA, VSL, [LDVSL], VSR, [LDVSR], 
 *       RCONDE, RCONDV, [WORK], [LWORK], [RWORK], [IWORK], [LIWORK], 
 *       [BWORK], [INFO])
  CHARACTER(LEN=1) :: JOBVSL, JOBVSR, SORT, SENSE
  COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX, DIMENSION(:,:) :: A, B, VSL, VSR
  INTEGER(8) :: N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  LOGICAL(8) :: SELCTG
  LOGICAL(8), DIMENSION(:) :: BWORK
  REAL, DIMENSION(:) :: RCONDE, RCONDV, RWORK

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

void cggesx_64(char jobvsl, char jobvsr, char sort, logical(*selctg)(complex,complex), char sense, long n, complex *a, long lda, complex *b, long ldb, long *sdim, complex *alpha, complex *beta, complex *vsl, long ldvsl, complex *vsr, long ldvsr, float *rconde, float *rcondv, long *info);

PURPOSE

cggesx computes for a pair of N-by-N complex nonsymmetric matrices (A,B), the generalized eigenvalues, the complex Schur form (S,T), and, optionally, the left and/or right matrices of 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; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

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 or for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if T is upper triangular with non-negative diagonal and S is upper triangular.

ARGUMENTS

JOBVSL (input)

 = 'N':  do not compute the left Schur vectors;

 = 'V':  compute the left Schur vectors.

JOBVSR (input)

 = 'N':  do not compute the right Schur vectors;

 = 'V':  compute the right Schur vectors.

SORT (input)
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 (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. 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+3 see INFO below).

SENSE (input)
Determines which reciprocal condition numbers are computed. = 'N' : None are computed;

 = 'E' : Computed for average of selected eigenvalues only;

 = 'V' : Computed for selected deflating subspaces only;

 = 'B' : Computed for both.
If SENSE  = 'E', 'V', or 'B', SORT must equal 'S'.

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) and BETA(j),j =1,...,N are the diagonals of the complex Schur form (S,T). 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 (output)
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 (output)
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.
RCONDE (output)
If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the reciprocal condition numbers for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.
RCONDV (output)
If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the reciprocal condition number for the selected deflating subspaces. Not referenced if SENSE = 'N' or 'E'.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
The dimension of the array WORK. LWORK > = 2*N. If SENSE = 'E', 'V', or 'B', LWORK > = MAX(2*N, 2*SDIM*(N-SDIM)).
RWORK (workspace)
dimension(8*N) Real workspace.
IWORK (workspace)
Not referenced if SENSE = 'N'. On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
LIWORK (input)
The dimension of the array WORK. LIWORK > = N+2.
BWORK (workspace)
dimension(N) Not referenced if SORT = 'N'.

INFO (output)

 = 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 CHGEQZ

 =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 CTGSEN.

NAME

SYNOPSIS

F95 INTERFACE

C INTERFACE

PURPOSE

ARGUMENTS