NAME

SYNOPSIS

  SUBROUTINE SGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, 
 *      B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, 
 *      RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
  INTEGER N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
  INTEGER IWORK(*)
  LOGICAL SELCTG
  LOGICAL BWORK(*)
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
 
  SUBROUTINE SGGESX_64( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, 
 *      LDA, B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, 
 *      RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
  CHARACTER * 1 JOBVSL, JOBVSR, SORT, SENSE
  INTEGER*8 N, LDA, LDB, SDIM, LDVSL, LDVSR, LWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  LOGICAL*8 SELCTG
  LOGICAL*8 BWORK(*)
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), VSL(LDVSL,*), VSR(LDVSR,*), RCONDE(*), RCONDV(*), WORK(*)
 

F95 INTERFACE

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

C INTERFACE

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

void sggesx_64(char jobvsl, char jobvsr, char sort, logical(*selctg)(float,float,float), char sense, long n, float *a, long lda, float *b, long ldb, long *sdim, float *alphar, float *alphai, float *beta, float *vsl, long ldvsl, float *vsr, long ldvsr, float *rconde, float *rcondv, long *info);

PURPOSE

A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-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 real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be ``standardized'' by making the corresponding elements of T have the form:

        [  a  0  ]
        [  0  b  ]

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

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 (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either one of a complex conjugate pair of eigenvalues is selected, then both complex eigenvalues are selected. Note that a selected complex eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(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.

* SENSE (input)

Determines which reciprocal condition numbers are computed.

* 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. (Complex conjugate pairs for which SELCTG is true for either eigenvalue count as 2.)

* ALPHAR (output)

On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigenvalues. ALPHAR(j) + ALPHAI(j)*i and BETA(j),j=1,...,N are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real Schur form of (A,B) were further reduced to triangular form using 2-by-2 complex unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio. However, ALPHAR and ALPHAI will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).

* ALPHAI (output)

See the description for ALPHAR.

* BETA (output)

See the description for ALPHAR.

* 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.

* 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 numbers 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 >= 8*(N+1)+16. If SENSE = 'E', 'V', or 'B', LWORK >= MAX( 8*(N+1)+16, 2*SDIM*(N-SDIM) ).

* IWORK (workspace)

Not referenced if SENSE = 'N'.

* LIWORK (input)

The dimension of the array WORK. LIWORK >= N+6.

* BWORK (workspace)

Not referenced if SORT = 'N'.

* INFO (output)