NAME

SYNOPSIS

  SUBROUTINE STGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, 
 *      ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, 
 *      LWORK, IWORK, LIWORK, INFO)
  INTEGER IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER IWORK(*)
  LOGICAL WANTQ, WANTZ
  LOGICAL SELECT(*)
  REAL PL, PR
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)
 
  SUBROUTINE STGSEN_64( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB, 
 *      ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF, WORK, 
 *      LWORK, IWORK, LIWORK, INFO)
  INTEGER*8 IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER*8 IWORK(*)
  LOGICAL*8 WANTQ, WANTZ
  LOGICAL*8 SELECT(*)
  REAL PL, PR
  REAL A(LDA,*), B(LDB,*), ALPHAR(*), ALPHAI(*), BETA(*), Q(LDQ,*), Z(LDZ,*), DIF(*), WORK(*)
 

F95 INTERFACE

  SUBROUTINE TGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, [LDA], B, [LDB], 
 *       ALPHAR, ALPHAI, BETA, Q, [LDQ], Z, [LDZ], M, PL, PR, DIF, [WORK], 
 *       [LWORK], [IWORK], [LIWORK], [INFO])
  INTEGER :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK
  LOGICAL :: WANTQ, WANTZ
  LOGICAL, DIMENSION(:) :: SELECT
  REAL :: PL, PR
  REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
  REAL, DIMENSION(:,:) :: A, B, Q, Z
 
  SUBROUTINE TGSEN_64( IJOB, WANTQ, WANTZ, SELECT, N, A, [LDA], B, 
 *       [LDB], ALPHAR, ALPHAI, BETA, Q, [LDQ], Z, [LDZ], M, PL, PR, DIF, 
 *       [WORK], [LWORK], [IWORK], [LIWORK], [INFO])
  INTEGER(8) :: IJOB, N, LDA, LDB, LDQ, LDZ, M, LWORK, LIWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK
  LOGICAL(8) :: WANTQ, WANTZ
  LOGICAL(8), DIMENSION(:) :: SELECT
  REAL :: PL, PR
  REAL, DIMENSION(:) :: ALPHAR, ALPHAI, BETA, DIF, WORK
  REAL, DIMENSION(:,:) :: A, B, Q, Z
 

C INTERFACE

void stgsen(int ijob, logical wantq, logical wantz, logical *select, int n, float *a, int lda, float *b, int ldb, float *alphar, float *alphai, float *beta, float *q, int ldq, float *z, int ldz, int *m, float *pl, float *pr, float *dif, int *info);

void stgsen_64(long ijob, logical wantq, logical wantz, logical *select, long n, float *a, long lda, float *b, long ldb, float *alphar, float *alphai, float *beta, float *q, long ldq, float *z, long ldz, long *m, float *pl, float *pr, float *dif, long *info);

PURPOSE

STGSEN also computes the generalized eigenvalues

            w(j) = (ALPHAR(j) + i*ALPHAI(j))/BETA(j)

of the reordered matrix pair (A, B).

Optionally, STGSEN computes the estimates of reciprocal condition numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11), (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s) between the matrix pairs (A11, B11) and (A22,B22) that correspond to the selected cluster and the eigenvalues outside the cluster, resp., and norms of ``projections'' onto left and right eigenspaces w.r.t. the selected cluster in the (1,1)-block.

ARGUMENTS

* IJOB (input)

Specifies whether condition numbers are required for the cluster of eigenvalues (PL and PR) or the deflating subspaces (Difu and Difl):

* WANTQ (input)

.TRUE. : update the left transformation matrix Q;

.FALSE.: do not update Q.

* WANTZ (input)

.TRUE. : update the right transformation matrix Z;

.FALSE.: do not update Z.

* SELECT (input)

SELECT specifies the eigenvalues in the selected cluster. To select a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select a complex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2 diagonal block, either SELECT(j) or SELECT(j+1) or both must be set to .TRUE.; a complex conjugate pair of eigenvalues must be either both included in the cluster or both excluded.

* N (input)

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

* A (input/output)

On entry, the upper quasi-triangular matrix A, with (A, B) in generalized real Schur canonical form. On exit, A is overwritten by the reordered matrix A.

* LDA (input)

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

* B (input/output)

On entry, the upper triangular matrix B, with (A, B) in generalized real Schur canonical form. On exit, B is overwritten by the reordered matrix B.

* LDB (input)

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

* 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 generalized Schur form of (A,B) were further reduced to triangular form using 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.

* ALPHAI (output)

See the description of ALPHAR.

* BETA (output)

See the description of ALPHAR.

* Q (input/output)

On entry, if WANTQ = .TRUE., Q is an N-by-N matrix. On exit, Q has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Q form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTQ = .FALSE., Q is not referenced.

* LDQ (input)

The leading dimension of the array Q. LDQ >= 1; and if WANTQ = .TRUE., LDQ >= N.

* Z (input/output)

On entry, if WANTZ = .TRUE., Z is an N-by-N matrix. On exit, Z has been postmultiplied by the left orthogonal transformation matrix which reorder (A, B); The leading M columns of Z form orthonormal bases for the specified pair of left eigenspaces (deflating subspaces). If WANTZ = .FALSE., Z is not referenced.

* LDZ (input)

The leading dimension of the array Z. LDZ >= 1; If WANTZ = .TRUE., LDZ >= N.

* M (output)

The dimension of the specified pair of left and right eigen- spaces (deflating subspaces). 0 <= M <= N.

* PL (output)

If IJOB = 1, 4 or 5, PL, PR are lower bounds on the reciprocal of the norm of ``projections'' onto left and right eigenspaces with respect to the selected cluster. 0 < PL, PR <= 1. If M = 0 or M = N, PL = PR = 1. If IJOB = 0, 2 or 3, PL and PR are not referenced.

* PR (output)

See the description of PL.

* DIF (output)

If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.

If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on

Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based estimates of Difu and Difl. If M = 0 or N, DIF(1:2) = F-norm([A, B]). If IJOB = 0 or 1, DIF is not referenced.

* WORK (workspace)

If IJOB = 0, WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1) returns the optimal LWORK.

* LWORK (input)

The dimension of the array WORK. LWORK >= 4*N+16. If IJOB = 1, 2 or 4, LWORK >= MAX(4*N+16, 2*M*(N-M)). If IJOB = 3 or 5, LWORK >= MAX(4*N+16, 4*M*(N-M)).

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.

* IWORK (workspace)

If IJOB = 0, IWORK is not referenced. Otherwise, on exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

* LIWORK (input)

The dimension of the array IWORK. LIWORK >= 1. If IJOB = 1, 2 or 4, LIWORK >= N+6. If IJOB = 3 or 5, LIWORK >= MAX(2*M*(N-M), N+6).

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA.

* INFO (output)