NAME

cgeesx - compute for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z

SYNOPSIS

  SUBROUTINE CGEESX( JOBZ, SORTEV, SELECT, SENSE, N, A, LDA, NOUT, W, 
 *      Z, LDZ, RCONE, RCONV, WORK, LDWORK, WORK2, BWORK3, INFO)
  CHARACTER * 1 JOBZ, SORTEV, SENSE
  COMPLEX A(LDA,*), W(*), Z(LDZ,*), WORK(*)
  INTEGER N, LDA, NOUT, LDZ, LDWORK, INFO
  LOGICAL SELECT
  LOGICAL BWORK3(*)
  REAL RCONE, RCONV
  REAL WORK2(*)

  SUBROUTINE CGEESX_64( JOBZ, SORTEV, SELECT, SENSE, N, A, LDA, NOUT, 
 *      W, Z, LDZ, RCONE, RCONV, WORK, LDWORK, WORK2, BWORK3, INFO)
  CHARACTER * 1 JOBZ, SORTEV, SENSE
  COMPLEX A(LDA,*), W(*), Z(LDZ,*), WORK(*)
  INTEGER*8 N, LDA, NOUT, LDZ, LDWORK, INFO
  LOGICAL*8 SELECT
  LOGICAL*8 BWORK3(*)
  REAL RCONE, RCONV
  REAL WORK2(*)

F95 INTERFACE

  SUBROUTINE GEESX( JOBZ, SORTEV, SELECT, SENSE, [N], A, [LDA], NOUT, 
 *       W, Z, [LDZ], RCONE, RCONV, [WORK], [LDWORK], [WORK2], [BWORK3], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOBZ, SORTEV, SENSE
  COMPLEX, DIMENSION(:) :: W, WORK
  COMPLEX, DIMENSION(:,:) :: A, Z
  INTEGER :: N, LDA, NOUT, LDZ, LDWORK, INFO
  LOGICAL :: SELECT
  LOGICAL, DIMENSION(:) :: BWORK3
  REAL :: RCONE, RCONV
  REAL, DIMENSION(:) :: WORK2

  SUBROUTINE GEESX_64( JOBZ, SORTEV, SELECT, SENSE, [N], A, [LDA], 
 *       NOUT, W, Z, [LDZ], RCONE, RCONV, [WORK], [LDWORK], [WORK2], 
 *       [BWORK3], [INFO])
  CHARACTER(LEN=1) :: JOBZ, SORTEV, SENSE
  COMPLEX, DIMENSION(:) :: W, WORK
  COMPLEX, DIMENSION(:,:) :: A, Z
  INTEGER(8) :: N, LDA, NOUT, LDZ, LDWORK, INFO
  LOGICAL(8) :: SELECT
  LOGICAL(8), DIMENSION(:) :: BWORK3
  REAL :: RCONE, RCONV
  REAL, DIMENSION(:) :: WORK2

C INTERFACE

#include <sunperf.h>

void cgeesx(char jobz, char sortev, logical(*select)(complex), char sense, int n, complex *a, int lda, int *nout, complex *w, complex *z, int ldz, float *rcone, float *rconv, int *info);

void cgeesx_64(char jobz, char sortev, logical(*select)(complex), char sense, long n, complex *a, long lda, long *nout, complex *w, complex *z, long ldz, float *rcone, float *rconv, long *info);

PURPOSE

cgeesx computes for an N-by-N complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (RCONDV). The leading columns of Z form an orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where these quantities are called s and sep respectively).

A complex matrix is in Schur form if it is upper triangular.

ARGUMENTS

JOBZ (input)

 = 'N': Schur vectors are not computed;

 = 'V': Schur vectors are computed.

SORTEV (input)
Specifies whether or not to order the eigenvalues on the diagonal of the Schur form. = 'N': Eigenvalues are not ordered;
```
 = 'S': Eigenvalues are ordered (see SELECT).
```
SELECT (input)
SELECT must be declared EXTERNAL in the calling subroutine. If SORTEV = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form. If SORTEV = 'N', SELECT is not referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is true.

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 right invariant subspace only;

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

N (input)
The order of the matrix A. N > = 0.
A (input/output)
On entry, the N-by-N matrix A. On exit, A is overwritten by its Schur form T.
LDA (input)
The leading dimension of the array A. LDA > = max(1,N).
NOUT (output)
If SORTEV = 'N', NOUT = 0. If SORTEV = 'S', NOUT = number of eigenvalues for which SELECT is true.
W (output)
W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
Z (output)
If JOBZ = 'V', Z contains the unitary matrix Z of Schur vectors. If JOBZ = 'N', Z is not referenced.
LDZ (input)
The leading dimension of the array Z. LDZ > = 1, and if JOBZ = 'V', LDZ > = N.
RCONE (output)
If SENSE = 'E' or 'B', RCONE contains the reciprocal condition number for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.
RCONV (output)
If SENSE = 'V' or 'B', RCONV contains the reciprocal condition number for the selected right invariant subspace. Not referenced if SENSE = 'N' or 'E'.
WORK (workspace)
dimension(LDWORK) On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.
LDWORK (input)
The dimension of the array WORK. LDWORK > = max(1,2*N). Also, if SENSE = 'E' or 'V' or 'B', LDWORK > = 2*NOUT*(N-NOUT), where NOUT is the number of selected eigenvalues computed by this routine. Note that 2*NOUT*(N-NOUT) < = N*N/2. For good performance, LDWORK must generally be larger.
WORK2 (workspace)
dimension(N)
BWORK3 (workspace)
dimension(N) Not referenced if SORTEV = 'N'.
INFO (output)
```
 = 0: successful exit
```
```
 < 0: if INFO  = -i, the i-th argument had an illegal value.
```
```
 > 0: if INFO  = i, and i is
```
```
 < = N: the QR algorithm failed to compute all the
```
eigenvalues; elements 1:ILO-1 and i+1:N of W contain those eigenvalues which have converged; if JOBZ = 'V', Z contains the transformation which reduces A to its partially converged Schur form. = N+1: the eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned); = N+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy SELECT =.TRUE. This could also be caused by underflow due to scaling.