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

SUBROUTINE CGEESX(

JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK, INFO )

CHARACTER JOBVS, SENSE, SORT

INTEGER INFO, LDA, LDVS, LWORK, N, SDIM

REAL RCONDE, RCONDV

LOGICAL BWORK( * )

REAL RWORK( * )

COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )

LOGICAL SELECT

EXTERNAL SELECT

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.

JOBVS (input) CHARACTER*1

= 'N': Schur vectors are not computed;

= 'V': Schur vectors are computed.

SORT (input) CHARACTER*1

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) LOGICAL FUNCTION of one COMPLEX argument

SELECT must be declared EXTERNAL in the calling subroutine. If SORT = 'S', SELECT is used to select eigenvalues to order to the top left of the Schur form. If SORT = 'N', SELECT is not referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is true.

SENSE (input) CHARACTER*1

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', SORT must equal 'S'.

N (input) INTEGER

The order of the matrix A. N >= 0.

A (input/output) COMPLEX array, dimension (LDA, N)

On entry, the N-by-N matrix A. On exit, A is overwritten by its Schur form T.

LDA (input) INTEGER

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

SDIM (output) INTEGER

If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of eigenvalues for which SELECT is true.

W (output) COMPLEX array, dimension (N)

W contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.

VS (output) COMPLEX array, dimension (LDVS,N)

If JOBVS = 'V', VS contains the unitary matrix Z of Schur vectors. If JOBVS = 'N', VS is not referenced.

LDVS (input) INTEGER

The leading dimension of the array VS. LDVS >= 1, and if JOBVS = 'V', LDVS >= N.

RCONDE (output) REAL

If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition number for the average of the selected eigenvalues. Not referenced if SENSE = 'N' or 'V'.

RCONDV (output) REAL

If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition number for the selected right invariant subspace. Not referenced if SENSE = 'N' or 'E'.

WORK (workspace/output) COMPLEX array, dimension (LWORK)

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

LWORK (input) INTEGER

The dimension of the array WORK. LWORK >= max(1,2*N). Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where SDIM is the number of selected eigenvalues computed by this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. For good performance, LWORK must generally be larger.

RWORK (workspace) REAL array, dimension (N)

BWORK (workspace) LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

INFO (output) INTEGER

= 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 JOBVS = 'V', VS 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.