ZGEESX(3F) ZGEESX(3F)
ZGEESX - 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 ZGEESX( 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
DOUBLE PRECISION RCONDE, RCONDV
LOGICAL BWORK( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
LOGICAL SELECT
EXTERNAL SELECT
ZGEESX 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.
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ZGEESX(3F) ZGEESX(3F)
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*16 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*16 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*16 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*16 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) DOUBLE PRECISION
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'.
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ZGEESX(3F) ZGEESX(3F)
RCONDV (output) DOUBLE PRECISION
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*16 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) DOUBLE PRECISION 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 illconditioned);
= 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.
ZGEESX(3F) ZGEESX(3F)
ZGEESX - 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 ZGEESX( 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
DOUBLE PRECISION RCONDE, RCONDV
LOGICAL BWORK( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
LOGICAL SELECT
EXTERNAL SELECT
ZGEESX 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.
Page 1
ZGEESX(3F) ZGEESX(3F)
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*16 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*16 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*16 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*16 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) DOUBLE PRECISION
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'.
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ZGEESX(3F) ZGEESX(3F)
RCONDV (output) DOUBLE PRECISION
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*16 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) DOUBLE PRECISION 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 illconditioned);
= 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.
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