SGEESX(3F) SGEESX(3F)
SGEESX - compute for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of Schur
vectors Z
SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
BWORK, INFO )
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
REAL RCONDE, RCONDV
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( *
)
LOGICAL SELECT
EXTERNAL SELECT
SGEESX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
Optionally, it also orders the eigenvalues on the diagonal of the real
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 real matrix is in real Schur form if it is upper quasi-triangular with
1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
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SGEESX(3F) SGEESX(3F)
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 two REAL arguments
SELECT must be declared EXTERNAL in the calling subroutine. If
SORT = 'S', SELECT is used to select eigenvalues to sort to the
top left of the Schur form. If SORT = 'N', SELECT is not
referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
conjugate pair of eigenvalues is selected, then both are. Note
that a selected complex eigenvalue may no longer satisfy
SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may
change the value of complex eigenvalues (especially if the
eigenvalue is ill-conditioned); in this case INFO may be set to
N+3 (see INFO below).
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) REAL array, dimension (LDA, N)
On entry, the N-by-N matrix A. On exit, A is overwritten by its
real 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 (after sorting) for which SELECT is true. (Complex
conjugate pairs for which SELECT is true for either eigenvalue
count as 2.)
WR (output) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and WI contain the
real and imaginary parts, respectively, of the computed
eigenvalues, in the same order that they appear on the diagonal
of the output Schur form T. Complex conjugate pairs of
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SGEESX(3F) SGEESX(3F)
eigenvalues appear consecutively with the eigenvalue having the
positive imaginary part first.
VS (output) REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal 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) REAL 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(1,3*N). Also, if
SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where SDIM
is the number of selected eigenvalues computed by this routine.
Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. For good performance,
LWORK must generally be larger.
IWORK (workspace) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V' or
'B', LIWORK >= SDIM*(N-SDIM).
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 WR and WI 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,
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SGEESX(3F) SGEESX(3F)
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.
SGEESX(3F) SGEESX(3F)
SGEESX - compute for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of Schur
vectors Z
SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK,
BWORK, INFO )
CHARACTER JOBVS, SENSE, SORT
INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
REAL RCONDE, RCONDV
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( *
)
LOGICAL SELECT
EXTERNAL SELECT
SGEESX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
Optionally, it also orders the eigenvalues on the diagonal of the real
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 real matrix is in real Schur form if it is upper quasi-triangular with
1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
Page 1
SGEESX(3F) SGEESX(3F)
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 two REAL arguments
SELECT must be declared EXTERNAL in the calling subroutine. If
SORT = 'S', SELECT is used to select eigenvalues to sort to the
top left of the Schur form. If SORT = 'N', SELECT is not
referenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
conjugate pair of eigenvalues is selected, then both are. Note
that a selected complex eigenvalue may no longer satisfy
SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may
change the value of complex eigenvalues (especially if the
eigenvalue is ill-conditioned); in this case INFO may be set to
N+3 (see INFO below).
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) REAL array, dimension (LDA, N)
On entry, the N-by-N matrix A. On exit, A is overwritten by its
real 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 (after sorting) for which SELECT is true. (Complex
conjugate pairs for which SELECT is true for either eigenvalue
count as 2.)
WR (output) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and WI contain the
real and imaginary parts, respectively, of the computed
eigenvalues, in the same order that they appear on the diagonal
of the output Schur form T. Complex conjugate pairs of
Page 2
SGEESX(3F) SGEESX(3F)
eigenvalues appear consecutively with the eigenvalue having the
positive imaginary part first.
VS (output) REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal 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) REAL 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(1,3*N). Also, if
SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where SDIM
is the number of selected eigenvalues computed by this routine.
Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. For good performance,
LWORK must generally be larger.
IWORK (workspace) INTEGER array, dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'.
LIWORK (input) INTEGER
The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V' or
'B', LIWORK >= SDIM*(N-SDIM).
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 WR and WI 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,
Page 3
SGEESX(3F) SGEESX(3F)
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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