DGEES(3F) DGEES(3F)
DGEES - 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 DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS,
WORK, LWORK, BWORK, INFO )
CHARACTER JOBVS, SORT
INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( *
), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
DGEES 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. The leading
columns of Z then form an orthonormal basis for the invariant subspace
corresponding to the selected eigenvalues.
A 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).
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).
Page 1
DGEES(3F) DGEES(3F)
SELECT (input) LOGICAL FUNCTION of two DOUBLE PRECISION 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 complex
eigenvalues are selected. 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 is set to N+2 (see INFO below).
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has been 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) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION 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
eigenvalues will appear consecutively with the eigenvalue having
the positive imaginary part first.
VS (output) DOUBLE PRECISION 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; if JOBVS =
'V', LDVS >= N.
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N). For good
performance, LWORK must generally be larger.
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DGEES(3F) DGEES(3F)
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 matrix 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.
DGEES(3F) DGEES(3F)
DGEES - 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 DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS, LDVS,
WORK, LWORK, BWORK, INFO )
CHARACTER JOBVS, SORT
INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( *
), WR( * )
LOGICAL SELECT
EXTERNAL SELECT
DGEES 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. The leading
columns of Z then form an orthonormal basis for the invariant subspace
corresponding to the selected eigenvalues.
A 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).
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).
Page 1
DGEES(3F) DGEES(3F)
SELECT (input) LOGICAL FUNCTION of two DOUBLE PRECISION 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 complex
eigenvalues are selected. 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 is set to N+2 (see INFO below).
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A. On exit, A has been 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) DOUBLE PRECISION array, dimension (N)
WI (output) DOUBLE PRECISION 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
eigenvalues will appear consecutively with the eigenvalue having
the positive imaginary part first.
VS (output) DOUBLE PRECISION 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; if JOBVS =
'V', LDVS >= N.
WORK (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N). For good
performance, LWORK must generally be larger.
Page 2
DGEES(3F) DGEES(3F)
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 matrix 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.
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