Contents
sgees - 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 SGEES(JOBZ, SORTEV, SELECT, N, A, LDA, NOUT, WR, WI, Z,
LDZ, WORK, LDWORK, WORK3, INFO)
CHARACTER * 1 JOBZ, SORTEV
INTEGER N, LDA, NOUT, LDZ, LDWORK, INFO
LOGICAL SELECT
LOGICAL WORK3(*)
REAL A(LDA,*), WR(*), WI(*), Z(LDZ,*), WORK(*)
SUBROUTINE SGEES_64(JOBZ, SORTEV, SELECT, N, A, LDA, NOUT, WR, WI, Z,
LDZ, WORK, LDWORK, WORK3, INFO)
CHARACTER * 1 JOBZ, SORTEV
INTEGER*8 N, LDA, NOUT, LDZ, LDWORK, INFO
LOGICAL*8 SELECT
LOGICAL*8 WORK3(*)
REAL A(LDA,*), WR(*), WI(*), Z(LDZ,*), WORK(*)
F95 INTERFACE
SUBROUTINE GEES(JOBZ, SORTEV, SELECT, [N], A, [LDA], NOUT, WR, WI, Z,
[LDZ], [WORK], [LDWORK], [WORK3], [INFO])
CHARACTER(LEN=1) :: JOBZ, SORTEV
INTEGER :: N, LDA, NOUT, LDZ, LDWORK, INFO
LOGICAL :: SELECT
LOGICAL, DIMENSION(:) :: WORK3
REAL, DIMENSION(:) :: WR, WI, WORK
REAL, DIMENSION(:,:) :: A, Z
SUBROUTINE GEES_64(JOBZ, SORTEV, SELECT, [N], A, [LDA], NOUT, WR, WI,
Z, [LDZ], [WORK], [LDWORK], [WORK3], [INFO])
CHARACTER(LEN=1) :: JOBZ, SORTEV
INTEGER(8) :: N, LDA, NOUT, LDZ, LDWORK, INFO
LOGICAL(8) :: SELECT
LOGICAL(8), DIMENSION(:) :: WORK3
REAL, DIMENSION(:) :: WR, WI, WORK
REAL, DIMENSION(:,:) :: A, Z
C INTERFACE
#include <sunperf.h>
void sgees(char jobz, char sortev,
int(*select)(float,float), int n, float *a, int
lda, int *nout, float *wr, float *wi, float *z,
int ldz, int *info);
void sgees_64(char jobz, char sortev,
long(*select)(float,float), long n, float *a, long
lda, long *nout, float *wr, float *wi, float *z,
long ldz, long *info);
sgees 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 factoriza-
tion 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 ortho-
normal 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).
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': Eigen-
values 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 SORTEV = 'S', SELECT is used to select eigen-
values to sort to the top left of the Schur form.
If SORTEV = '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) 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 has been overwritten by its
real 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 (after sorting) for which
SELECT is true. (Complex conjugate pairs for which
SELECT is true for either eigenvalue count as 2.)
WR (output)
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.
WI (output)
See the description for WR.
Z (output)
If JOBZ = 'V', Z contains the orthogonal matrix Z
of Schur vectors. If JOBZ = 'N', Z is not refer-
enced.
LDZ (input)
The leading dimension of the array Z. LDZ >= 1;
if JOBZ = 'V', LDZ >= N.
WORK (workspace)
On exit, if INFO = 0, WORK(1) contains the optimal
LDWORK.
LDWORK (input)
The dimension of the array WORK. LDWORK >=
max(1,3*N). For good performance, LDWORK must
generally be larger.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of
the WORK array, returns this value as the first
entry of the WORK array, and no error message
related to LDWORK is issued by XERBLA.
WORK3 (workspace)
dimension(N) Not referenced if SORTEV = 'N'.
INFO (output)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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 JOBZ = 'V', Z 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 com-
plex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy SELECT=.TRUE.
This could also be caused by underflow due to
scaling.