Contents
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(JOBZ, SORTEV, SELECT, SENSE, N, A, LDA, NOUT, WR,
WI, Z, LDZ, SRCONE, RCONV, WORK, LDWORK, IWORK2, LDWRK2, BWORK3,
INFO)
CHARACTER * 1 JOBZ, SORTEV, SENSE
INTEGER N, LDA, NOUT, LDZ, LDWORK, LDWRK2, INFO
INTEGER IWORK2(*)
LOGICAL SELECT
LOGICAL BWORK3(*)
REAL SRCONE, RCONV
REAL A(LDA,*), WR(*), WI(*), Z(LDZ,*), WORK(*)
SUBROUTINE SGEESX_64(JOBZ, SORTEV, SELECT, SENSE, N, A, LDA, NOUT,
WR, WI, Z, LDZ, SRCONE, RCONV, WORK, LDWORK, IWORK2, LDWRK2,
BWORK3, INFO)
CHARACTER * 1 JOBZ, SORTEV, SENSE
INTEGER*8 N, LDA, NOUT, LDZ, LDWORK, LDWRK2, INFO
INTEGER*8 IWORK2(*)
LOGICAL*8 SELECT
LOGICAL*8 BWORK3(*)
REAL SRCONE, RCONV
REAL A(LDA,*), WR(*), WI(*), Z(LDZ,*), WORK(*)
F95 INTERFACE
SUBROUTINE GEESX(JOBZ, SORTEV, SELECT, SENSE, [N], A, [LDA], NOUT,
WR, WI, Z, [LDZ], SRCONE, RCONV, [WORK], [LDWORK], [IWORK2],
[LDWRK2], [BWORK3], [INFO])
CHARACTER(LEN=1) :: JOBZ, SORTEV, SENSE
INTEGER :: N, LDA, NOUT, LDZ, LDWORK, LDWRK2, INFO
INTEGER, DIMENSION(:) :: IWORK2
LOGICAL :: SELECT
LOGICAL, DIMENSION(:) :: BWORK3
REAL :: SRCONE, RCONV
REAL, DIMENSION(:) :: WR, WI, WORK
REAL, DIMENSION(:,:) :: A, Z
SUBROUTINE GEESX_64(JOBZ, SORTEV, SELECT, SENSE, [N], A, [LDA], NOUT,
WR, WI, Z, [LDZ], SRCONE, RCONV, [WORK], [LDWORK], [IWORK2],
[LDWRK2], [BWORK3], [INFO])
CHARACTER(LEN=1) :: JOBZ, SORTEV, SENSE
INTEGER(8) :: N, LDA, NOUT, LDZ, LDWORK, LDWRK2, INFO
INTEGER(8), DIMENSION(:) :: IWORK2
LOGICAL(8) :: SELECT
LOGICAL(8), DIMENSION(:) :: BWORK3
REAL :: SRCONE, RCONV
REAL, DIMENSION(:) :: WR, WI, WORK
REAL, DIMENSION(:,:) :: A, Z
C INTERFACE
#include <sunperf.h>
void sgeesx(char jobz, char sortev,
int(*select)(float,float), char sense, int n,
float *a, int lda, int *nout, float *wr, float
*wi, float *z, int ldz, float *srcone, float
*rconv, int *info);
void sgeesx_64(char jobz, char sortev,
long(*select)(float,float), char sense, long n,
float *a, long lda, long *nout, float *wr, float
*wi, float *z, long ldz, float *srcone, float
*rconv, long *info);
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 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; 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 respec-
tively).
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).
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 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 eigen-
values (especially if the eigenvalue is ill-
conditioned); in this case INFO may be set to N+3
(see INFO below).
SENSE (input)
Determines which reciprocal condition numbers are
computed. = 'N': None are computed;
= 'E': Computed for average of selected eigen-
values only;
= 'V': Computed for selected right invariant sub-
space only;
= 'B': Computed for both. If SENSE = 'E', 'V' or
'B', SORTEV must equal 'S'.
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 is 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 appear consecutively with the eigen-
value 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,
and if JOBZ = 'V', LDZ >= N.
SRCONE (output)
If SENSE = 'E' or 'B', SRCONE contains the
reciprocal condition number for the average of the
selected eigenvalues. Not referenced if SENSE =
'N' or 'V'.
RCONV (output)
If SENSE = 'V' or 'B', RCONV contains the recipro-
cal condition number for the selected right
invariant subspace. Not referenced if SENSE = 'N'
or 'E'.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LDWORK.
LDWORK (input)
The dimension of the array WORK. LDWORK >=
max(1,3*N). Also, if SENSE = 'E' or 'V' or 'B',
LDWORK >= N+2*NOUT*(N-NOUT), where NOUT is the
number of selected eigenvalues computed by this
routine. Note that N+2*NOUT*(N-NOUT) <= N+N*N/2.
For good performance, LDWORK must generally be
larger.
IWORK2 (workspace/output)
Not referenced if SENSE = 'N' or 'E'. On exit, if
INFO = 0, IWORK2(1) returns the optimal LDWRK2.
LDWRK2 (input)
The dimension of the array IWORK2. LDWRK2 >= 1;
if SENSE = 'V' or 'B', LDWRK2 >= NOUT*(N-NOUT).
BWORK3 (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 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 eigen-
values in the Schur form no longer satisfy
SELECT=.TRUE. This could also be caused by under-
flow due to scaling.