sgeesx.f man page

sgeesx.f —

Synopsis

Functions/Subroutines

subroutine sgeesx (JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK, LIWORK, BWORK, INFO)
SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Function/Subroutine Documentation

subroutine sgeesx (characterJOBVS, characterSORT, logical, externalSELECT, characterSENSE, integerN, real, dimension( lda, * )A, integerLDA, integerSDIM, real, dimension( * )WR, real, dimension( * )WI, real, dimension( ldvs, * )VS, integerLDVS, realRCONDE, realRCONDV, real, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerLIWORK, logical, dimension( * )BWORK, integerINFO)

SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Purpose:

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).

Parameters:

JOBVS

JOBVS is CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

SORT

SORT is 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

SELECT is procedure) 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

SENSE is 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

N is INTEGER
The order of the matrix A. N >= 0.

A

A is 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

LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).

SDIM

SDIM is 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

WR is REAL array, dimension (N)

WI

WI is 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 eigenvalues appear consecutively with the
eigenvalue having the positive imaginary part first.

VS

VS is 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

LDVS is INTEGER
The leading dimension of the array VS.  LDVS >= 1, and if
JOBVS = 'V', LDVS >= N.

RCONDE

RCONDE is 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

RCONDV is 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

WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

LWORK is 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. Note also that an error is only
returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
'B' this may not be large enough.
For good performance, LWORK must generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine
only calculates upper bounds on the optimal sizes of the
arrays WORK and IWORK, returns these values as the first
entries of the WORK and IWORK arrays, and no error messages
related to LWORK or LIWORK are issued by XERBLA.

IWORK

IWORK is INTEGER array, dimension (MAX(1,LIWORK))
On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
may not be large enough.

If LIWORK = -1, then a workspace query is assumed; the
routine only calculates upper bounds on the optimal sizes of
the arrays WORK and IWORK, returns these values as the first
entries of the WORK and IWORK arrays, and no error messages
related to LWORK or LIWORK are issued by XERBLA.

BWORK

BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.

INFO

INFO is 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, 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.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Definition at line 280 of file sgeesx.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

sgeesx(3) is an alias of sgeesx.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK