Man page - gees(3)

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Manual

gees

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgees (character jobvs, character sort, external select, integern, complex, dimension( lda, * ) a, integer lda, integer sdim, complex,dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, logical, dimension( * ) bwork, integer info)
subroutine dgees (character jobvs, character sort, external select, integern, double precision, dimension( lda, * ) a, integer lda, integer sdim,double precision, dimension( * ) wr, double precision, dimension( * )wi, double precision, dimension( ldvs, * ) vs, integer ldvs, doubleprecision, dimension( * ) work, integer lwork, logical, dimension( * )bwork, integer info)
subroutine sgees (character jobvs, character sort, external select, integern, real, dimension( lda, * ) a, integer lda, integer sdim, real,dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvs, * )vs, integer ldvs, real, dimension( * ) work, integer lwork, logical,dimension( * ) bwork, integer info)
subroutine zgees (character jobvs, character sort, external select, integern, complex*16, dimension( lda, * ) a, integer lda, integer sdim,complex*16, dimension( * ) w, complex*16, dimension( ldvs, * ) vs,integer ldvs, complex*16, dimension( * ) work, integer lwork, doubleprecision, dimension( * ) rwork, logical, dimension( * ) bwork, integerinfo)
Author

NAME

gees - gees: Schur form

SYNOPSIS

Functions

subroutine cgees (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)
CGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine dgees (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)
DGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine sgees (jobvs, sort, select, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info)
SGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices
subroutine zgees (jobvs, sort, select, n, a, lda, sdim, w, vs, ldvs, work, lwork, rwork, bwork, info)
ZGEES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices

Detailed Description

Function Documentation

subroutine cgees (character jobvs, character sort, external select, integern, complex, dimension( lda, * ) a, integer lda, integer sdim, complex,dimension( * ) w, complex, dimension( ldvs, * ) vs, integer ldvs,complex, dimension( * ) work, integer lwork, real, dimension( * )rwork, logical, dimension( * ) bwork, integer info)

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

Purpose:

CGEES computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal of the
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 complex matrix is in Schur form if it is upper triangular.

Parameters

JOBVS

JOBVS is CHARACTER*1
= ā€™Nā€™: Schur vectors are not computed;
= ā€™Vā€™: Schur vectors are computed.

SORT

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= ā€™Nā€™: Eigenvalues are not ordered:
= ā€™Sā€™: Eigenvalues are ordered (see SELECT).

SELECT

SELECT is a LOGICAL FUNCTION of one COMPLEX argument
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = ā€™Sā€™, SELECT is used to select eigenvalues to order
to the top left of the Schur form.
IF SORT = ā€™Nā€™, SELECT is not referenced.
The eigenvalue W(j) is selected if SELECT(W(j)) is true.

N

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

A

A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its Schur form T.

LDA

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

SDIM

SDIM is INTEGER
If SORT = ā€™Nā€™, SDIM = 0.
If SORT = ā€™Sā€™, SDIM = number of eigenvalues for which
SELECT is true.

W

W is COMPLEX array, dimension (N)
W contains the computed eigenvalues, in the same order that
they appear on the diagonal of the output Schur form T.

VS

VS is COMPLEX array, dimension (LDVS,N)
If JOBVS = ā€™Vā€™, VS contains the unitary matrix Z of Schur
vectors.
If JOBVS = ā€™Nā€™, VS is not referenced.

LDVS

LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = ā€™Vā€™, LDVS >= N.

WORK

WORK is COMPLEX 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(1,2*N).
For good performance, LWORK must generally be larger.

If LWORK = -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 LWORK is issued by XERBLA.

RWORK

RWORK is REAL array, dimension (N)

BWORK

BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = ā€™Nā€™.

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 W
contain those eigenvalues which have converged;
if JOBVS = ā€™Vā€™, 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgees (character jobvs, character sort, external select, integern, double precision, dimension( lda, * ) a, integer lda, integer sdim,double precision, dimension( * ) wr, double precision, dimension( * )wi, double precision, dimension( ldvs, * ) vs, integer ldvs, doubleprecision, dimension( * ) work, integer lwork, logical, dimension( * )bwork, integer info)

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

Purpose:

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).

Parameters

JOBVS

JOBVS is CHARACTER*1
= ā€™Nā€™: Schur vectors are not computed;
= ā€™Vā€™: Schur vectors are computed.

SORT

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= ā€™Nā€™: Eigenvalues are not ordered;
= ā€™Sā€™: Eigenvalues are ordered (see SELECT).

SELECT

SELECT is a LOGICAL FUNCTION of two DOUBLE PRECISION arguments
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = ā€™Sā€™, SELECT is used to select eigenvalues to sort
to the top left of the Schur form.
If SORT = ā€™Nā€™, 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

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

A

A is 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

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

SDIM

SDIM is INTEGER
If SORT = ā€™Nā€™, SDIM = 0.
If SORT = ā€™Sā€™, 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 DOUBLE PRECISION array, dimension (N)

WI

WI is 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

VS is DOUBLE PRECISION array, dimension (LDVS,N)
If JOBVS = ā€™Vā€™, VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = ā€™Nā€™, VS is not referenced.

LDVS

LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = ā€™Vā€™, LDVS >= N.

WORK

WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N).
For good performance, LWORK must generally be larger.

If LWORK = -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 LWORK is issued by XERBLA.

BWORK

BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = ā€™Nā€™.

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 = ā€™Vā€™, 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sgees (character jobvs, character sort, external select, integern, real, dimension( lda, * ) a, integer lda, integer sdim, real,dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldvs, * )vs, integer ldvs, real, dimension( * ) work, integer lwork, logical,dimension( * ) bwork, integer info)

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

Purpose:

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 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).

Parameters

JOBVS

JOBVS is CHARACTER*1
= ā€™Nā€™: Schur vectors are not computed;
= ā€™Vā€™: Schur vectors are computed.

SORT

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= ā€™Nā€™: Eigenvalues are not ordered;
= ā€™Sā€™: Eigenvalues are ordered (see SELECT).

SELECT

SELECT is a LOGICAL FUNCTION of two REAL arguments
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = ā€™Sā€™, SELECT is used to select eigenvalues to sort
to the top left of the Schur form.
If SORT = ā€™Nā€™, 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

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 has been 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 = ā€™Nā€™, SDIM = 0.
If SORT = ā€™Sā€™, 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 will appear
consecutively with the eigenvalue having the positive
imaginary part first.

VS

VS is REAL array, dimension (LDVS,N)
If JOBVS = ā€™Vā€™, VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = ā€™Nā€™, VS is not referenced.

LDVS

LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = ā€™Vā€™, LDVS >= N.

WORK

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

LWORK

LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N).
For good performance, LWORK must generally be larger.

If LWORK = -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 LWORK is issued by XERBLA.

BWORK

BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = ā€™Nā€™.

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 = ā€™Vā€™, 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zgees (character jobvs, character sort, external select, integern, complex*16, dimension( lda, * ) a, integer lda, integer sdim,complex*16, dimension( * ) w, complex*16, dimension( ldvs, * ) vs,integer ldvs, complex*16, dimension( * ) work, integer lwork, doubleprecision, dimension( * ) rwork, logical, dimension( * ) bwork, integerinfo)

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

Purpose:

ZGEES computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues, the Schur form T, and, optionally, the matrix of Schur
vectors Z. This gives the Schur factorization A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal of the
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 complex matrix is in Schur form if it is upper triangular.

Parameters

JOBVS

JOBVS is CHARACTER*1
= ā€™Nā€™: Schur vectors are not computed;
= ā€™Vā€™: Schur vectors are computed.

SORT

SORT is CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= ā€™Nā€™: Eigenvalues are not ordered:
= ā€™Sā€™: Eigenvalues are ordered (see SELECT).

SELECT

SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument
SELECT must be declared EXTERNAL in the calling subroutine.
If SORT = ā€™Sā€™, SELECT is used to select eigenvalues to order
to the top left of the Schur form.
IF SORT = ā€™Nā€™, SELECT is not referenced.
The eigenvalue W(j) is selected if SELECT(W(j)) is true.

N

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

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its Schur form T.

LDA

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

SDIM

SDIM is INTEGER
If SORT = ā€™Nā€™, SDIM = 0.
If SORT = ā€™Sā€™, SDIM = number of eigenvalues for which
SELECT is true.

W

W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues, in the same order that
they appear on the diagonal of the output Schur form T.

VS

VS is COMPLEX*16 array, dimension (LDVS,N)
If JOBVS = ā€™Vā€™, VS contains the unitary matrix Z of Schur
vectors.
If JOBVS = ā€™Nā€™, VS is not referenced.

LDVS

LDVS is INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = ā€™Vā€™, LDVS >= N.

WORK

WORK is COMPLEX*16 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(1,2*N).
For good performance, LWORK must generally be larger.

If LWORK = -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 LWORK is issued by XERBLA.

RWORK

RWORK is DOUBLE PRECISION array, dimension (N)

BWORK

BWORK is LOGICAL array, dimension (N)
Not referenced if SORT = ā€™Nā€™.

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 W
contain those eigenvalues which have converged;
if JOBVS = ā€™Vā€™, 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.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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