Man page - trsen(3)

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Manual

trsen

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine ctrsen (character job, character compq, logical, dimension( * )select, integer n, complex, dimension( ldt, * ) t, integer ldt,complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) w,integer m, real s, real sep, complex, dimension( * ) work, integerlwork, integer info)
subroutine dtrsen (character job, character compq, logical, dimension( * )select, integer n, double precision, dimension( ldt, * ) t, integerldt, double precision, dimension( ldq, * ) q, integer ldq, doubleprecision, dimension( * ) wr, double precision, dimension( * ) wi,integer m, double precision s, double precision sep, double precision,dimension( * ) work, integer lwork, integer, dimension( * ) iwork,integer liwork, integer info)
subroutine strsen (character job, character compq, logical, dimension( * )select, integer n, real, dimension( ldt, * ) t, integer ldt, real,dimension( ldq, * ) q, integer ldq, real, dimension( * ) wr, real,dimension( * ) wi, integer m, real s, real sep, real, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer liwork,integer info)
subroutine ztrsen (character job, character compq, logical, dimension( * )select, integer n, complex*16, dimension( ldt, * ) t, integer ldt,complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension(* ) w, integer m, double precision s, double precision sep, complex*16,dimension( * ) work, integer lwork, integer info)
Author

NAME

trsen - trsen: reorder Schur form

SYNOPSIS

Functions

subroutine ctrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)
CTRSEN
subroutine dtrsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
DTRSEN
subroutine strsen (job, compq, select, n, t, ldt, q, ldq, wr, wi, m, s, sep, work, lwork, iwork, liwork, info)
STRSEN
subroutine ztrsen (job, compq, select, n, t, ldt, q, ldq, w, m, s, sep, work, lwork, info)
ZTRSEN

Detailed Description

Function Documentation

subroutine ctrsen (character job, character compq, logical, dimension( * )select, integer n, complex, dimension( ldt, * ) t, integer ldt,complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) w,integer m, real s, real sep, complex, dimension( * ) work, integerlwork, integer info)

CTRSEN

Purpose:

CTRSEN reorders the Schur factorization of a complex matrix
A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
the leading positions on the diagonal of the upper triangular matrix
T, and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
SEP).

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select the j-th eigenvalue, SELECT(j) must be set to .TRUE..

N

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

T

T is COMPLEX array, dimension (LDT,N)
On entry, the upper triangular matrix T.
On exit, T is overwritten by the reordered matrix T, with the
selected eigenvalues as the leading diagonal elements.

LDT

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

Q

Q is COMPLEX array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
unitary transformation matrix which reorders T; the leading M
columns of Q form an orthonormal basis for the specified
invariant subspace.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

W

W is COMPLEX array, dimension (N)
The reordered eigenvalues of T, in the same order as they
appear on the diagonal of T.

M

M is INTEGER
The dimension of the specified invariant subspace.
0 <= M <= N.

S

S is REAL
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.

SEP

SEP is REAL
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.

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.
If JOB = ’N’, LWORK >= 1;
if JOB = ’E’, LWORK = max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

CTRSEN first collects the selected eigenvalues by computing a unitary
transformation Z to move them to the top left corner of T. In other
words, the selected eigenvalues are the eigenvalues of T11 in:

Z**H * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2

where N = n1+n2. The first
n1 columns of Z span the specified invariant subspace of T.

If T has been obtained from the Schur factorization of a matrix
A = Q*T*Q**H, then the reordered Schur factorization of A is given by
A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the
corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I R ) n1
( 0 0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP

subroutine dtrsen (character job, character compq, logical, dimension( * )select, integer n, double precision, dimension( ldt, * ) t, integerldt, double precision, dimension( ldq, * ) q, integer ldq, doubleprecision, dimension( * ) wr, double precision, dimension( * ) wi,integer m, double precision s, double precision sep, double precision,dimension( * ) work, integer lwork, integer, dimension( * ) iwork,integer liwork, integer info)

DTRSEN

Purpose:

DTRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

T must be in Schur canonical form (as returned by DHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
SEP).

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.

N

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

T

T is DOUBLE PRECISION array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, T is overwritten by the reordered matrix T, again in
Schur canonical form, with the selected eigenvalues in the
leading diagonal blocks.

LDT

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

Q

Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix which reorders T; the
leading M columns of Q form an orthonormal basis for the
specified invariant subspace.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

WR

WR is DOUBLE PRECISION array, dimension (N)

WI

WI is DOUBLE PRECISION array, dimension (N)

The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is
sufficiently ill-conditioned, then its value may differ
significantly from its value before reordering.

M

M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.

S

S is DOUBLE PRECISION
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.

SEP

SEP is DOUBLE PRECISION
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.

WORK

WORK is DOUBLE PRECISION 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.
If JOB = ’N’, LWORK >= max(1,N);
if JOB = ’E’, LWORK >= max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).

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.

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.
If JOB = ’N’ or ’E’, LIWORK >= 1;
if JOB = ’V’ or ’B’, LIWORK >= max(1,M*(N-M)).

If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned);
T may have been partially reordered, and WR and WI
contain the eigenvalues in the same order as in T; S and
SEP (if requested) are set to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

DTRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
in:

Z**T * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2

where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.

If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I R ) n1
( 0 0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP

subroutine strsen (character job, character compq, logical, dimension( * )select, integer n, real, dimension( ldt, * ) t, integer ldt, real,dimension( ldq, * ) q, integer ldq, real, dimension( * ) wr, real,dimension( * ) wi, integer m, real s, real sep, real, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer liwork,integer info)

STRSEN

Purpose:

STRSEN reorders the real Schur factorization of a real matrix
A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
the leading diagonal blocks of the upper quasi-triangular matrix T,
and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

T must be in Schur canonical form (as returned by SHSEQR), that is,
block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
2-by-2 diagonal block has its diagonal elements equal and its
off-diagonal elements of opposite sign.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
SEP).

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select a real eigenvalue w(j), SELECT(j) must be set to
.TRUE.. To select a complex conjugate pair of eigenvalues
w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
either SELECT(j) or SELECT(j+1) or both must be set to
.TRUE.; a complex conjugate pair of eigenvalues must be
either both included in the cluster or both excluded.

N

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

T

T is REAL array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, T is overwritten by the reordered matrix T, again in
Schur canonical form, with the selected eigenvalues in the
leading diagonal blocks.

LDT

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

Q

Q is REAL array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
orthogonal transformation matrix which reorders T; the
leading M columns of Q form an orthonormal basis for the
specified invariant subspace.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

WR

WR is REAL array, dimension (N)

WI

WI is REAL array, dimension (N)

The real and imaginary parts, respectively, of the reordered
eigenvalues of T. The eigenvalues are stored in the same
order as on the diagonal of T, with WR(i) = T(i,i) and, if
T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
WI(i+1) = -WI(i). Note that if a complex eigenvalue is
sufficiently ill-conditioned, then its value may differ
significantly from its value before reordering.

M

M is INTEGER
The dimension of the specified invariant subspace.
0 < = M <= N.

S

S is REAL
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.

SEP

SEP is REAL
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.

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.
If JOB = ’N’, LWORK >= max(1,N);
if JOB = ’E’, LWORK >= max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).

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.

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.
If JOB = ’N’ or ’E’, LIWORK >= 1;
if JOB = ’V’ or ’B’, LIWORK >= max(1,M*(N-M)).

If LIWORK = -1, then a workspace query is assumed; the
routine only calculates the optimal size of the IWORK array,
returns this value as the first entry of the IWORK array, and
no error message related to LIWORK is issued by XERBLA.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
= 1: reordering of T failed because some eigenvalues are too
close to separate (the problem is very ill-conditioned);
T may have been partially reordered, and WR and WI
contain the eigenvalues in the same order as in T; S and
SEP (if requested) are set to zero.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

STRSEN first collects the selected eigenvalues by computing an
orthogonal transformation Z to move them to the top left corner of T.
In other words, the selected eigenvalues are the eigenvalues of T11
in:

Z**T * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2

where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
of Z span the specified invariant subspace of T.

If T has been obtained from the real Schur factorization of a matrix
A = Q*T*Q**T, then the reordered real Schur factorization of A is given
by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
the corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I R ) n1
( 0 0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP

subroutine ztrsen (character job, character compq, logical, dimension( * )select, integer n, complex*16, dimension( ldt, * ) t, integer ldt,complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension(* ) w, integer m, double precision s, double precision sep, complex*16,dimension( * ) work, integer lwork, integer info)

ZTRSEN

Purpose:

ZTRSEN reorders the Schur factorization of a complex matrix
A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
the leading positions on the diagonal of the upper triangular matrix
T, and the leading columns of Q form an orthonormal basis of the
corresponding right invariant subspace.

Optionally the routine computes the reciprocal condition numbers of
the cluster of eigenvalues and/or the invariant subspace.

Parameters

JOB

JOB is CHARACTER*1
Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= ’N’: none;
= ’E’: for eigenvalues only (S);
= ’V’: for invariant subspace only (SEP);
= ’B’: for both eigenvalues and invariant subspace (S and
SEP).

COMPQ

COMPQ is CHARACTER*1
= ’V’: update the matrix Q of Schur vectors;
= ’N’: do not update Q.

SELECT

SELECT is LOGICAL array, dimension (N)
SELECT specifies the eigenvalues in the selected cluster. To
select the j-th eigenvalue, SELECT(j) must be set to .TRUE..

N

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

T

T is COMPLEX*16 array, dimension (LDT,N)
On entry, the upper triangular matrix T.
On exit, T is overwritten by the reordered matrix T, with the
selected eigenvalues as the leading diagonal elements.

LDT

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

Q

Q is COMPLEX*16 array, dimension (LDQ,N)
On entry, if COMPQ = ’V’, the matrix Q of Schur vectors.
On exit, if COMPQ = ’V’, Q has been postmultiplied by the
unitary transformation matrix which reorders T; the leading M
columns of Q form an orthonormal basis for the specified
invariant subspace.
If COMPQ = ’N’, Q is not referenced.

LDQ

LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if COMPQ = ’V’, LDQ >= N.

W

W is COMPLEX*16 array, dimension (N)
The reordered eigenvalues of T, in the same order as they
appear on the diagonal of T.

M

M is INTEGER
The dimension of the specified invariant subspace.
0 <= M <= N.

S

S is DOUBLE PRECISION
If JOB = ’E’ or ’B’, S is a lower bound on the reciprocal
condition number for the selected cluster of eigenvalues.
S cannot underestimate the true reciprocal condition number
by more than a factor of sqrt(N). If M = 0 or N, S = 1.
If JOB = ’N’ or ’V’, S is not referenced.

SEP

SEP is DOUBLE PRECISION
If JOB = ’V’ or ’B’, SEP is the estimated reciprocal
condition number of the specified invariant subspace. If
M = 0 or N, SEP = norm(T).
If JOB = ’N’ or ’E’, SEP is not referenced.

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.
If JOB = ’N’, LWORK >= 1;
if JOB = ’E’, LWORK = max(1,M*(N-M));
if JOB = ’V’ or ’B’, LWORK >= max(1,2*M*(N-M)).

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.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

ZTRSEN first collects the selected eigenvalues by computing a unitary
transformation Z to move them to the top left corner of T. In other
words, the selected eigenvalues are the eigenvalues of T11 in:

Z**H * T * Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2

where N = n1+n2. The first
n1 columns of Z span the specified invariant subspace of T.

If T has been obtained from the Schur factorization of a matrix
A = Q*T*Q**H, then the reordered Schur factorization of A is given by
A = (Q*Z)*(Z**H*T*Z)*(Q*Z)**H, and the first n1 columns of Q*Z span the
corresponding invariant subspace of A.

The reciprocal condition number of the average of the eigenvalues of
T11 may be returned in S. S lies between 0 (very badly conditioned)
and 1 (very well conditioned). It is computed as follows. First we
compute R so that

P = ( I R ) n1
( 0 0 ) n2
n1 n2

is the projector on the invariant subspace associated with T11.
R is the solution of the Sylvester equation:

T11*R - R*T22 = T12.

Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
the two-norm of M. Then S is computed as the lower bound

(1 + F-norm(R)**2)**(-1/2)

on the reciprocal of 2-norm(P), the true reciprocal condition number.
S cannot underestimate 1 / 2-norm(P) by more than a factor of
sqrt(N).

An approximate error bound for the computed average of the
eigenvalues of T11 is

EPS * norm(T) / S

where EPS is the machine precision.

The reciprocal condition number of the right invariant subspace
spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
SEP is defined as the separation of T11 and T22:

sep( T11, T22 ) = sigma-min( C )

where sigma-min(C) is the smallest singular value of the
n1*n2-by-n1*n2 matrix

C = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

I(m) is an m by m identity matrix, and kprod denotes the Kronecker
product. We estimate sigma-min(C) by the reciprocal of an estimate of
the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

When SEP is small, small changes in T can cause large changes in
the invariant subspace. An approximate bound on the maximum angular
error in the computed right invariant subspace is

EPS * norm(T) / SEP

Author

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