dtrsen - reorder 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,
SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, * S, SEP, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER * 1 JOB, COMPQ INTEGER N, LDT, LDQ, M, LWORK, LIWORK, INFO INTEGER IWORK(*) LOGICAL SELECT(*) DOUBLE PRECISION S, SEP DOUBLE PRECISION T(LDT,*), Q(LDQ,*), WR(*), WI(*), WORK(*)
SUBROUTINE DTRSEN_64( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, * M, S, SEP, WORK, LWORK, IWORK, LIWORK, INFO) CHARACTER * 1 JOB, COMPQ INTEGER*8 N, LDT, LDQ, M, LWORK, LIWORK, INFO INTEGER*8 IWORK(*) LOGICAL*8 SELECT(*) DOUBLE PRECISION S, SEP DOUBLE PRECISION T(LDT,*), Q(LDQ,*), WR(*), WI(*), WORK(*)
SUBROUTINE TRSEN( JOB, COMPQ, SELECT, [N], T, [LDT], Q, [LDQ], WR, * WI, M, S, SEP, WORK, [LWORK], [IWORK], [LIWORK], [INFO]) CHARACTER(LEN=1) :: JOB, COMPQ INTEGER :: N, LDT, LDQ, M, LWORK, LIWORK, INFO INTEGER, DIMENSION(:) :: IWORK LOGICAL, DIMENSION(:) :: SELECT REAL(8) :: S, SEP REAL(8), DIMENSION(:) :: WR, WI, WORK REAL(8), DIMENSION(:,:) :: T, Q
SUBROUTINE TRSEN_64( JOB, COMPQ, SELECT, [N], T, [LDT], Q, [LDQ], * WR, WI, M, S, SEP, WORK, [LWORK], [IWORK], [LIWORK], [INFO]) CHARACTER(LEN=1) :: JOB, COMPQ INTEGER(8) :: N, LDT, LDQ, M, LWORK, LIWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK LOGICAL(8), DIMENSION(:) :: SELECT REAL(8) :: S, SEP REAL(8), DIMENSION(:) :: WR, WI, WORK REAL(8), DIMENSION(:,:) :: T, Q
#include <sunperf.h>
void dtrsen(char job, char compq, logical *select, int n, double *t, int ldt, double *q, int ldq, double *wr, double *wi, int *m, double *s, double *sep, double *work, int lwork, int *info);
void dtrsen_64(char job, char compq, logical *select, long n, double *t, long ldt, double *q, long ldq, double *wr, double *wi, long *m, double *s, double *sep, double *work, long lwork, long *info);
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 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 elemnts equal and its off-diagonal elements of opposite sign.
= 'N': none;
= 'E': for eigenvalues only (S);
= 'V': for invariant subspace only (SEP);
= 'B': for both eigenvalues and invariant subspace (S and SEP).
= 'V': update the matrix Q of Schur vectors;
= 'N': do not update Q.
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.
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.
WORK(1) returns the optimal LWORK.
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.
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.
= 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.
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*Z = ( T11 T12 ) n1
( 0 T22 ) n2
n1 n2
where N = n1+n2 and Z' 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', then the reordered real Schur factorization of A is given by A = (Q*Z)*(Z'*T*Z)*(Q*Z)', 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