ctrsen (l)  Linux Man Pages
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
NAME
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 subspaceSYNOPSIS
 SUBROUTINE CTRSEN(
 JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO )
 CHARACTER COMPQ, JOB
 INTEGER INFO, LDQ, LDT, LWORK, M, N
 REAL S, SEP
 LOGICAL SELECT( * )
 COMPLEX Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
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.ARGUMENTS
 JOB (input) CHARACTER*1

Specifies whether condition numbers are required for the
cluster of eigenvalues (S) or the invariant subspace (SEP):
= aqNaq: none;
= aqEaq: for eigenvalues only (S);
= aqVaq: for invariant subspace only (SEP);
= aqBaq: for both eigenvalues and invariant subspace (S and SEP).  COMPQ (input) CHARACTER*1

= aqVaq: update the matrix Q of Schur vectors;
= aqNaq: do not update Q.  SELECT (input) LOGICAL array, dimension (N)
 SELECT specifies the eigenvalues in the selected cluster. To select the jth eigenvalue, SELECT(j) must be set to .TRUE..
 N (input) INTEGER
 The order of the matrix T. N >= 0.
 T (input/output) 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 (input) INTEGER
 The leading dimension of the array T. LDT >= max(1,N).
 Q (input/output) COMPLEX array, dimension (LDQ,N)
 On entry, if COMPQ = aqVaq, the matrix Q of Schur vectors. On exit, if COMPQ = aqVaq, 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 = aqNaq, Q is not referenced.
 LDQ (input) INTEGER
 The leading dimension of the array Q. LDQ >= 1; and if COMPQ = aqVaq, LDQ >= N.
 W (output) COMPLEX array, dimension (N)
 The reordered eigenvalues of T, in the same order as they appear on the diagonal of T.
 M (output) INTEGER
 The dimension of the specified invariant subspace. 0 <= M <= N.
 S (output) REAL
 If JOB = aqEaq or aqBaq, 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 = aqNaq or aqVaq, S is not referenced.
 SEP (output) REAL
 If JOB = aqVaq or aqBaq, SEP is the estimated reciprocal condition number of the specified invariant subspace. If M = 0 or N, SEP = norm(T). If JOB = aqNaq or aqEaq, SEP is not referenced.
 WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. If JOB = aqNaq, LWORK >= 1; if JOB = aqEaq, LWORK = max(1,M*(NM)); if JOB = aqVaq or aqBaq, LWORK >= max(1,2*M*(NM)). 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 (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
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:where N = n1+n2 and Zaq means the conjugate transpose of Z. 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*Qaq, then the reordered Schur factorization of A is given by A = (Q*Z)*(Zaq*T*Z)*(Q*Z)aq, 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
is the projector on the invariant subspace associated with T11. R is the solution of the Sylvester equation:
Let Fnorm(M) denote the Frobeniusnorm of M and 2norm(M) denote the twonorm of M. Then S is computed as the lower bound
on the reciprocal of 2norm(P), the true reciprocal condition number. S cannot underestimate 1 / 2norm(P) by more than a factor of sqrt(N).
An approximate error bound for the computed average of the eigenvalues of T11 is
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:
where sigmamin(C) is the smallest singular value of the
n1*n2byn1*n2 matrix
C