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ztrsen.f(3)				      LAPACK				      ztrsen.f(3)

NAME
       ztrsen.f -

SYNOPSIS
   Functions/Subroutines
       subroutine ztrsen (JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S, SEP, WORK, LWORK, INFO)
	   ZTRSEN

Function/Subroutine Documentation
   subroutine ztrsen (characterJOB, characterCOMPQ, logical, dimension( * )SELECT, integerN,
       complex*16, dimension( ldt, * )T, integerLDT, complex*16, dimension( ldq, * )Q,
       integerLDQ, complex*16, dimension( * )W, integerM, double precisionS, double precisionSEP,
       complex*16, dimension( * )WORK, integerLWORK, integerINFO)
       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.

       Date:
	   November 2011

       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

       Definition at line 264 of file ztrsen.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      ztrsen.f(3)
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