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RedHat 9 (Linux i386) - man page for ztrsen (redhat section l)

ZTRSEN(l)					)					ZTRSEN(l)

NAME
       ZTRSEN  -  reorder  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

SYNOPSIS
       SUBROUTINE ZTRSEN( 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

	   DOUBLE	  PRECISION S, SEP

	   LOGICAL	  SELECT( * )

	   COMPLEX*16	  Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )

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.

ARGUMENTS
       JOB     (input) 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   (input) CHARACTER*1
	       = 'V': update the matrix Q of Schur vectors;
	       = 'N': do not update Q.

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

       N       (input) INTEGER
	       The order of the matrix T. N >= 0.

       T       (input/output) 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  ele-
	       ments.

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

       Q       (input/output) 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     (input) INTEGER
	       The leading dimension of the array Q.  LDQ >= 1; and if COMPQ = 'V', LDQ >= N.

       W       (output) COMPLEX*16 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) 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  con-
	       dition  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     (output) DOUBLE PRECISION
	       If JOB = 'V' or 'B', SEP is the estimated reciprocal condition number of the spec-
	       ified  invariant subspace. If M = 0 or N, SEP = norm(T).  If JOB = 'N' or 'E', SEP
	       is not referenced.

       WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
	       If JOB = 'N', WORK is not referenced.  Otherwise, on exit, if INFO  =  0,  WORK(1)
	       returns the optimal LWORK.

       LWORK   (input) INTEGER
	       The  dimension of the array WORK.  If JOB = 'N', LWORK >= 1; if JOB = 'E', LWORK =
	       M*(N-M); if JOB = 'V' or 'B', LWORK >= 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    (output) INTEGER
	       = 0:  successful exit
	       < 0:  if INFO = -i, the i-th argument had an illegal value

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'*T*Z = ( T11 T12 ) n1
			      (  0  T22 ) n2
				 n1  n2

       where N = n1+n2 and Z' 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*Q', then the
       reordered 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  com-
       puted 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 underesti-
       mate 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 recip-
       rocal 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

LAPACK version 3.0			   15 June 2000 				ZTRSEN(l)


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