DTRSEN man page on Scientific

Printed from http://www.polarhome.com/service/man/?qf=DTRSEN&af=0&tf=2&of=Scientific

DTRSEN(1)		 LAPACK routine (version 3.2)		     DTRSEN(1)

NAME
       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,

SYNOPSIS
       SUBROUTINE DTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, WR, WI, M, S,
			  SEP, WORK, LWORK, IWORK, LIWORK, INFO )

	   CHARACTER	  COMPQ, JOB

	   INTEGER	  INFO, LDQ, LDT, LIWORK, LWORK, M, N

	   DOUBLE	  PRECISION S, SEP

	   LOGICAL	  SELECT( * )

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( *
			  ), WR( * )

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 tri‐
       angular 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.

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies whether condition numbers are required for the	 clus‐
	       ter 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 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 com‐
	       plex conjugate pair of eigenvalues must be either both included
	       in the cluster or both excluded.

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

       T       (input/output) DOUBLE PRECISION array, dimension (LDT,N)
	       On entry, the upper quasi-triangular matrix T, in Schur canoni‐
	       cal 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     (input) INTEGER
	       The leading dimension of the array T. LDT >= max(1,N).

       Q       (input/output) 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 orthogo‐
	       nal transformation matrix which reorders T; the leading M  col‐
	       umns 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.

       WR      (output) DOUBLE PRECISION array, dimension (N)
	       WI      (output) 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       (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 con‐
	       dition number for the selected cluster of eigenvalues.  S  can‐
	       not  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     (output) 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	  (workspace/output)   DOUBLE	PRECISION   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  =  '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   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
	       On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

       LIWORK  (input) 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  cal‐
	       culates 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    (output) 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 eigen‐
	       values in the same order as in T; S and SEP (if requested)  are
	       set to zero.

FURTHER DETAILS
       DTRSEN first collects the selected eigenvalues by computing an orthogo‐
       nal 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

 LAPACK routine (version 3.2)	 November 2008			     DTRSEN(1)
[top]

List of man pages available for Scientific

Copyright (c) for man pages and the logo by the respective OS vendor.

For those who want to learn more, the polarhome community provides shell access and support.

[legal] [privacy] [GNU] [policy] [cookies] [netiquette] [sponsors] [FAQ]
Tweet
Polarhome, production since 1999.
Member of Polarhome portal.
Based on Fawad Halim's script.
....................................................................
Vote for polarhome
Free Shell Accounts :: the biggest list on the net