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

CTRSNA(l)					)					CTRSNA(l)

NAME
       CTRSNA  -  estimate  reciprocal	condition  numbers for specified eigenvalues and/or right
       eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q uni-
       tary)

SYNOPSIS
       SUBROUTINE CTRSNA( JOB,	HOWMNY,  SELECT,  N,  T,  LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M,
			  WORK, LDWORK, RWORK, INFO )

	   CHARACTER	  HOWMNY, JOB

	   INTEGER	  INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N

	   LOGICAL	  SELECT( * )

	   REAL 	  RWORK( * ), S( * ), SEP( * )

	   COMPLEX	  T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

PURPOSE
       CTRSNA estimates reciprocal condition  numbers  for  specified  eigenvalues  and/or  right
       eigenvectors of a complex upper triangular matrix T (or of any matrix Q*T*Q**H with Q uni-
       tary).

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies whether condition numbers are required for eigenvalues (S) or	eigenvec-
	       tors (SEP):
	       = 'E': for eigenvalues only (S);
	       = 'V': for eigenvectors only (SEP);
	       = 'B': for both eigenvalues and eigenvectors (S and SEP).

       HOWMNY  (input) CHARACTER*1
	       = 'A': compute condition numbers for all eigenpairs;
	       =  'S':	compute  condition numbers for selected eigenpairs specified by the array
	       SELECT.

       SELECT  (input) LOGICAL array, dimension (N)
	       If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition  numbers  are
	       required.  To  select  condition numbers for the j-th eigenpair, SELECT(j) must be
	       set to .TRUE..  If HOWMNY = 'A', SELECT is not referenced.

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

       T       (input) COMPLEX array, dimension (LDT,N)
	       The upper triangular matrix T.

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

       VL      (input) COMPLEX array, dimension (LDVL,M)
	       If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or	of  any  Q*T*Q**H
	       with  Q	unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT.
	       The eigenvectors must be stored in consecutive  columns	of  VL,  as  returned  by
	       CHSEIN or CTREVC.  If JOB = 'V', VL is not referenced.

       LDVL    (input) INTEGER
	       The  leading  dimension of the array VL.  LDVL >= 1; and if JOB = 'E' or 'B', LDVL
	       >= N.

       VR      (input) COMPLEX array, dimension (LDVR,M)
	       If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of  any  Q*T*Q**H
	       with  Q	unitary), corresponding to the eigenpairs specified by HOWMNY and SELECT.
	       The eigenvectors must be stored in consecutive  columns	of  VR,  as  returned  by
	       CHSEIN or CTREVC.  If JOB = 'V', VR is not referenced.

       LDVR    (input) INTEGER
	       The  leading  dimension of the array VR.  LDVR >= 1; and if JOB = 'E' or 'B', LDVR
	       >= N.

       S       (output) REAL array, dimension (MM)
	       If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues,
	       stored  in consecutive elements of the array. Thus S(j), SEP(j), and the j-th col-
	       umns of VL and VR all correspond to the same eigenpair (but not in general the  j-
	       th  eigenpair, unless all eigenpairs are selected).  If JOB = 'V', S is not refer-
	       enced.

       SEP     (output) REAL array, dimension (MM)
	       If JOB = 'V' or 'B', the estimated reciprocal condition numbers	of  the  selected
	       eigenvectors,  stored  in consecutive elements of the array.  If JOB = 'E', SEP is
	       not referenced.

       MM      (input) INTEGER
	       The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB	=
	       'V' or 'B'). MM >= M.

       M       (output) INTEGER
	       The number of elements of the arrays S and/or SEP actually used to store the esti-
	       mated condition numbers.  If HOWMNY = 'A', M is set to N.

       WORK    (workspace) COMPLEX array, dimension (LDWORK,N+1)
	       If JOB = 'E', WORK is not referenced.

       LDWORK  (input) INTEGER
	       The leading dimension of the array WORK.  LDWORK >= 1; and if JOB =  'V'  or  'B',
	       LDWORK >= N.

       RWORK   (workspace) REAL array, dimension (N)
	       If JOB = 'E', RWORK is not referenced.

       INFO    (output) INTEGER
	       = 0: successful exit
	       < 0: if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS
       The reciprocal of the condition number of an eigenvalue lambda is defined as

	       S(lambda) = |v'*u| / (norm(u)*norm(v))

       where u and v are the right and left eigenvectors of T corresponding to lambda; v' denotes
       the conjugate transpose of v, and norm(u) denotes the  Euclidean  norm.	These  reciprocal
       condition numbers always lie between zero (very badly conditioned) and one (very well con-
       ditioned). If n = 1, S(lambda) is defined to be 1.

       An approximate error bound for a computed eigenvalue W(i) is given by

			   EPS * norm(T) / S(i)

       where EPS is the machine precision.

       The reciprocal of the condition number of the right eigenvector u corresponding to  lambda
       is defined as follows. Suppose

		   T = ( lambda  c  )
		       (   0	T22 )

       Then the reciprocal condition number is

	       SEP( lambda, T22 ) = sigma-min( T22 - lambda*I )

       where  sigma-min denotes the smallest singular value. We approximate the smallest singular
       value by the reciprocal of an estimate of the one-norm of the inverse of T22  -	lambda*I.
       If n = 1, SEP(1) is defined to be abs(T(1,1)).

       An approximate error bound for a computed right eigenvector VR(i) is given by

			   EPS * norm(T) / SEP(i)

LAPACK version 3.0			   15 June 2000 				CTRSNA(l)


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