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CTRSNA(l)			       )			     CTRSNA(l)

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

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 unitary).

ARGUMENTS
       JOB     (input) CHARACTER*1
	       Specifies  whether condition numbers are required for eigenval‐
	       ues (S) or eigenvectors (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 speci‐
	       fied by the array SELECT.

       SELECT  (input) LOGICAL array, dimension (N)
	       If HOWMNY = 'S', SELECT specifies the eigenpairs for which con‐
	       dition 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 eigen‐
	       pairs 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 columns 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 referenced.

       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 estimated 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 conditioned).
       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  cor‐
       responding 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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