ZTRSNA(l)								 )								 ZTRSNA(l)

NAME

       ZTRSNA  -  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 unitary)

SYNOPSIS

       SUBROUTINE ZTRSNA( 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( * )

	   DOUBLE	  PRECISION RWORK( * ), S( * ), SEP( * )

	   COMPLEX*16	  T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

PURPOSE

       ZTRSNA 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 eigenvalues (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 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*16 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*16 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 ZHSEIN or ZTREVC.	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*16 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 ZHSEIN or ZTREVC.	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) DOUBLE PRECISION 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) DOUBLE PRECISION 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*16 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) DOUBLE PRECISION 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 condi-
       tioned). 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 							 ZTRSNA(l)