DTRSNA(l)								 )								 DTRSNA(l)

NAME

       DTRSNA  - estimate reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix
       T (or of any matrix Q*T*Q**T with Q orthogonal)

SYNOPSIS

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

	   CHARACTER	  HOWMNY, JOB

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

	   LOGICAL	  SELECT( * )

	   INTEGER	  IWORK( * )

	   DOUBLE	  PRECISION S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

PURPOSE

       DTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T
       (or  of any matrix Q*T*Q**T with Q orthogonal).	T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular
       with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements  of  oppo-
       site sign.

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
	       eigenpair corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers corresponding to  a
	       complex conjugate pair of eigenvalues w(j) and w(j+1), either SELECT(j) or SELECT(j+1) or both, 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) DOUBLE PRECISION array, dimension (LDT,N)
	       The upper quasi-triangular matrix T, in Schur canonical form.

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

       VL      (input) DOUBLE PRECISION array, dimension (LDVL,M)
	       If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs
	       specified  by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by DHSEIN or DTREVC.	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) DOUBLE PRECISION array, dimension (LDVR,M)
	       If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding  to  the  eigen-
	       pairs  specified  by  HOWMNY  and  SELECT.  The	eigenvectors must be stored in consecutive columns of VR, as returned by DHSEIN or
	       DTREVC.	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. For
	       a  complex  conjugate  pair of eigenvalues two consecutive elements of S are set to the same value. 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. For a complex eigenvector two consecutive elements of SEP are set to the same value. If the eigenvalues cannot be  reordered
	       to  compute  SEP(j),  SEP(j) is set to 0; this can only occur when the true value would be very small anyway.  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) DOUBLE PRECISION 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.

       IWORK   (workspace) INTEGER array, dimension (N)
	       If JOB = 'E', IWORK 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 							 DTRSNA(l)