DTRSNA(l)		LAPACK routine (version	1.1)		    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 diago-
  nal block has	its diagonal elements equal and	its off-diagonal elements of
  opposite 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 condi-
	  tion 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	eigenpairs
	  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 ele-
	  ments	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 and SEP. MM >=	M.

  M	  (output) INTEGER
	  The number of	elements of the	arrays S and SEP used to store the
	  specified condition numbers; for each	selected real eigenvalue one
	  element is used, and for each	selected complex conjugate pair	of
	  eigenvalues, two elements are	used. 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 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
  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)