CTRSNA man page on IRIX

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CTRSNA(3F)							    CTRSNA(3F)

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

									Page 1

CTRSNA(3F)							    CTRSNA(3F)

     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 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.

									Page 2

CTRSNA(3F)							    CTRSNA(3F)

     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
     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)

									Page 3

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