ctrsna.f(3)							      LAPACK							       ctrsna.f(3)

NAME

       ctrsna.f -

SYNOPSIS

   Functions/Subroutines
       subroutine ctrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, RWORK, INFO)
	   CTRSNA

Function/Subroutine Documentation
   subroutine ctrsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, complex, dimension( ldt, * )T, integerLDT, complex,
       dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( * )SEP,
       integerMM, integerM, complex, dimension( ldwork, * )WORK, integerLDWORK, real, dimension( * )RWORK, integerINFO)
       CTRSNA

       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).

       Parameters:
	   JOB

		     JOB is 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

		     HOWMNY is CHARACTER*1
		     = 'A': compute condition numbers for all eigenpairs;
		     = 'S': compute condition numbers for selected eigenpairs
			    specified by the array SELECT.

	   SELECT

		     SELECT is 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

		     N is INTEGER
		     The order of the matrix T. N >= 0.

	   T

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

	   LDT

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

	   VL

		     VL is 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

		     LDVL is INTEGER
		     The leading dimension of the array VL.
		     LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N.

	   VR

		     VR is 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

		     LDVR is INTEGER
		     The leading dimension of the array VR.
		     LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

	   S

		     S is 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

		     SEP is 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

		     MM is 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

		     M is 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

		     WORK is COMPLEX array, dimension (LDWORK,N+6)
		     If JOB = 'E', WORK is not referenced.

	   LDWORK

		     LDWORK is INTEGER
		     The leading dimension of the array WORK.
		     LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

	   RWORK

		     RWORK is REAL array, dimension (N)
		     If JOB = 'E', RWORK is not referenced.

	   INFO

		     INFO is INTEGER
		     = 0: successful exit
		     < 0: if INFO = -i, the i-th argument had an illegal value

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Further Details:

	     The reciprocal of the condition number of an eigenvalue lambda is
	     defined as

		     S(lambda) = |v**H*u| / (norm(u)*norm(v))

	     where u and v are the right and left eigenvectors of T corresponding
	     to lambda; v**H 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)

       Definition at line 248 of file ctrsna.f.

Author
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Version 3.4.2							  Tue Sep 25 2012						       ctrsna.f(3)