strsna.f(3)							      LAPACK							       strsna.f(3)

NAME

       strsna.f -

SYNOPSIS

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

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

       Purpose:

	    STRSNA 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 SHSEQR), 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 opposite sign.

       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 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

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

	   T

		     T is REAL array, dimension (LDT,N)
		     The upper quasi-triangular matrix T, in Schur canonical form.

	   LDT

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

	   VL

		     VL is REAL 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
		     SHSEIN or STREVC.
		     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 REAL 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
		     SHSEIN or STREVC.
		     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. 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

		     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. 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

		     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 REAL 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.

	   IWORK

		     IWORK is INTEGER array, dimension (2*(N-1))
		     If JOB = 'E', IWORK 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**T*u| / (norm(u)*norm(v))

	     where u and v are the right and left eigenvectors of T corresponding
	     to lambda; v**T denotes the 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 264 of file strsna.f.

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