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dtrsna examples
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Man Page Details
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NAME
       DTRSNA  - 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 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
          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
          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 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
  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)

