Previous: ctrsen Up: ../lapack-c.html Next: ctrsyl


ctrsna


 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 uni-
      tary)

 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 uni-
      tary).

 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 eigen-
              pairs 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 con-
              dition 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.

      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 eigenvec-
              tors 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 con-
              secutive 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 eigenvec-
              tors 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 con-
              secutive 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 con-
              secutive 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 condi-
              tion 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 and SEP. MM
              >= M.

      M       (output) INTEGER
              The number of elements of the arrays S and SEP used

              to store the specified 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.

      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 recipro-
      cal 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 eigen-
      vector u corresponding to lambda is defined as follows. Sup-
      pose

                  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)