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.
     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.

     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)