strsna(3F)

NAME
     STRSNA - estimate 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 STRSNA( 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( * )

         REAL           S( * ), SEP( * ), T( LDT, * ), VL( LDVL, * ), VR(
                        LDVR, * ), WORK( LDWORK, * )

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.


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

     VR      (input) 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    (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. 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) 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      (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) REAL 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)