STRSNA (3lapack)

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)