dtrsna.f(3)

NAME
dtrsna.f -

SYNOPSIS

  Functions/Subroutines
subroutine dtrsna (JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M, WORK, LDWORK, IWORK, INFO)
DTRSNA

Function/Subroutine Documentation

  subroutine dtrsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, double precision, dimension( ldt, * )T, integerLDT, double precision, dimension( ldvl, * )VL, integerLDVL, double precision, dimension( ldvr, * )VR, integerLDVR, double precision, dimension( * )S, double precision, dimension( * )SEP, integerMM, integerM, double precision, dimension( ldwork, * )WORK, integerLDWORK, integer, dimension( * )IWORK, integerINFO)

DTRSNA

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.

Parameters:
    JOB

              JOB is 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

              HOWMNY is CHARACTER*1
              = 'A': compute condition numbers for all eigenpairs;
              = 'S': compute condition numbers for selected eigenpairs
                     specified by the array SELECT.

    SELECT

              SELECT is 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

              N is INTEGER
              The order of the matrix T. N >= 0.

    T

              T is DOUBLE PRECISION array, dimension (LDT,N)
              The upper quasi-triangular matrix T, in Schur canonical form.

    LDT

              LDT is INTEGER
              The leading dimension of the array T. LDT >= max(1,N).

    VL

              VL is 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

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

    VR

              VR is 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

              LDVR is INTEGER
              The leading dimension of the array VR.
              LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N.

    S

              S is 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

              SEP is 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

              MM is 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

              M is 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

              WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
              If JOB = 'E', WORK is not referenced.

    LDWORK

              LDWORK is INTEGER
              The leading dimension of the array WORK.
              LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

    IWORK

              IWORK is INTEGER array, dimension (2*(N-1))
              If JOB = 'E', IWORK is not referenced.

    INFO

              INFO is INTEGER
              = 0: successful exit
              < 0: if INFO = -i, the i-th argument had an illegal value

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    November 2011

Further Details:

      The reciprocal of the condition number of an eigenvalue lambda is
      defined as
              S(lambda) = |v**T*u| / (norm(u)*norm(v))
      where u and v are the right and left eigenvectors of T corresponding
      to lambda; v**T denotes the 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)

Definition at line 264 of file dtrsna.f.

Author

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