SRC/dtrsna.f(3)

NAME
SRC/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 (character job, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) sep, integer mm, integer m, double precision, dimension( ldwork, * ) work, integer ldwork, integer, dimension( * ) iwork, integer info)

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.

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 262 of file dtrsna.f.

Author

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