dtrsna.f man page

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.

Date:

November 2017

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 267 of file dtrsna.f.

Author

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Referenced By

The man page dtrsna(3) is an alias of dtrsna.f(3).

Tue Nov 14 2017 Version 3.8.0 LAPACK