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dtrsna examples
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Man Page Details
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NAME
       DTRSNA  - 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 DTRSNA( 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( * )

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

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-diag-
       onal elements of opposite sign.


ARGUMENTS
       JOB     (input) CHARACTER*1
               Specifies whether condition numbers are required for  eigenval-
               ues (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 speci-
               fied by the array SELECT.

       SELECT  (input) LOGICAL array, dimension (N)
               If HOWMNY = 'S', SELECT specifies the eigenpairs for which con-
               dition  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) DOUBLE PRECISION 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) 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 eigen-
               pairs 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    (input) INTEGER
               The leading dimension of the array VL.  LDVL >= 1; and if JOB =
               'E' or 'B', LDVL >= N.

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

       S       (output) 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 consecu-
               tive  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  refer-
               enced.

       SEP     (output) 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      (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) DOUBLE PRECISION array, dimension (LDWORK,N+6)
               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 (2*(N-1))
               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  cor-
       responding 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)

