NAME

       ztrsna.f -

SYNOPSIS

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

Function/Subroutine Documentation

   subroutine ztrsna (character JOB, character HOWMNY, logical, dimension( * ) SELECT, integer N,
       complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldvl, * ) VL,
       integer LDVL, complex*16, dimension( ldvr, * ) VR, integer LDVR, double precision,
       dimension( * ) S, double precision, dimension( * ) SEP, integer MM, integer M, complex*16,
       dimension( ldwork, * ) WORK, integer LDWORK, double precision, dimension( * ) RWORK,
       integer INFO)
       ZTRSNA

       Purpose:

            ZTRSNA estimates reciprocal condition numbers for specified
            eigenvalues and/or right eigenvectors of a complex upper triangular
            matrix T (or of any matrix Q*T*Q**H with Q unitary).

       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 j-th eigenpair, SELECT(j) 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 COMPLEX*16 array, dimension (LDT,N)
                     The upper triangular matrix T.

           LDT

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

           VL

                     VL is COMPLEX*16 array, dimension (LDVL,M)
                     If JOB = 'E' or 'B', VL must contain left eigenvectors of T
                     (or of any Q*T*Q**H with Q unitary), corresponding to the
                     eigenpairs specified by HOWMNY and SELECT. The eigenvectors
                     must be stored in consecutive columns of VL, as returned by
                     ZHSEIN or ZTREVC.
                     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 COMPLEX*16 array, dimension (LDVR,M)
                     If JOB = 'E' or 'B', VR must contain right eigenvectors of T
                     (or of any Q*T*Q**H with Q unitary), corresponding to the
                     eigenpairs specified by HOWMNY and SELECT. The eigenvectors
                     must be stored in consecutive columns of VR, as returned by
                     ZHSEIN or ZTREVC.
                     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. 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.
                     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 COMPLEX*16 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.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (N)
                     If JOB = 'E', RWORK 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**H*u| / (norm(u)*norm(v))

             where u and v are the right and left eigenvectors of T corresponding
             to lambda; v**H 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)

Author

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