Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  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)

SYNOPSIS

       SUBROUTINE ZTRSNA( JOB,  HOWMNY,  SELECT,  N,  T,  LDT, VL, LDVL, VR, LDVR, S, SEP, MM, M,
                          WORK, LDWORK, RWORK, INFO )

           CHARACTER      HOWMNY, JOB

           INTEGER        INFO, LDT, LDVL, LDVR, LDWORK, M, MM, N

           LOGICAL        SELECT( * )

           DOUBLE         PRECISION RWORK( * ), S( * ), SEP( * )

           COMPLEX*16     T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( LDWORK, * )

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).

ARGUMENTS

        JOB     (input) 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  (input) CHARACTER*1
                = 'A': compute condition numbers for all eigenpairs;
                = 'S': compute condition numbers for selected eigenpairs
                specified by the array SELECT.

        SELECT  (input) 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       (input) INTEGER
                The order of the matrix T. N >= 0.

        T       (input) COMPLEX*16 array, dimension (LDT,N)
                The upper triangular matrix T.

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

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

        VR      (input) 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    (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. 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     (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.
                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) COMPLEX*16 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.

        RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
                If JOB = 'E', RWORK 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**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)

 LAPACK routine (version 3.3.1)             April 2011                            ZTRSNA(3lapack)