DTRSNA(3)      LAPACK routine of NEC Numeric Library Collection      DTRSNA(3)



NAME
       DTRSNA

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



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.




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

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

           T         (input)
                     T is DOUBLE PRECISION array, dimension (LDT,N)
                     The upper quasi-triangular matrix T, in Schur canonical form.

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

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

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

           S         (output)
                     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       (output)
                     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        (input)
                     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         (output)
                     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      (output)
                     WORK is DOUBLE PRECISION array, dimension (LDWORK,N+6)
                     If JOB = 'E', WORK is not referenced.

           LDWORK    (input)
                     LDWORK is INTEGER
                     The leading dimension of the array WORK.
                     LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N.

           IWORK     (output)
                     IWORK is INTEGER array, dimension (2*(N-1))
                     If JOB = 'E', IWORK is not referenced.

           INFO      (output)
                     INFO is 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**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)



LAPACK routine                  31 October 2017                      DTRSNA(3)