CLAR1V(3)      LAPACK routine of NEC Numeric Library Collection      CLAR1V(3)



NAME
       CLAR1V

SYNOPSIS
       SUBROUTINE CLAR1V (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z,
           WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR,
           WORK)



PURPOSE
            CLAR1V computes the (scaled) r-th column of the inverse of
            the sumbmatrix in rows B1 through BN of the tridiagonal matrix
            L D L**T - sigma I. When sigma is close to an eigenvalue, the
            computed vector is an accurate eigenvector. Usually, r corresponds
            to the index where the eigenvector is largest in magnitude.
            The following steps accomplish this computation :
            (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
            (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
            (c) Computation of the diagonal elements of the inverse of
                L D L**T - sigma I by combining the above transforms, and choosing
                r as the index where the diagonal of the inverse is (one of the)
                largest in magnitude.
            (d) Computation of the (scaled) r-th column of the inverse using the
                twisted factorization obtained by combining the top part of the
                the stationary and the bottom part of the progressive transform.




ARGUMENTS
           N         (input)
                     N is INTEGER
                      The order of the matrix L D L**T.

           B1        (input)
                     B1 is INTEGER
                      First index of the submatrix of L D L**T.

           BN        (input)
                     BN is INTEGER
                      Last index of the submatrix of L D L**T.

           LAMBDA    (input)
                     LAMBDA is REAL
                      The shift. In order to compute an accurate eigenvector,
                      LAMBDA should be a good approximation to an eigenvalue
                      of L D L**T.

           L         (input)
                     L is REAL array, dimension (N-1)
                      The (n-1) subdiagonal elements of the unit bidiagonal matrix
                      L, in elements 1 to N-1.

           D         (input)
                     D is REAL array, dimension (N)
                      The n diagonal elements of the diagonal matrix D.

           LD        (input)
                     LD is REAL array, dimension (N-1)
                      The n-1 elements L(i)*D(i).

           LLD       (input)
                     LLD is REAL array, dimension (N-1)
                      The n-1 elements L(i)*L(i)*D(i).

           PIVMIN    (input)
                     PIVMIN is REAL
                      The minimum pivot in the Sturm sequence.

           GAPTOL    (input)
                     GAPTOL is REAL
                      Tolerance that indicates when eigenvector entries are negligible
                      w.r.t. their contribution to the residual.

           Z         (input/output)
                     Z is COMPLEX array, dimension (N)
                      On input, all entries of Z must be set to 0.
                      On output, Z contains the (scaled) r-th column of the
                      inverse. The scaling is such that Z(R) equals 1.

           WANTNC    (input)
                     WANTNC is LOGICAL
                      Specifies whether NEGCNT has to be computed.

           NEGCNT    (output)
                     NEGCNT is INTEGER
                      If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
                      in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

           ZTZ       (output)
                     ZTZ is REAL
                      The square of the 2-norm of Z.

           MINGMA    (output)
                     MINGMA is REAL
                      The reciprocal of the largest (in magnitude) diagonal
                      element of the inverse of L D L**T - sigma I.

           R         (input/output)
                     R is INTEGER
                      The twist index for the twisted factorization used to
                      compute Z.
                      On input, 0 <= R <= N. If R is input as 0, R is set to
                      the index where (L D L**T - sigma I)^{-1} is largest
                      in magnitude. If 1 <= R <= N, R is unchanged.
                      On output, R contains the twist index used to compute Z.
                      Ideally, R designates the position of the maximum entry in the
                      eigenvector.

           ISUPPZ    (output)
                     ISUPPZ is INTEGER array, dimension (2)
                      The support of the vector in Z, i.e., the vector Z is
                      nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

           NRMINV    (output)
                     NRMINV is REAL
                      NRMINV = 1/SQRT( ZTZ )

           RESID     (output)
                     RESID is REAL
                      The residual of the FP vector.
                      RESID = ABS( MINGMA )/SQRT( ZTZ )

           RQCORR    (output)
                     RQCORR is REAL
                      The Rayleigh Quotient correction to LAMBDA.
                      RQCORR = MINGMA*TMP

           WORK      (output)
                     WORK is REAL array, dimension (4*N)



LAPACK routine                  31 October 2017                      CLAR1V(3)