NAME

       slar1v.f -

SYNOPSIS

   Functions/Subroutines
       subroutine slar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT,
           ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
           SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
           through bn of the tridiagonal matrix LDLT - λI.

Function/Subroutine Documentation

   subroutine slar1v (integerN, integerB1, integerBN, realLAMBDA, real, dimension( * )D, real,
       dimension( * )L, real, dimension( * )LD, real, dimension( * )LLD, realPIVMIN, realGAPTOL,
       real, dimension( * )Z, logicalWANTNC, integerNEGCNT, realZTZ, realMINGMA, integerR,
       integer, dimension( * )ISUPPZ, realNRMINV, realRESID, realRQCORR, real, dimension( *
       )WORK)
       SLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
       through bn of the tridiagonal matrix LDLT - λI.

       Purpose:

            SLAR1V 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.

       Parameters:
           N

                     N is INTEGER
                      The order of the matrix L D L**T.

           B1

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

           BN

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

           LAMBDA

                     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

                     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

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

           LD

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

           LLD

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

           PIVMIN

                     PIVMIN is REAL
                      The minimum pivot in the Sturm sequence.

           GAPTOL

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

           Z

                     Z is REAL 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

                     WANTNC is LOGICAL
                      Specifies whether NEGCNT has to be computed.

           NEGCNT

                     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

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

           MINGMA

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

           R

                     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

                     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

                     NRMINV is REAL
                      NRMINV = 1/SQRT( ZTZ )

           RESID

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

           RQCORR

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

           WORK

                     WORK is REAL array, dimension (4*N)

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Contributors:
           Beresford Parlett, University of California, Berkeley, USA
            Jim Demmel, University of California, Berkeley, USA
            Inderjit Dhillon, University of Texas, Austin, USA
            Osni Marques, LBNL/NERSC, USA
            Christof Voemel, University of California, Berkeley, USA

       Definition at line 229 of file slar1v.f.

Author

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