SRC/dlarrv.f(3)

NAME
SRC/dlarrv.f

SYNOPSIS

  Functions/Subroutines
subroutine dlarrv (n, vl, vu, d, l, pivmin, isplit, m, dol, dou, minrgp, rtol1, rtol2, w, werr, wgap, iblock, indexw, gers, z, ldz, isuppz, work, iwork, info)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Function/Subroutine Documentation

  subroutine dlarrv (integer n, double precision vl, double precision vu, double precision, dimension( * ) d, double precision, dimension( * ) l, double precision pivmin, integer, dimension( * ) isplit, integer m, integer dol, integer dou, double precision minrgp, double precision rtol1, double precision rtol2, double precision, dimension( * ) w, double precision, dimension( * ) werr, double precision, dimension( * ) wgap, integer, dimension( * ) iblock, integer, dimension( * ) indexw, double precision, dimension( * ) gers, double precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.

Purpose:

     DLARRV computes the eigenvectors of the tridiagonal matrix
     T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
     The input eigenvalues should have been computed by DLARRE.

Parameters
    N

              N is INTEGER
              The order of the matrix.  N >= 0.

    VL

              VL is DOUBLE PRECISION
              Lower bound of the interval that contains the desired
              eigenvalues. VL < VU. Needed to compute gaps on the left or right
              end of the extremal eigenvalues in the desired RANGE.

    VU

              VU is DOUBLE PRECISION
              Upper bound of the interval that contains the desired
              eigenvalues. VL < VU.
              Note: VU is currently not used by this implementation of DLARRV, VU is
              passed to DLARRV because it could be used compute gaps on the right end
              of the extremal eigenvalues. However, with not much initial accuracy in
              LAMBDA and VU, the formula can lead to an overestimation of the right gap
              and thus to inadequately early RQI 'convergence'. This is currently
              prevented this by forcing a small right gap. And so it turns out that VU
              is currently not used by this implementation of DLARRV.

    D

              D is DOUBLE PRECISION array, dimension (N)
              On entry, the N diagonal elements of the diagonal matrix D.
              On exit, D may be overwritten.

    L

              L is DOUBLE PRECISION array, dimension (N)
              On entry, the (N-1) subdiagonal elements of the unit
              bidiagonal matrix L are in elements 1 to N-1 of L
              (if the matrix is not split.) At the end of each block
              is stored the corresponding shift as given by DLARRE.
              On exit, L is overwritten.

    PIVMIN

              PIVMIN is DOUBLE PRECISION
              The minimum pivot allowed in the Sturm sequence.

    ISPLIT

              ISPLIT is INTEGER array, dimension (N)
              The splitting points, at which T breaks up into blocks.
              The first block consists of rows/columns 1 to
              ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
              through ISPLIT( 2 ), etc.

    M

              M is INTEGER
              The total number of input eigenvalues.  0 <= M <= N.

    DOL

              DOL is INTEGER

    DOU

              DOU is INTEGER
              If the user wants to compute only selected eigenvectors from all
              the eigenvalues supplied, he can specify an index range DOL:DOU.
              Or else the setting DOL=1, DOU=M should be applied.
              Note that DOL and DOU refer to the order in which the eigenvalues
              are stored in W.
              If the user wants to compute only selected eigenpairs, then
              the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
              computed eigenvectors. All other columns of Z are set to zero.

    MINRGP

              MINRGP is DOUBLE PRECISION

    RTOL1

              RTOL1 is DOUBLE PRECISION

    RTOL2

              RTOL2 is DOUBLE PRECISION
               Parameters for bisection.
               An interval [LEFT,RIGHT] has converged if
               RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

    W

              W is DOUBLE PRECISION array, dimension (N)
              The first M elements of W contain the APPROXIMATE eigenvalues for
              which eigenvectors are to be computed.  The eigenvalues
              should be grouped by split-off block and ordered from
              smallest to largest within the block ( The output array
              W from DLARRE is expected here ). Furthermore, they are with
              respect to the shift of the corresponding root representation
              for their block. On exit, W holds the eigenvalues of the
              UNshifted matrix.

    WERR

              WERR is DOUBLE PRECISION array, dimension (N)
              The first M elements contain the semiwidth of the uncertainty
              interval of the corresponding eigenvalue in W

    WGAP

              WGAP is DOUBLE PRECISION array, dimension (N)
              The separation from the right neighbor eigenvalue in W.

    IBLOCK

              IBLOCK is INTEGER array, dimension (N)
              The indices of the blocks (submatrices) associated with the
              corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
              W(i) belongs to the first block from the top, =2 if W(i)
              belongs to the second block, etc.

    INDEXW

              INDEXW is INTEGER array, dimension (N)
              The indices of the eigenvalues within each block (submatrix);
              for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
              i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

    GERS

              GERS is DOUBLE PRECISION array, dimension (2*N)
              The N Gerschgorin intervals (the i-th Gerschgorin interval
              is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
              be computed from the original UNshifted matrix.

    Z

              Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
              If INFO = 0, the first M columns of Z contain the
              orthonormal eigenvectors of the matrix T
              corresponding to the input eigenvalues, with the i-th
              column of Z holding the eigenvector associated with W(i).
              Note: the user must ensure that at least max(1,M) columns are
              supplied in the array Z.

    LDZ

              LDZ is INTEGER
              The leading dimension of the array Z.  LDZ >= 1, and if
              JOBZ = 'V', LDZ >= max(1,N).

    ISUPPZ

              ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
              The support of the eigenvectors in Z, i.e., the indices
              indicating the nonzero elements in Z. The I-th eigenvector
              is nonzero only in elements ISUPPZ( 2*I-1 ) through
              ISUPPZ( 2*I ).

    WORK

              WORK is DOUBLE PRECISION array, dimension (12*N)

    IWORK

              IWORK is INTEGER array, dimension (7*N)

    INFO

              INFO is INTEGER
              = 0:  successful exit
              > 0:  A problem occurred in DLARRV.
              < 0:  One of the called subroutines signaled an internal problem.
                    Needs inspection of the corresponding parameter IINFO
                    for further information.
              =-1:  Problem in DLARRB when refining a child's eigenvalues.
              =-2:  Problem in DLARRF when computing the RRR of a child.
                    When a child is inside a tight cluster, it can be difficult
                    to find an RRR. A partial remedy from the user's point of
                    view is to make the parameter MINRGP smaller and recompile.
                    However, as the orthogonality of the computed vectors is
                    proportional to 1/MINRGP, the user should be aware that
                    he might be trading in precision when he decreases MINRGP.
              =-3:  Problem in DLARRB when refining a single eigenvalue
                    after the Rayleigh correction was rejected.
              = 5:  The Rayleigh Quotient Iteration failed to converge to
                    full accuracy in MAXITR steps.

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

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 287 of file dlarrv.f.

Author

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