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



NAME
       ZLARRV

SYNOPSIS
       SUBROUTINE ZLARRV (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)



PURPOSE
            ZLARRV 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.




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

           VL        (input)
                     VL is DOUBLE PRECISION

           VU        (input)
                     VU is DOUBLE PRECISION
                     Lower and upper bounds 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.

           D         (input/output)
                     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         (input/output)
                     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 splitted.) At the end of each block
                     is stored the corresponding shift as given by DLARRE.
                     On exit, L is overwritten.

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

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

           DOL       (input)
                     DOL is INTEGER

           DOU       (input)
                     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    (input)
                     MINRGP is DOUBLE PRECISION

           RTOL1     (input)
                     RTOL1 is DOUBLE PRECISION

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

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

           IBLOCK    (input)
                     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    (input)
                     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      (input)
                     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         (output)
                     Z is COMPLEX*16 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       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z.  LDZ >= 1, and if
                     JOBZ = 'V', LDZ >= max(1,N).

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

           IWORK     (output)
                     IWORK is INTEGER array, dimension (7*N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit

                     > 0:  A problem occured in ZLARRV.
                     < 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.



LAPACK routine                  31 October 2017                      ZLARRV(3)