Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  find  the desired eigenvalues of a given real symmetric tridiagonal matrix T,
       DLARRE sets any "small" off-diagonal elements to zero, and for each unreduced  block  T_i,
       it finds (a) a suitable shift at one end of the block's spectrum,

SYNOPSIS

       SUBROUTINE DLARRE( RANGE,  N,  VL,  VU,  IL,  IU,  D, E, E2, RTOL1, RTOL2, SPLTOL, NSPLIT,
                          ISPLIT, M, W, WERR, WGAP, IBLOCK, INDEXW, GERS,  PIVMIN,  WORK,  IWORK,
                          INFO )

           IMPLICIT       NONE

           CHARACTER      RANGE

           INTEGER        IL, INFO, IU, M, N, NSPLIT

           DOUBLE         PRECISION PIVMIN, RTOL1, RTOL2, SPLTOL, VL, VU

           INTEGER        IBLOCK( * ), ISPLIT( * ), IWORK( * ), INDEXW( * )

           DOUBLE         PRECISION D( * ), E( * ), E2( * ), GERS( * ), W( * ),WERR( * ), WGAP( *
                          ), WORK( * )

PURPOSE

       To find the desired eigenvalues of a given real symmetric  tridiagonal  matrix  T,  DLARRE
       sets any "small" off-diagonal elements to zero, and for each unreduced block T_i, it finds
       (a) a suitable shift at one end of the block's spectrum,
        (b) the base representation, T_i - sigma_i I = L_i D_i L_i^T, and
        (c) eigenvalues of each L_i D_i L_i^T.
        The representations and eigenvalues found are then used by
        DSTEMR to compute the eigenvectors of T.
        The accuracy varies depending on whether bisection is used to
        find a few eigenvalues or the dqds algorithm (subroutine DLASQ2) to
        conpute all and then discard any unwanted one.
        As an added benefit, DLARRE also outputs the n
        Gerschgorin intervals for the matrices L_i D_i L_i^T.

ARGUMENTS

        RANGE   (input) CHARACTER*1
                = 'A': ("All")   all eigenvalues will be found.
                = 'V': ("Value") all eigenvalues in the half-open interval
                (VL, VU] will be found.
                = 'I': ("Index") the IL-th through IU-th eigenvalues (of the
                entire matrix) will be found.

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

        VL      (input/output) DOUBLE PRECISION
                VU      (input/output) DOUBLE PRECISION
                If RANGE='V', the lower and upper bounds for the eigenvalues.
                Eigenvalues less than or equal to VL, or greater than VU,
                will not be returned.  VL < VU.
                If RANGE='I' or ='A', DLARRE computes bounds on the desired
                part of the spectrum.

        IL      (input) INTEGER
                IU      (input) INTEGER
                If RANGE='I', the indices (in ascending order) of the
                smallest and largest eigenvalues to be returned.
                1 <= IL <= IU <= N.

        D       (input/output) DOUBLE PRECISION array, dimension (N)
                On entry, the N diagonal elements of the tridiagonal
                matrix T.
                On exit, the N diagonal elements of the diagonal
                matrices D_i.

        E       (input/output) DOUBLE PRECISION array, dimension (N)
                On entry, the first (N-1) entries contain the subdiagonal
                elements of the tridiagonal matrix T; E(N) need not be set.
                On exit, E contains the subdiagonal elements of the unit
                bidiagonal matrices L_i. The entries E( ISPLIT( I ) ),
                1 <= I <= NSPLIT, contain the base points sigma_i on output.

        E2      (input/output) DOUBLE PRECISION array, dimension (N)
                On entry, the first (N-1) entries contain the SQUARES of the
                subdiagonal elements of the tridiagonal matrix T;
                E2(N) need not be set.
                On exit, the entries E2( ISPLIT( I ) ),
                1 <= I <= NSPLIT, have been set to zero

        RTOL1   (input) DOUBLE PRECISION
                RTOL2   (input) DOUBLE PRECISION
                Parameters for bisection.
                RIGHT-LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

        SPLTOL  (input) DOUBLE PRECISION
                The threshold for splitting.

        NSPLIT  (output) INTEGER
                The number of blocks T splits into. 1 <= NSPLIT <= N.

        ISPLIT  (output) 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., and the NSPLIT-th consists of rows/columns
                ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.

        M       (output) INTEGER
                The total number of eigenvalues (of all L_i D_i L_i^T)
                found.

        W       (output) DOUBLE PRECISION array, dimension (N)
                The first M elements contain the eigenvalues. The
                eigenvalues of each of the blocks, L_i D_i L_i^T, are
                sorted in ascending order ( DLARRE may use the
                remaining N-M elements as workspace).

        WERR    (output) DOUBLE PRECISION array, dimension (N)
                The error bound on the corresponding eigenvalue in W.

        WGAP    (output) DOUBLE PRECISION array, dimension (N)
                The separation from the right neighbor eigenvalue in W.
                The gap is only with respect to the eigenvalues of the same block
                as each block has its own representation tree.
                Exception: at the right end of a block we store the left gap

        IBLOCK  (output) 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  (output) 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 block 2

        GERS    (output) DOUBLE PRECISION array, dimension (2*N)
                The N Gerschgorin intervals (the i-th Gerschgorin interval
                is (GERS(2*i-1), GERS(2*i)).

        PIVMIN  (output) DOUBLE PRECISION
                The minimum pivot in the Sturm sequence for T.

        WORK    (workspace) DOUBLE PRECISION array, dimension (6*N)
                Workspace.

        IWORK   (workspace) INTEGER array, dimension (5*N)
                Workspace.

        INFO    (output) INTEGER
                = 0:  successful exit
                > 0:  A problem occured in DLARRE.
                < 0:  One of the called subroutines signaled an internal problem.
                Needs inspection of the corresponding parameter IINFO
                for further information.

        =-1:  Problem in DLARRD.
              = 2:  No base representation could be found in MAXTRY iterations.
              Increasing MAXTRY and recompilation might be a remedy.
              =-3:  Problem in DLARRB when computing the refined root
              representation for DLASQ2.
              =-4:  Problem in DLARRB when preforming bisection on the
              desired part of the spectrum.
              =-5:  Problem in DLASQ2.
              =-6:  Problem in DLASQ2.
              Further Details
              element growth and consequently define all their eigenvalues to
              high relative accuracy.
              ===============
              Based on contributions by
              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

 LAPACK auxiliary routine (version 3.3.1)   April 2011                            DLARRE(3lapack)