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

NAME

       LAPACK-3  -  computes  the  eigenvalues  of  a  symmetric tridiagonal matrix T to suitable
       accuracy

SYNOPSIS

       SUBROUTINE DLARRD( RANGE, ORDER, N, VL, VU, IL,  IU,  GERS,  RELTOL,  D,  E,  E2,  PIVMIN,
                          NSPLIT, ISPLIT, M, W, WERR, WL, WU, IBLOCK, INDEXW, WORK, IWORK, INFO )

           CHARACTER      ORDER, RANGE

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

           DOUBLE         PRECISION PIVMIN, RELTOL, VL, VU, WL, WU

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

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

PURPOSE

       DLARRD computes the eigenvalues of a symmetric tridiagonal matrix T to suitable  accuracy.
       This is an auxiliary code to be
        called from DSTEMR.
        The user may ask for all eigenvalues, all eigenvalues
        in the half-open interval (VL, VU], or the IL-th through IU-th
        eigenvalues.
        To avoid overflow, the matrix must be scaled so that its
        largest element is no greater than overflow**(1/2) *
        underflow**(1/4) in absolute value, and for greatest
        accuracy, it should not be much smaller than that.
        See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
        Matrix", Report CS41, Computer Science Dept., Stanford
        University, July 21, 1966.

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.

        ORDER   (input) CHARACTER*1
                = 'B': ("By Block") the eigenvalues will be grouped by
                split-off block (see IBLOCK, ISPLIT) and
                ordered from smallest to largest within
                the block.
                = 'E': ("Entire matrix")
                the eigenvalues for the entire matrix
                will be ordered from smallest to
                largest.

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

        VL      (input) DOUBLE PRECISION
                VU      (input) DOUBLE PRECISION
                If RANGE='V', the lower and upper bounds of the interval to
                be searched for eigenvalues.  Eigenvalues less than or equal
                to VL, or greater than VU, will not be returned.  VL < VU.
                Not referenced if RANGE = 'A' or 'I'.

        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, if N > 0; IL = 1 and IU = 0 if N = 0.
                Not referenced if RANGE = 'A' or 'V'.

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

        RELTOL  (input) DOUBLE PRECISION
                The minimum relative width of an interval.  When an interval
                is narrower than RELTOL times the larger (in
                magnitude) endpoint, then it is considered to be
                sufficiently small, i.e., converged.  Note: this should
                always be at least radix*machine epsilon.

        D       (input) DOUBLE PRECISION array, dimension (N)
                The n diagonal elements of the tridiagonal matrix T.

        E       (input) DOUBLE PRECISION array, dimension (N-1)
                The (n-1) off-diagonal elements of the tridiagonal matrix T.

        E2      (input) DOUBLE PRECISION array, dimension (N-1)
                The (n-1) squared off-diagonal elements of the tridiagonal matrix T.

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

        NSPLIT  (input) INTEGER
                The number of diagonal blocks in the matrix T.
                1 <= NSPLIT <= N.

        ISPLIT  (input) INTEGER array, dimension (N)
                The splitting points, at which T breaks up into submatrices.
                The first submatrix 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.
                (Only the first NSPLIT elements will actually be used, but
                since the user cannot know a priori what value NSPLIT will
                have, N words must be reserved for ISPLIT.)

        M       (output) INTEGER
                The actual number of eigenvalues found. 0 <= M <= N.
                (See also the description of INFO=2,3.)

        W       (output) DOUBLE PRECISION array, dimension (N)
                On exit, the first M elements of W will contain the
                eigenvalue approximations. DLARRD computes an interval
                I_j = (a_j, b_j] that includes eigenvalue j. The eigenvalue
                approximation is given as the interval midpoint
                W(j)= ( a_j + b_j)/2. The corresponding error is bounded by
                WERR(j) = abs( a_j - b_j)/2

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

        WL      (output) DOUBLE PRECISION
                WU      (output) DOUBLE PRECISION
                The interval (WL, WU] contains all the wanted eigenvalues.
                If RANGE='V', then WL=VL and WU=VU.
                If RANGE='A', then WL and WU are the global Gerschgorin bounds
                on the spectrum.
                If RANGE='I', then WL and WU are computed by DLAEBZ from the
                index range specified.

        IBLOCK  (output) INTEGER array, dimension (N)
                At each row/column j where E(j) is zero or small, the
                matrix T is considered to split into a block diagonal
                matrix.  On exit, if INFO = 0, IBLOCK(i) specifies to which
                block (from 1 to the number of blocks) the eigenvalue W(i)
                belongs.  (DLARRD may use the remaining N-M elements as
                workspace.)

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

        WORK    (workspace) DOUBLE PRECISION array, dimension (4*N)

        IWORK   (workspace) INTEGER array, dimension (3*N)

        INFO    (output) INTEGER
                = 0:  successful exit
                < 0:  if INFO = -i, the i-th argument had an illegal value
                > 0:  some or all of the eigenvalues failed to converge or
                were not computed:
                =1 or 3: Bisection failed to converge for some
                eigenvalues; these eigenvalues are flagged by a
                negative block number.  The effect is that the
                eigenvalues may not be as accurate as the
                absolute and relative tolerances.  This is
                generally caused by unexpectedly inaccurate
                arithmetic.
                =2 or 3: RANGE='I' only: Not all of the eigenvalues
                IL:IU were found.
                Effect: M < IU+1-IL
                Cause:  non-monotonic arithmetic, causing the
                Sturm sequence to be non-monotonic.
                Cure:   recalculate, using RANGE='A', and pick
                out eigenvalues IL:IU.  In some cases,
                increasing the PARAMETER "FUDGE" may
                make things work.
                = 4:    RANGE='I', and the Gershgorin interval
                initially used was too small.  No eigenvalues
                were computed.
                Probable cause: your machine has sloppy
                floating-point arithmetic.
                Cure: Increase the PARAMETER "FUDGE",
                recompile, and try again.

PARAMETERS

        FUDGE   DOUBLE PRECISION, default = 2
                A "fudge factor" to widen the Gershgorin intervals.  Ideally,
                a value of 1 should work, but on machines with sloppy
                arithmetic, this needs to be larger.  The default for
                publicly released versions should be large enough to handle
                the worst machine around.  Note that this has no effect
                on accuracy of the solution.
                Based on contributions by
                W. Kahan, University of California, Berkeley, USA
                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 3.3.0)   April 2011                            DLARRD(3lapack)