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



NAME
       DLARRD

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)



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.





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

           VL        (input)
                     VL is DOUBLE PRECISION

           VU        (input)
                     VU is 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)
                     IL is INTEGER

           IU        (input)
                     IU is 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)
                     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)).

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

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

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

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

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

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

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

           WL        (output)
                     WL is DOUBLE PRECISION

           WU        (output)
                     WU is 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)
                     IBLOCK is 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)
                     INDEXW is 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      (output)
                     WORK is DOUBLE PRECISION array, dimension (4*N)

           IWORK     (output)
                     IWORK is INTEGER array, dimension (3*N)

           INFO      (output)
                     INFO is 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.



       Internal 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.



LAPACK routine                  31 October 2017                      DLARRD(3)