dlarrd.f

          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 )
    *
    *  -- LAPACK auxiliary routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          ORDER, RANGE
          INTEGER            IL, INFO, IU, M, N, NSPLIT
          DOUBLE PRECISION    PIVMIN, RELTOL, VL, VU, WL, WU
    *     ..
    *     .. Array Arguments ..
          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
    *          = '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
    *          = '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.
    *
    *  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.
    *
    *  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
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE, TWO, HALF, FUDGE
          PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0,
         $                     TWO = 2.0D0, HALF = ONE/TWO,
         $                     FUDGE = TWO )
          INTEGER   ALLRNG, VALRNG, INDRNG
          PARAMETER ( ALLRNG = 1, VALRNG = 2, INDRNG = 3 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            NCNVRG, TOOFEW
          INTEGER            I, IB, IBEGIN, IDISCL, IDISCU, IE, IEND, IINFO,
         $                   IM, IN, IOFF, IOUT, IRANGE, ITMAX, ITMP1,
         $                   ITMP2, IW, IWOFF, J, JBLK, JDISC, JE, JEE, NB,
         $                   NWL, NWU
          DOUBLE PRECISION   ATOLI, EPS, GL, GU, RTOLI, SPDIAM, TMP1, TMP2,
         $                   TNORM, UFLOW, WKILL, WLU, WUL

    *     ..
    *     .. Local Arrays ..
          INTEGER            IDUMMA( 1 )
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH
          EXTERNAL           LSAME, ILAENV, DLAMCH
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DLAEBZ
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, INT, LOG, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
          INFO = 0
    *
    *     Decode RANGE
    *
          IF( LSAME( RANGE, 'A' ) ) THEN
             IRANGE = ALLRNG
          ELSE IF( LSAME( RANGE, 'V' ) ) THEN
             IRANGE = VALRNG
          ELSE IF( LSAME( RANGE, 'I' ) ) THEN
             IRANGE = INDRNG
          ELSE
             IRANGE = 0
          END IF
    *
    *     Check for Errors
    *
          IF( IRANGE.LE.0 ) THEN
             INFO = -1
          ELSE IF( .NOT.(LSAME(ORDER,'B').OR.LSAME(ORDER,'E')) ) THEN
             INFO = -2
          ELSE IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( IRANGE.EQ.VALRNG ) THEN
             IF( VL.GE.VU )
         $      INFO = -5
          ELSE IF( IRANGE.EQ.INDRNG .AND.
         $        ( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) ) THEN
             INFO = -6
          ELSE IF( IRANGE.EQ.INDRNG .AND.
         $        ( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) ) THEN
             INFO = -7
          END IF
    *
          IF( INFO.NE.0 ) THEN
             RETURN
          END IF

    *     Initialize error flags
          INFO = 0
          NCNVRG = .FALSE.
          TOOFEW = .FALSE.

    *     Quick return if possible
          M = 0
          IF( N.EQ.0 ) RETURN

    *     Simplification:
          IF( IRANGE.EQ.INDRNG .AND. IL.EQ.1 .AND. IU.EQ.N ) IRANGE = 1

    *     Get machine constants
          EPS = DLAMCH( 'P' )
          UFLOW = DLAMCH( 'U' )

    *     Special Case when N=1
    *     Treat case of 1x1 matrix for quick return
          IF( N.EQ.1 ) THEN
             IF( (IRANGE.EQ.ALLRNG).OR.
         $       ((IRANGE.EQ.VALRNG).AND.(D(1).GT.VL).AND.(D(1).LE.VU)).OR.
         $       ((IRANGE.EQ.INDRNG).AND.(IL.EQ.1).AND.(IU.EQ.1)) ) THEN
                M = 1
                W(1) = D(1)
    *           The computation error of the eigenvalue is zero
                WERR(1) = ZERO
                IBLOCK( 1 ) = 1
                INDEXW( 1 ) = 1
             ENDIF
             RETURN
          END IF

    *     NB is the minimum vector length for vector bisection, or 0
    *     if only scalar is to be done.
          NB = ILAENV( 1, 'DSTEBZ', ' ', N, -1, -1, -1 )
          IF( NB.LE.1 ) NB = 0

    *     Find global spectral radius
          GL = D(1)
          GU = D(1)
          DO 5 I = 1,N
             GL =  MIN( GL, GERS( 2*I - 1))
             GU = MAX( GU, GERS(2*I) )
     5    CONTINUE
    *     Compute global Gerschgorin bounds and spectral diameter
          TNORM = MAX( ABS( GL ), ABS( GU ) )
          GL = GL - FUDGE*TNORM*EPS*N - FUDGE*TWO*PIVMIN
          GU = GU + FUDGE*TNORM*EPS*N + FUDGE*TWO*PIVMIN
          SPDIAM = GU - GL
    *     Input arguments for DLAEBZ:
    *     The relative tolerance.  An interval (a,b] lies within
    *     "relative tolerance" if  b-a < RELTOL*max(|a|,|b|),
          RTOLI = RELTOL
    *     Set the absolute tolerance for interval convergence to zero to force
    *     interval convergence based on relative size of the interval.
    *     This is dangerous because intervals might not converge when RELTOL is
    *     small. But at least a very small number should be selected so that for
    *     strongly graded matrices, the code can get relatively accurate
    *     eigenvalues.
          ATOLI = FUDGE*TWO*UFLOW + FUDGE*TWO*PIVMIN

          IF( IRANGE.EQ.INDRNG ) THEN

    *        RANGE='I': Compute an interval containing eigenvalues
    *        IL through IU. The initial interval [GL,GU] from the global
    *        Gerschgorin bounds GL and GU is refined by DLAEBZ.
             ITMAX = INT( ( LOG( TNORM+PIVMIN )-LOG( PIVMIN ) ) /
         $           LOG( TWO ) ) + 2
             WORK( N+1 ) = GL
             WORK( N+2 ) = GL
             WORK( N+3 ) = GU
             WORK( N+4 ) = GU
             WORK( N+5 ) = GL
             WORK( N+6 ) = GU
             IWORK( 1 ) = -1
             IWORK( 2 ) = -1
             IWORK( 3 ) = N + 1
             IWORK( 4 ) = N + 1
             IWORK( 5 ) = IL - 1
             IWORK( 6 ) = IU
    *
             CALL DLAEBZ( 3, ITMAX, N, 2, 2, NB, ATOLI, RTOLI, PIVMIN,
         $         D, E, E2, IWORK( 5 ), WORK( N+1 ), WORK( N+5 ), IOUT,
         $                IWORK, W, IBLOCK, IINFO )
             IF( IINFO .NE. 0 ) THEN
                INFO = IINFO
                RETURN
             END IF
    *        On exit, output intervals may not be ordered by ascending negcount
             IF( IWORK( 6 ).EQ.IU ) THEN
                WL = WORK( N+1 )
                WLU = WORK( N+3 )
                NWL = IWORK( 1 )
                WU = WORK( N+4 )
                WUL = WORK( N+2 )
                NWU = IWORK( 4 )
             ELSE
                WL = WORK( N+2 )
                WLU = WORK( N+4 )
                NWL = IWORK( 2 )
                WU = WORK( N+3 )
                WUL = WORK( N+1 )
                NWU = IWORK( 3 )
             END IF
    *        On exit, the interval [WL, WLU] contains a value with negcount NWL,
    *        and [WUL, WU] contains a value with negcount NWU.
             IF( NWL.LT.0 .OR. NWL.GE.N .OR. NWU.LT.1 .OR. NWU.GT.N ) THEN
                INFO = 4
                RETURN
             END IF

          ELSEIF( IRANGE.EQ.VALRNG ) THEN
             WL = VL
             WU = VU

          ELSEIF( IRANGE.EQ.ALLRNG ) THEN
             WL = GL
             WU = GU
          ENDIF

    *     Find Eigenvalues -- Loop Over blocks and recompute NWL and NWU.
    *     NWL accumulates the number of eigenvalues .le. WL,
    *     NWU accumulates the number of eigenvalues .le. WU
          M = 0
          IEND = 0
          INFO = 0
          NWL = 0
          NWU = 0
    *
          DO 70 JBLK = 1, NSPLIT
             IOFF = IEND
             IBEGIN = IOFF + 1
             IEND = ISPLIT( JBLK )
             IN = IEND - IOFF
    *
             IF( IN.EQ.1 ) THEN
    *           1x1 block
                IF( WL.GE.D( IBEGIN )-PIVMIN )
         $         NWL = NWL + 1
                IF( WU.GE.D( IBEGIN )-PIVMIN )
         $         NWU = NWU + 1
                IF( IRANGE.EQ.ALLRNG .OR.
         $           ( WL.LT.D( IBEGIN )-PIVMIN
         $             .AND. WU.GE. D( IBEGIN )-PIVMIN ) ) THEN
                   M = M + 1
                   W( M ) = D( IBEGIN )
                   WERR(M) = ZERO
    *              The gap for a single block doesn't matter for the later
    *              algorithm and is assigned an arbitrary large value
                   IBLOCK( M ) = JBLK
                   INDEXW( M ) = 1
                END IF

    *        Disabled 2x2 case because of a failure on the following matrix
    *        RANGE = 'I', IL = IU = 4
    *          Original Tridiagonal, d = [
    *           -0.150102010615740E+00
    *           -0.849897989384260E+00
    *           -0.128208148052635E-15
    *            0.128257718286320E-15
    *          ];
    *          e = [
    *           -0.357171383266986E+00
    *           -0.180411241501588E-15
    *           -0.175152352710251E-15
    *          ];
    *
    *         ELSE IF( IN.EQ.2 ) THEN
    **           2x2 block
    *            DISC = SQRT( (HALF*(D(IBEGIN)-D(IEND)))**2 + E(IBEGIN)**2 )
    *            TMP1 = HALF*(D(IBEGIN)+D(IEND))
    *            L1 = TMP1 - DISC
    *            IF( WL.GE. L1-PIVMIN )
    *     $         NWL = NWL + 1
    *            IF( WU.GE. L1-PIVMIN )
    *     $         NWU = NWU + 1
    *            IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L1-PIVMIN .AND. WU.GE.
    *     $          L1-PIVMIN ) ) THEN
    *               M = M + 1
    *               W( M ) = L1
    **              The uncertainty of eigenvalues of a 2x2 matrix is very small
    *               WERR( M ) = EPS * ABS( W( M ) ) * TWO
    *               IBLOCK( M ) = JBLK
    *               INDEXW( M ) = 1
    *            ENDIF
    *            L2 = TMP1 + DISC
    *            IF( WL.GE. L2-PIVMIN )
    *     $         NWL = NWL + 1
    *            IF( WU.GE. L2-PIVMIN )
    *     $         NWU = NWU + 1
    *            IF( IRANGE.EQ.ALLRNG .OR. ( WL.LT.L2-PIVMIN .AND. WU.GE.
    *     $          L2-PIVMIN ) ) THEN
    *               M = M + 1
    *               W( M ) = L2
    **              The uncertainty of eigenvalues of a 2x2 matrix is very small
    *               WERR( M ) = EPS * ABS( W( M ) ) * TWO
    *               IBLOCK( M ) = JBLK
    *               INDEXW( M ) = 2
    *            ENDIF
             ELSE
    *           General Case - block of size IN >= 2
    *           Compute local Gerschgorin interval and use it as the initial
    *           interval for DLAEBZ
                GU = D( IBEGIN )
                GL = D( IBEGIN )
                TMP1 = ZERO

                DO 40 J = IBEGIN, IEND
                   GL =  MIN( GL, GERS( 2*J - 1))
                   GU = MAX( GU, GERS(2*J) )
       40       CONTINUE
                SPDIAM = GU - GL
                GL = GL - FUDGE*SPDIAM*EPS*IN - FUDGE*PIVMIN
                GU = GU + FUDGE*SPDIAM*EPS*IN + FUDGE*PIVMIN
    *
                IF( IRANGE.GT.1 ) THEN
                   IF( GU.LT.WL ) THEN
    *                 the local block contains none of the wanted eigenvalues
                      NWL = NWL + IN
                      NWU = NWU + IN
                      GO TO 70
                   END IF
    *              refine search interval if possible, only range (WL,WU] matters
                   GL = MAX( GL, WL )
                   GU = MIN( GU, WU )
                   IF( GL.GE.GU )
         $            GO TO 70
                END IF

    *           Find negcount of initial interval boundaries GL and GU
                WORK( N+1 ) = GL
                WORK( N+IN+1 ) = GU
                CALL DLAEBZ( 1, 0, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
         $                   D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
         $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IM,
         $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
                IF( IINFO .NE. 0 ) THEN
                   INFO = IINFO
                   RETURN
                END IF
    *
                NWL = NWL + IWORK( 1 )
                NWU = NWU + IWORK( IN+1 )
                IWOFF = M - IWORK( 1 )

    *           Compute Eigenvalues
                ITMAX = INT( ( LOG( GU-GL+PIVMIN )-LOG( PIVMIN ) ) /
         $              LOG( TWO ) ) + 2
                CALL DLAEBZ( 2, ITMAX, IN, IN, 1, NB, ATOLI, RTOLI, PIVMIN,
         $                   D( IBEGIN ), E( IBEGIN ), E2( IBEGIN ),
         $                   IDUMMA, WORK( N+1 ), WORK( N+2*IN+1 ), IOUT,
         $                   IWORK, W( M+1 ), IBLOCK( M+1 ), IINFO )
                IF( IINFO .NE. 0 ) THEN
                   INFO = IINFO
                   RETURN
                END IF
    *
    *           Copy eigenvalues into W and IBLOCK
    *           Use -JBLK for block number for unconverged eigenvalues.
    *           Loop over the number of output intervals from DLAEBZ
                DO 60 J = 1, IOUT
    *              eigenvalue approximation is middle point of interval
                   TMP1 = HALF*( WORK( J+N )+WORK( J+IN+N ) )
    *              semi length of error interval
                   TMP2 = HALF*ABS( WORK( J+N )-WORK( J+IN+N ) )
                   IF( J.GT.IOUT-IINFO ) THEN
    *                 Flag non-convergence.
                      NCNVRG = .TRUE.
                      IB = -JBLK
                   ELSE
                      IB = JBLK
                   END IF
                   DO 50 JE = IWORK( J ) + 1 + IWOFF,
         $                 IWORK( J+IN ) + IWOFF
                      W( JE ) = TMP1
                      WERR( JE ) = TMP2
                      INDEXW( JE ) = JE - IWOFF
                      IBLOCK( JE ) = IB
       50          CONTINUE
       60       CONTINUE
    *
                M = M + IM
             END IF
       70 CONTINUE

    *     If RANGE='I', then (WL,WU) contains eigenvalues NWL+1,...,NWU
    *     If NWL+1 < IL or NWU > IU, discard extra eigenvalues.
          IF( IRANGE.EQ.INDRNG ) THEN
             IDISCL = IL - 1 - NWL
             IDISCU = NWU - IU
    *
             IF( IDISCL.GT.0 ) THEN
                IM = 0
                DO 80 JE = 1, M
    *              Remove some of the smallest eigenvalues from the left so that
    *              at the end IDISCL =0. Move all eigenvalues up to the left.
                   IF( W( JE ).LE.WLU .AND. IDISCL.GT.0 ) THEN
                      IDISCL = IDISCL - 1
                   ELSE
                      IM = IM + 1
                      W( IM ) = W( JE )
                      WERR( IM ) = WERR( JE )
                      INDEXW( IM ) = INDEXW( JE )
                      IBLOCK( IM ) = IBLOCK( JE )
                   END IF
     80         CONTINUE
                M = IM
             END IF
             IF( IDISCU.GT.0 ) THEN
    *           Remove some of the largest eigenvalues from the right so that
    *           at the end IDISCU =0. Move all eigenvalues up to the left.
                IM=M+1
                DO 81 JE = M, 1, -1
                   IF( W( JE ).GE.WUL .AND. IDISCU.GT.0 ) THEN
                      IDISCU = IDISCU - 1
                   ELSE
                      IM = IM - 1
                      W( IM ) = W( JE )
                      WERR( IM ) = WERR( JE )
                      INDEXW( IM ) = INDEXW( JE )
                      IBLOCK( IM ) = IBLOCK( JE )
                   END IF
     81         CONTINUE
                JEE = 0
                DO 82 JE = IM, M
                   JEE = JEE + 1
                   W( JEE ) = W( JE )
                   WERR( JEE ) = WERR( JE )
                   INDEXW( JEE ) = INDEXW( JE )
                   IBLOCK( JEE ) = IBLOCK( JE )
     82         CONTINUE
                M = M-IM+1
             END IF

             IF( IDISCL.GT.0 .OR. IDISCU.GT.0 ) THEN
    *           Code to deal with effects of bad arithmetic. (If N(w) is
    *           monotone non-decreasing, this should never happen.)
    *           Some low eigenvalues to be discarded are not in (WL,WLU],
    *           or high eigenvalues to be discarded are not in (WUL,WU]
    *           so just kill off the smallest IDISCL/largest IDISCU
    *           eigenvalues, by marking the corresponding IBLOCK = 0
                IF( IDISCL.GT.0 ) THEN
                   WKILL = WU
                   DO 100 JDISC = 1, IDISCL
                      IW = 0
                      DO 90 JE = 1, M
                         IF( IBLOCK( JE ).NE.0 .AND.
         $                    ( W( JE ).LT.WKILL .OR. IW.EQ.0 ) ) THEN
                            IW = JE
                            WKILL = W( JE )
                         END IF
     90               CONTINUE
                      IBLOCK( IW ) = 0
     100           CONTINUE
                END IF
                IF( IDISCU.GT.0 ) THEN
                   WKILL = WL
                   DO 120 JDISC = 1, IDISCU
                      IW = 0
                      DO 110 JE = 1, M
                         IF( IBLOCK( JE ).NE.0 .AND.
         $                    ( W( JE ).GE.WKILL .OR. IW.EQ.0 ) ) THEN
                            IW = JE
                            WKILL = W( JE )
                         END IF
     110              CONTINUE
                      IBLOCK( IW ) = 0
     120           CONTINUE
                END IF
    *           Now erase all eigenvalues with IBLOCK set to zero
                IM = 0
                DO 130 JE = 1, M
                   IF( IBLOCK( JE ).NE.0 ) THEN
                      IM = IM + 1
                      W( IM ) = W( JE )
                      WERR( IM ) = WERR( JE )
                      INDEXW( IM ) = INDEXW( JE )
                      IBLOCK( IM ) = IBLOCK( JE )
                   END IF
     130        CONTINUE
                M = IM
             END IF
             IF( IDISCL.LT.0 .OR. IDISCU.LT.0 ) THEN
                TOOFEW = .TRUE.
             END IF
          END IF
    *
          IF(( IRANGE.EQ.ALLRNG .AND. M.NE.N ).OR.
         $   ( IRANGE.EQ.INDRNG .AND. M.NE.IU-IL+1 ) ) THEN
             TOOFEW = .TRUE.
          END IF

    *     If ORDER='B', do nothing the eigenvalues are already sorted by
    *        block.
    *     If ORDER='E', sort the eigenvalues from smallest to largest

          IF( LSAME(ORDER,'E') .AND. NSPLIT.GT.1 ) THEN
             DO 150 JE = 1, M - 1
                IE = 0
                TMP1 = W( JE )
                DO 140 J = JE + 1, M
                   IF( W( J ).LT.TMP1 ) THEN
                      IE = J
                      TMP1 = W( J )
                   END IF
      140       CONTINUE
                IF( IE.NE.0 ) THEN
                   TMP2 = WERR( IE )
                   ITMP1 = IBLOCK( IE )
                   ITMP2 = INDEXW( IE )
                   W( IE ) = W( JE )
                   WERR( IE ) = WERR( JE )
                   IBLOCK( IE ) = IBLOCK( JE )
                   INDEXW( IE ) = INDEXW( JE )
                   W( JE ) = TMP1
                   WERR( JE ) = TMP2
                   IBLOCK( JE ) = ITMP1
                   INDEXW( JE ) = ITMP2
                END IF
      150    CONTINUE
          END IF
    *
          INFO = 0
          IF( NCNVRG )
         $   INFO = INFO + 1
          IF( TOOFEW )
         $   INFO = INFO + 2
          RETURN
    *
    *     End of DLARRD
    *
          END