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    1: *> \brief \b DLARRA computes the splitting points with the specified threshold.
    2: *
    3: *  =========== DOCUMENTATION ===========
    4: *
    5: * Online html documentation available at
    6: *            http://www.netlib.org/lapack/explore-html/
    7: *
    8: *> \htmlonly
    9: *> Download DLARRA + dependencies
   10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarra.f">
   11: *> [TGZ]</a>
   12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarra.f">
   13: *> [ZIP]</a>
   14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarra.f">
   15: *> [TXT]</a>
   16: *> \endhtmlonly
   17: *
   18: *  Definition:
   19: *  ===========
   20: *
   21: *       SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
   22: *                           NSPLIT, ISPLIT, INFO )
   23: *
   24: *       .. Scalar Arguments ..
   25: *       INTEGER            INFO, N, NSPLIT
   26: *       DOUBLE PRECISION    SPLTOL, TNRM
   27: *       ..
   28: *       .. Array Arguments ..
   29: *       INTEGER            ISPLIT( * )
   30: *       DOUBLE PRECISION   D( * ), E( * ), E2( * )
   31: *       ..
   32: *
   33: *
   34: *> \par Purpose:
   35: *  =============
   36: *>
   37: *> \verbatim
   38: *>
   39: *> Compute the splitting points with threshold SPLTOL.
   40: *> DLARRA sets any "small" off-diagonal elements to zero.
   41: *> \endverbatim
   42: *
   43: *  Arguments:
   44: *  ==========
   45: *
   46: *> \param[in] N
   47: *> \verbatim
   48: *>          N is INTEGER
   49: *>          The order of the matrix. N > 0.
   50: *> \endverbatim
   51: *>
   52: *> \param[in] D
   53: *> \verbatim
   54: *>          D is DOUBLE PRECISION array, dimension (N)
   55: *>          On entry, the N diagonal elements of the tridiagonal
   56: *>          matrix T.
   57: *> \endverbatim
   58: *>
   59: *> \param[in,out] E
   60: *> \verbatim
   61: *>          E is DOUBLE PRECISION array, dimension (N)
   62: *>          On entry, the first (N-1) entries contain the subdiagonal
   63: *>          elements of the tridiagonal matrix T; E(N) need not be set.
   64: *>          On exit, the entries E( ISPLIT( I ) ), 1 <= I <= NSPLIT,
   65: *>          are set to zero, the other entries of E are untouched.
   66: *> \endverbatim
   67: *>
   68: *> \param[in,out] E2
   69: *> \verbatim
   70: *>          E2 is DOUBLE PRECISION array, dimension (N)
   71: *>          On entry, the first (N-1) entries contain the SQUARES of the
   72: *>          subdiagonal elements of the tridiagonal matrix T;
   73: *>          E2(N) need not be set.
   74: *>          On exit, the entries E2( ISPLIT( I ) ),
   75: *>          1 <= I <= NSPLIT, have been set to zero
   76: *> \endverbatim
   77: *>
   78: *> \param[in] SPLTOL
   79: *> \verbatim
   80: *>          SPLTOL is DOUBLE PRECISION
   81: *>          The threshold for splitting. Two criteria can be used:
   82: *>          SPLTOL<0 : criterion based on absolute off-diagonal value
   83: *>          SPLTOL>0 : criterion that preserves relative accuracy
   84: *> \endverbatim
   85: *>
   86: *> \param[in] TNRM
   87: *> \verbatim
   88: *>          TNRM is DOUBLE PRECISION
   89: *>          The norm of the matrix.
   90: *> \endverbatim
   91: *>
   92: *> \param[out] NSPLIT
   93: *> \verbatim
   94: *>          NSPLIT is INTEGER
   95: *>          The number of blocks T splits into. 1 <= NSPLIT <= N.
   96: *> \endverbatim
   97: *>
   98: *> \param[out] ISPLIT
   99: *> \verbatim
  100: *>          ISPLIT is INTEGER array, dimension (N)
  101: *>          The splitting points, at which T breaks up into blocks.
  102: *>          The first block consists of rows/columns 1 to ISPLIT(1),
  103: *>          the second of rows/columns ISPLIT(1)+1 through ISPLIT(2),
  104: *>          etc., and the NSPLIT-th consists of rows/columns
  105: *>          ISPLIT(NSPLIT-1)+1 through ISPLIT(NSPLIT)=N.
  106: *> \endverbatim
  107: *>
  108: *> \param[out] INFO
  109: *> \verbatim
  110: *>          INFO is INTEGER
  111: *>          = 0:  successful exit
  112: *> \endverbatim
  113: *
  114: *  Authors:
  115: *  ========
  116: *
  117: *> \author Univ. of Tennessee
  118: *> \author Univ. of California Berkeley
  119: *> \author Univ. of Colorado Denver
  120: *> \author NAG Ltd.
  121: *
  122: *> \date June 2017
  123: *
  124: *> \ingroup OTHERauxiliary
  125: *
  126: *> \par Contributors:
  127: *  ==================
  128: *>
  129: *> Beresford Parlett, University of California, Berkeley, USA \n
  130: *> Jim Demmel, University of California, Berkeley, USA \n
  131: *> Inderjit Dhillon, University of Texas, Austin, USA \n
  132: *> Osni Marques, LBNL/NERSC, USA \n
  133: *> Christof Voemel, University of California, Berkeley, USA
  134: *
  135: *  =====================================================================
  136:       SUBROUTINE DLARRA( N, D, E, E2, SPLTOL, TNRM,
  137:      $                    NSPLIT, ISPLIT, INFO )
  138: *
  139: *  -- LAPACK auxiliary routine (version 3.7.1) --
  140: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
  141: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
  142: *     June 2017
  143: *
  144: *     .. Scalar Arguments ..
  145:       INTEGER            INFO, N, NSPLIT
  146:       DOUBLE PRECISION    SPLTOL, TNRM
  147: *     ..
  148: *     .. Array Arguments ..
  149:       INTEGER            ISPLIT( * )
  150:       DOUBLE PRECISION   D( * ), E( * ), E2( * )
  151: *     ..
  152: *
  153: *  =====================================================================
  154: *
  155: *     .. Parameters ..
  156:       DOUBLE PRECISION   ZERO
  157:       PARAMETER          ( ZERO = 0.0D0 )
  158: *     ..
  159: *     .. Local Scalars ..
  160:       INTEGER            I
  161:       DOUBLE PRECISION   EABS, TMP1
  162: 
  163: *     ..
  164: *     .. Intrinsic Functions ..
  165:       INTRINSIC          ABS
  166: *     ..
  167: *     .. Executable Statements ..
  168: *
  169:       INFO = 0
  170: *
  171: *     Quick return if possible
  172: *
  173:       IF( N.LE.0 ) THEN
  174:          RETURN
  175:       END IF
  176: *
  177: *     Compute splitting points
  178:       NSPLIT = 1
  179:       IF(SPLTOL.LT.ZERO) THEN
  180: *        Criterion based on absolute off-diagonal value
  181:          TMP1 = ABS(SPLTOL)* TNRM
  182:          DO 9 I = 1, N-1
  183:             EABS = ABS( E(I) )
  184:             IF( EABS .LE. TMP1) THEN
  185:                E(I) = ZERO
  186:                E2(I) = ZERO
  187:                ISPLIT( NSPLIT ) = I
  188:                NSPLIT = NSPLIT + 1
  189:             END IF
  190:  9       CONTINUE
  191:       ELSE
  192: *        Criterion that guarantees relative accuracy
  193:          DO 10 I = 1, N-1
  194:             EABS = ABS( E(I) )
  195:             IF( EABS .LE. SPLTOL * SQRT(ABS(D(I)))*SQRT(ABS(D(I+1))) )
  196:      $      THEN
  197:                E(I) = ZERO
  198:                E2(I) = ZERO
  199:                ISPLIT( NSPLIT ) = I
  200:                NSPLIT = NSPLIT + 1
  201:             END IF
  202:  10      CONTINUE
  203:       ENDIF
  204:       ISPLIT( NSPLIT ) = N
  205: 
  206:       RETURN
  207: *
  208: *     End of DLARRA
  209: *
  210:       END