1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227
|
*> \brief \b DLANEG
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLANEG + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaneg.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaneg.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaneg.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* .. Scalar Arguments ..
* INTEGER N, R
* DOUBLE PRECISION PIVMIN, SIGMA
* ..
* .. Array Arguments ..
* DOUBLE PRECISION D( * ), LLD( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLANEG computes the Sturm count, the number of negative pivots
*> encountered while factoring tridiagonal T - sigma I = L D L^T.
*> This implementation works directly on the factors without forming
*> the tridiagonal matrix T. The Sturm count is also the number of
*> eigenvalues of T less than sigma.
*>
*> This routine is called from DLARRB.
*>
*> The current routine does not use the PIVMIN parameter but rather
*> requires IEEE-754 propagation of Infinities and NaNs. This
*> routine also has no input range restrictions but does require
*> default exception handling such that x/0 produces Inf when x is
*> non-zero, and Inf/Inf produces NaN. For more information, see:
*>
*> Marques, Riedy, and Voemel, "Benefits of IEEE-754 Features in
*> Modern Symmetric Tridiagonal Eigensolvers," SIAM Journal on
*> Scientific Computing, v28, n5, 2006. DOI 10.1137/050641624
*> (Tech report version in LAWN 172 with the same title.)
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is DOUBLE PRECISION array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is DOUBLE PRECISION array, dimension (N-1)
*> The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is DOUBLE PRECISION
*> Shift amount in T - sigma I = L D L^T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is DOUBLE PRECISION
*> The minimum pivot in the Sturm sequence. May be used
*> when zero pivots are encountered on non-IEEE-754
*> architectures.
*> \endverbatim
*>
*> \param[in] R
*> \verbatim
*> R is INTEGER
*> The twist index for the twisted factorization that is used
*> for the negcount.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
*> \par Contributors:
* ==================
*>
*> Osni Marques, LBNL/NERSC, USA \n
*> Christof Voemel, University of California, Berkeley, USA \n
*> Jason Riedy, University of California, Berkeley, USA \n
*>
* =====================================================================
INTEGER FUNCTION DLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* -- LAPACK auxiliary routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER N, R
DOUBLE PRECISION PIVMIN, SIGMA
* ..
* .. Array Arguments ..
DOUBLE PRECISION D( * ), LLD( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* Some architectures propagate Infinities and NaNs very slowly, so
* the code computes counts in BLKLEN chunks. Then a NaN can
* propagate at most BLKLEN columns before being detected. This is
* not a general tuning parameter; it needs only to be just large
* enough that the overhead is tiny in common cases.
INTEGER BLKLEN
PARAMETER ( BLKLEN = 128 )
* ..
* .. Local Scalars ..
INTEGER BJ, J, NEG1, NEG2, NEGCNT
DOUBLE PRECISION BSAV, DMINUS, DPLUS, GAMMA, P, T, TMP
LOGICAL SAWNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN, MAX
* ..
* .. External Functions ..
LOGICAL DISNAN
EXTERNAL DISNAN
* ..
* .. Executable Statements ..
NEGCNT = 0
* I) upper part: L D L^T - SIGMA I = L+ D+ L+^T
T = -SIGMA
DO 210 BJ = 1, R-1, BLKLEN
NEG1 = 0
BSAV = T
DO 21 J = BJ, MIN(BJ+BLKLEN-1, R-1)
DPLUS = D( J ) + T
IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
TMP = T / DPLUS
T = TMP * LLD( J ) - SIGMA
21 CONTINUE
SAWNAN = DISNAN( T )
* Run a slower version of the above loop if a NaN is detected.
* A NaN should occur only with a zero pivot after an infinite
* pivot. In that case, substituting 1 for T/DPLUS is the
* correct limit.
IF( SAWNAN ) THEN
NEG1 = 0
T = BSAV
DO 22 J = BJ, MIN(BJ+BLKLEN-1, R-1)
DPLUS = D( J ) + T
IF( DPLUS.LT.ZERO ) NEG1 = NEG1 + 1
TMP = T / DPLUS
IF (DISNAN(TMP)) TMP = ONE
T = TMP * LLD(J) - SIGMA
22 CONTINUE
END IF
NEGCNT = NEGCNT + NEG1
210 CONTINUE
*
* II) lower part: L D L^T - SIGMA I = U- D- U-^T
P = D( N ) - SIGMA
DO 230 BJ = N-1, R, -BLKLEN
NEG2 = 0
BSAV = P
DO 23 J = BJ, MAX(BJ-BLKLEN+1, R), -1
DMINUS = LLD( J ) + P
IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
TMP = P / DMINUS
P = TMP * D( J ) - SIGMA
23 CONTINUE
SAWNAN = DISNAN( P )
* As above, run a slower version that substitutes 1 for Inf/Inf.
*
IF( SAWNAN ) THEN
NEG2 = 0
P = BSAV
DO 24 J = BJ, MAX(BJ-BLKLEN+1, R), -1
DMINUS = LLD( J ) + P
IF( DMINUS.LT.ZERO ) NEG2 = NEG2 + 1
TMP = P / DMINUS
IF (DISNAN(TMP)) TMP = ONE
P = TMP * D(J) - SIGMA
24 CONTINUE
END IF
NEGCNT = NEGCNT + NEG2
230 CONTINUE
*
* III) Twist index
* T was shifted by SIGMA initially.
GAMMA = (T + SIGMA) + P
IF( GAMMA.LT.ZERO ) NEGCNT = NEGCNT+1
DLANEG = NEGCNT
END
|