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
|
*> \brief \b SLANEG
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition
* ==========
*
* INTEGER FUNCTION SLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* .. Scalar Arguments ..
* INTEGER N, R
* REAL PIVMIN, SIGMA
* ..
* .. Array Arguments ..
* REAL D( * ), LLD( * )
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> SLANEG 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 SLARRB.
*>
*> 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 REAL array, dimension (N)
*> The N diagonal elements of the diagonal matrix D.
*> \endverbatim
*>
*> \param[in] LLD
*> \verbatim
*> LLD is REAL array, dimension (N-1)
*> The (N-1) elements L(i)*L(i)*D(i).
*> \endverbatim
*>
*> \param[in] SIGMA
*> \verbatim
*> SIGMA is REAL
*> Shift amount in T - sigma I = L D L^T.
*> \endverbatim
*>
*> \param[in] PIVMIN
*> \verbatim
*> PIVMIN is REAL
*> 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
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> Based on contributions by
*> Osni Marques, LBNL/NERSC, USA
*> Christof Voemel, University of California, Berkeley, USA
*> Jason Riedy, University of California, Berkeley, USA
*>
*> \endverbatim
*>
* =====================================================================
INTEGER FUNCTION SLANEG( N, D, LLD, SIGMA, PIVMIN, R )
*
* -- LAPACK auxiliary routine (version 3.2.2) --
* -- 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
REAL PIVMIN, SIGMA
* ..
* .. Array Arguments ..
REAL D( * ), LLD( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* 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
REAL BSAV, DMINUS, DPLUS, GAMMA, P, T, TMP
LOGICAL SAWNAN
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN, MAX
* ..
* .. External Functions ..
LOGICAL SISNAN
EXTERNAL SISNAN
* ..
* .. 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 = SISNAN( 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 (SISNAN(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 = SISNAN( 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 (SISNAN(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
SLANEG = NEGCNT
END
|
