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
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
|
*> \brief \b DLA_SYRPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric indefinite matrix.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLA_SYRPVGRW + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dla_syrpvgrw.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dla_syrpvgrw.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dla_syrpvgrw.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* DOUBLE PRECISION FUNCTION DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
* LDAF, IPIV, WORK )
*
* .. Scalar Arguments ..
* CHARACTER*1 UPLO
* INTEGER N, INFO, LDA, LDAF
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*>
*> DLA_SYRPVGRW computes the reciprocal pivot growth factor
*> norm(A)/norm(U). The "max absolute element" norm is used. If this is
*> much less than 1, the stability of the LU factorization of the
*> (equilibrated) matrix A could be poor. This also means that the
*> solution X, estimated condition numbers, and error bounds could be
*> unreliable.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of linear equations, i.e., the order of the
*> matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] INFO
*> \verbatim
*> INFO is INTEGER
*> The value of INFO returned from DSYTRF, .i.e., the pivot in
*> column INFO is exactly 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the N-by-N matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] AF
*> \verbatim
*> AF is DOUBLE PRECISION array, dimension (LDAF,N)
*> The block diagonal matrix D and the multipliers used to
*> obtain the factor U or L as computed by DSYTRF.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> Details of the interchanges and the block structure of D
*> as determined by DSYTRF.
*> \endverbatim
*>
*> \param[in] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (2*N)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup doubleSYcomputational
*
* =====================================================================
DOUBLE PRECISION FUNCTION DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF,
$ LDAF, IPIV, WORK )
*
* -- LAPACK computational 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 ..
CHARACTER*1 UPLO
INTEGER N, INFO, LDA, LDAF
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER NCOLS, I, J, K, KP
DOUBLE PRECISION AMAX, UMAX, RPVGRW, TMP
LOGICAL UPPER
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. External Functions ..
EXTERNAL LSAME, DLASET
LOGICAL LSAME
* ..
* .. Executable Statements ..
*
UPPER = LSAME( 'Upper', UPLO )
IF ( INFO.EQ.0 ) THEN
IF ( UPPER ) THEN
NCOLS = 1
ELSE
NCOLS = N
END IF
ELSE
NCOLS = INFO
END IF
RPVGRW = 1.0D+0
DO I = 1, 2*N
WORK( I ) = 0.0D+0
END DO
*
* Find the max magnitude entry of each column of A. Compute the max
* for all N columns so we can apply the pivot permutation while
* looping below. Assume a full factorization is the common case.
*
IF ( UPPER ) THEN
DO J = 1, N
DO I = 1, J
WORK( N+I ) = MAX( ABS( A( I, J ) ), WORK( N+I ) )
WORK( N+J ) = MAX( ABS( A( I, J ) ), WORK( N+J ) )
END DO
END DO
ELSE
DO J = 1, N
DO I = J, N
WORK( N+I ) = MAX( ABS( A( I, J ) ), WORK( N+I ) )
WORK( N+J ) = MAX( ABS( A( I, J ) ), WORK( N+J ) )
END DO
END DO
END IF
*
* Now find the max magnitude entry of each column of U or L. Also
* permute the magnitudes of A above so they're in the same order as
* the factor.
*
* The iteration orders and permutations were copied from dsytrs.
* Calls to SSWAP would be severe overkill.
*
IF ( UPPER ) THEN
K = N
DO WHILE ( K .LT. NCOLS .AND. K.GT.0 )
IF ( IPIV( K ).GT.0 ) THEN
! 1x1 pivot
KP = IPIV( K )
IF ( KP .NE. K ) THEN
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
END IF
DO I = 1, K
WORK( K ) = MAX( ABS( AF( I, K ) ), WORK( K ) )
END DO
K = K - 1
ELSE
! 2x2 pivot
KP = -IPIV( K )
TMP = WORK( N+K-1 )
WORK( N+K-1 ) = WORK( N+KP )
WORK( N+KP ) = TMP
DO I = 1, K-1
WORK( K ) = MAX( ABS( AF( I, K ) ), WORK( K ) )
WORK( K-1 ) = MAX( ABS( AF( I, K-1 ) ), WORK( K-1 ) )
END DO
WORK( K ) = MAX( ABS( AF( K, K ) ), WORK( K ) )
K = K - 2
END IF
END DO
K = NCOLS
DO WHILE ( K .LE. N )
IF ( IPIV( K ).GT.0 ) THEN
KP = IPIV( K )
IF ( KP .NE. K ) THEN
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
END IF
K = K + 1
ELSE
KP = -IPIV( K )
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
K = K + 2
END IF
END DO
ELSE
K = 1
DO WHILE ( K .LE. NCOLS )
IF ( IPIV( K ).GT.0 ) THEN
! 1x1 pivot
KP = IPIV( K )
IF ( KP .NE. K ) THEN
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
END IF
DO I = K, N
WORK( K ) = MAX( ABS( AF( I, K ) ), WORK( K ) )
END DO
K = K + 1
ELSE
! 2x2 pivot
KP = -IPIV( K )
TMP = WORK( N+K+1 )
WORK( N+K+1 ) = WORK( N+KP )
WORK( N+KP ) = TMP
DO I = K+1, N
WORK( K ) = MAX( ABS( AF( I, K ) ), WORK( K ) )
WORK( K+1 ) = MAX( ABS( AF(I, K+1 ) ), WORK( K+1 ) )
END DO
WORK( K ) = MAX( ABS( AF( K, K ) ), WORK( K ) )
K = K + 2
END IF
END DO
K = NCOLS
DO WHILE ( K .GE. 1 )
IF ( IPIV( K ).GT.0 ) THEN
KP = IPIV( K )
IF ( KP .NE. K ) THEN
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
END IF
K = K - 1
ELSE
KP = -IPIV( K )
TMP = WORK( N+K )
WORK( N+K ) = WORK( N+KP )
WORK( N+KP ) = TMP
K = K - 2
ENDIF
END DO
END IF
*
* Compute the *inverse* of the max element growth factor. Dividing
* by zero would imply the largest entry of the factor's column is
* zero. Than can happen when either the column of A is zero or
* massive pivots made the factor underflow to zero. Neither counts
* as growth in itself, so simply ignore terms with zero
* denominators.
*
IF ( UPPER ) THEN
DO I = NCOLS, N
UMAX = WORK( I )
AMAX = WORK( N+I )
IF ( UMAX /= 0.0D+0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
ELSE
DO I = 1, NCOLS
UMAX = WORK( I )
AMAX = WORK( N+I )
IF ( UMAX /= 0.0D+0 ) THEN
RPVGRW = MIN( AMAX / UMAX, RPVGRW )
END IF
END DO
END IF
DLA_SYRPVGRW = RPVGRW
END
|
