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
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
|
SUBROUTINE DSYTRS( UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* DSYTRS solves a system of linear equations A*X = B with a real
* symmetric matrix A using the factorization A = U*D*U**T or
* A = L*D*L**T computed by DSYTRF.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the details of the factorization are stored
* as an upper or lower triangular matrix.
* = 'U': Upper triangular, form is A = U*D*U**T;
* = 'L': Lower triangular, form is A = L*D*L**T.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrix B. NRHS >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The block diagonal matrix D and the multipliers used to
* obtain the factor U or L as computed by DSYTRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* Details of the interchanges and the block structure of D
* as determined by DSYTRF.
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the solution matrix X.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UPPER
INTEGER J, K, KP
DOUBLE PRECISION AK, AKM1, AKM1K, BK, BKM1, DENOM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DSYTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( UPPER ) THEN
*
* Solve A*X = B, where A = U*D*U'.
*
* First solve U*D*X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 30
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGER( K-1, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K-1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K-1 )
$ CALL DSWAP( NRHS, B( K-1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(U(K)), where U(K) is the transformation
* stored in columns K-1 and K of A.
*
CALL DGER( K-2, NRHS, -ONE, A( 1, K ), 1, B( K, 1 ), LDB,
$ B( 1, 1 ), LDB )
CALL DGER( K-2, NRHS, -ONE, A( 1, K-1 ), 1, B( K-1, 1 ),
$ LDB, B( 1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = A( K-1, K )
AKM1 = A( K-1, K-1 ) / AKM1K
AK = A( K, K ) / AKM1K
DENOM = AKM1*AK - ONE
DO 20 J = 1, NRHS
BKM1 = B( K-1, J ) / AKM1K
BK = B( K, J ) / AKM1K
B( K-1, J ) = ( AK*BKM1-BK ) / DENOM
B( K, J ) = ( AKM1*BK-BKM1 ) / DENOM
20 CONTINUE
K = K - 2
END IF
*
GO TO 10
30 CONTINUE
*
* Next solve U'*X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
40 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 50
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(U'(K)), where U(K) is the transformation
* stored in column K of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ),
$ 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(U'(K+1)), where U(K+1) is the transformation
* stored in columns K and K+1 of A.
*
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB, A( 1, K ),
$ 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', K-1, NRHS, -ONE, B, LDB,
$ A( 1, K+1 ), 1, ONE, B( K+1, 1 ), LDB )
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K + 2
END IF
*
GO TO 40
50 CONTINUE
*
ELSE
*
* Solve A*X = B, where A = L*D*L'.
*
* First solve L*D*X = B, overwriting B with X.
*
* K is the main loop index, increasing from 1 to N in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = 1
60 CONTINUE
*
* If K > N, exit from loop.
*
IF( K.GT.N )
$ GO TO 80
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGER( N-K, NRHS, -ONE, A( K+1, K ), 1, B( K, 1 ),
$ LDB, B( K+1, 1 ), LDB )
*
* Multiply by the inverse of the diagonal block.
*
CALL DSCAL( NRHS, ONE / A( K, K ), B( K, 1 ), LDB )
K = K + 1
ELSE
*
* 2 x 2 diagonal block
*
* Interchange rows K+1 and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K+1 )
$ CALL DSWAP( NRHS, B( K+1, 1 ), LDB, B( KP, 1 ), LDB )
*
* Multiply by inv(L(K)), where L(K) is the transformation
* stored in columns K and K+1 of A.
*
IF( K.LT.N-1 ) THEN
CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K ), 1, B( K, 1 ),
$ LDB, B( K+2, 1 ), LDB )
CALL DGER( N-K-1, NRHS, -ONE, A( K+2, K+1 ), 1,
$ B( K+1, 1 ), LDB, B( K+2, 1 ), LDB )
END IF
*
* Multiply by the inverse of the diagonal block.
*
AKM1K = A( K+1, K )
AKM1 = A( K, K ) / AKM1K
AK = A( K+1, K+1 ) / AKM1K
DENOM = AKM1*AK - ONE
DO 70 J = 1, NRHS
BKM1 = B( K, J ) / AKM1K
BK = B( K+1, J ) / AKM1K
B( K, J ) = ( AK*BKM1-BK ) / DENOM
B( K+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
70 CONTINUE
K = K + 2
END IF
*
GO TO 60
80 CONTINUE
*
* Next solve L'*X = B, overwriting B with X.
*
* K is the main loop index, decreasing from N to 1 in steps of
* 1 or 2, depending on the size of the diagonal blocks.
*
K = N
90 CONTINUE
*
* If K < 1, exit from loop.
*
IF( K.LT.1 )
$ GO TO 100
*
IF( IPIV( K ).GT.0 ) THEN
*
* 1 x 1 diagonal block
*
* Multiply by inv(L'(K)), where L(K) is the transformation
* stored in column K of A.
*
IF( K.LT.N )
$ CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
*
* Interchange rows K and IPIV(K).
*
KP = IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K - 1
ELSE
*
* 2 x 2 diagonal block
*
* Multiply by inv(L'(K-1)), where L(K-1) is the transformation
* stored in columns K-1 and K of A.
*
IF( K.LT.N ) THEN
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K ), 1, ONE, B( K, 1 ), LDB )
CALL DGEMV( 'Transpose', N-K, NRHS, -ONE, B( K+1, 1 ),
$ LDB, A( K+1, K-1 ), 1, ONE, B( K-1, 1 ),
$ LDB )
END IF
*
* Interchange rows K and -IPIV(K).
*
KP = -IPIV( K )
IF( KP.NE.K )
$ CALL DSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
K = K - 2
END IF
*
GO TO 90
100 CONTINUE
END IF
*
RETURN
*
* End of DSYTRS
*
END
|
