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
|
SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
$ X, LDX, FERR, BERR, WORK, IWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* DGERFS improves the computed solution to a system of linear
* equations and provides error bounds and backward error estimates for
* the solution.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A**T * X = B (Transpose)
* = 'C': A**H * X = B (Conjugate transpose = Transpose)
*
* 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 matrices B and X. NRHS >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The original N-by-N matrix A.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
* The factors L and U from the factorization A = P*L*U
* as computed by DGETRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from DGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by DGETRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) DOUBLE PRECISION array, dimension (NRHS)
* The estimated forward error bound for each solution vector
* X(j) (the j-th column of the solution matrix X).
* If XTRUE is the true solution corresponding to X(j), FERR(j)
* is an estimated upper bound for the magnitude of the largest
* element in (X(j) - XTRUE) divided by the magnitude of the
* largest element in X(j). The estimate is as reliable as
* the estimate for RCOND, and is almost always a slight
* overestimate of the true error.
*
* BERR (output) DOUBLE PRECISION array, dimension (NRHS)
* The componentwise relative backward error of each solution
* vector X(j) (i.e., the smallest relative change in
* any element of A or B that makes X(j) an exact solution).
*
* WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
*
* IWORK (workspace) INTEGER array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Internal Parameters
* ===================
*
* ITMAX is the maximum number of steps of iterative refinement.
*
* =====================================================================
*
* .. Parameters ..
INTEGER ITMAX
PARAMETER ( ITMAX = 5 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D+0 )
DOUBLE PRECISION THREE
PARAMETER ( THREE = 3.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
CHARACTER TRANST
INTEGER COUNT, I, J, K, KASE, NZ
DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
* ..
* .. Local Arrays ..
INTEGER ISAVE( 3 )
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DCOPY, DGEMV, DGETRS, DLACN2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL LSAME, DLAMCH
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) 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( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -12
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGERFS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
DO 10 J = 1, NRHS
FERR( J ) = ZERO
BERR( J ) = ZERO
10 CONTINUE
RETURN
END IF
*
IF( NOTRAN ) THEN
TRANST = 'T'
ELSE
TRANST = 'N'
END IF
*
* NZ = maximum number of nonzero elements in each row of A, plus 1
*
NZ = N + 1
EPS = DLAMCH( 'Epsilon' )
SAFMIN = DLAMCH( 'Safe minimum' )
SAFE1 = NZ*SAFMIN
SAFE2 = SAFE1 / EPS
*
* Do for each right hand side
*
DO 140 J = 1, NRHS
*
COUNT = 1
LSTRES = THREE
20 CONTINUE
*
* Loop until stopping criterion is satisfied.
*
* Compute residual R = B - op(A) * X,
* where op(A) = A, A**T, or A**H, depending on TRANS.
*
CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
$ WORK( N+1 ), 1 )
*
* Compute componentwise relative backward error from formula
*
* max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
*
* where abs(Z) is the componentwise absolute value of the matrix
* or vector Z. If the i-th component of the denominator is less
* than SAFE2, then SAFE1 is added to the i-th components of the
* numerator and denominator before dividing.
*
DO 30 I = 1, N
WORK( I ) = ABS( B( I, J ) )
30 CONTINUE
*
* Compute abs(op(A))*abs(X) + abs(B).
*
IF( NOTRAN ) THEN
DO 50 K = 1, N
XK = ABS( X( K, J ) )
DO 40 I = 1, N
WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
40 CONTINUE
50 CONTINUE
ELSE
DO 70 K = 1, N
S = ZERO
DO 60 I = 1, N
S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
60 CONTINUE
WORK( K ) = WORK( K ) + S
70 CONTINUE
END IF
S = ZERO
DO 80 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
ELSE
S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
$ ( WORK( I )+SAFE1 ) )
END IF
80 CONTINUE
BERR( J ) = S
*
* Test stopping criterion. Continue iterating if
* 1) The residual BERR(J) is larger than machine epsilon, and
* 2) BERR(J) decreased by at least a factor of 2 during the
* last iteration, and
* 3) At most ITMAX iterations tried.
*
IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
$ COUNT.LE.ITMAX ) THEN
*
* Update solution and try again.
*
CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
$ INFO )
CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
LSTRES = BERR( J )
COUNT = COUNT + 1
GO TO 20
END IF
*
* Bound error from formula
*
* norm(X - XTRUE) / norm(X) .le. FERR =
* norm( abs(inv(op(A)))*
* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
*
* where
* norm(Z) is the magnitude of the largest component of Z
* inv(op(A)) is the inverse of op(A)
* abs(Z) is the componentwise absolute value of the matrix or
* vector Z
* NZ is the maximum number of nonzeros in any row of A, plus 1
* EPS is machine epsilon
*
* The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
* is incremented by SAFE1 if the i-th component of
* abs(op(A))*abs(X) + abs(B) is less than SAFE2.
*
* Use DLACN2 to estimate the infinity-norm of the matrix
* inv(op(A)) * diag(W),
* where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
*
DO 90 I = 1, N
IF( WORK( I ).GT.SAFE2 ) THEN
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
ELSE
WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
END IF
90 CONTINUE
*
KASE = 0
100 CONTINUE
CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
$ KASE, ISAVE )
IF( KASE.NE.0 ) THEN
IF( KASE.EQ.1 ) THEN
*
* Multiply by diag(W)*inv(op(A)**T).
*
CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
$ N, INFO )
DO 110 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
110 CONTINUE
ELSE
*
* Multiply by inv(op(A))*diag(W).
*
DO 120 I = 1, N
WORK( N+I ) = WORK( I )*WORK( N+I )
120 CONTINUE
CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
$ INFO )
END IF
GO TO 100
END IF
*
* Normalize error.
*
LSTRES = ZERO
DO 130 I = 1, N
LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
130 CONTINUE
IF( LSTRES.NE.ZERO )
$ FERR( J ) = FERR( J ) / LSTRES
*
140 CONTINUE
*
RETURN
*
* End of DGERFS
*
END
|
