1 SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
2 $ LDX, FERR, BERR, WORK, IWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
10 *
11 * .. Scalar Arguments ..
12 CHARACTER UPLO
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DPORFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is symmetric positive definite,
26 * and provides error bounds and backward error estimates for the
27 * solution.
28 *
29 * Arguments
30 * =========
31 *
32 * UPLO (input) CHARACTER*1
33 * = 'U': Upper triangle of A is stored;
34 * = 'L': Lower triangle of A is stored.
35 *
36 * N (input) INTEGER
37 * The order of the matrix A. N >= 0.
38 *
39 * NRHS (input) INTEGER
40 * The number of right hand sides, i.e., the number of columns
41 * of the matrices B and X. NRHS >= 0.
42 *
43 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
44 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
45 * upper triangular part of A contains the upper triangular part
46 * of the matrix A, and the strictly lower triangular part of A
47 * is not referenced. If UPLO = 'L', the leading N-by-N lower
48 * triangular part of A contains the lower triangular part of
49 * the matrix A, and the strictly upper triangular part of A is
50 * not referenced.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,N).
54 *
55 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
56 * The triangular factor U or L from the Cholesky factorization
57 * A = U**T*U or A = L*L**T, as computed by DPOTRF.
58 *
59 * LDAF (input) INTEGER
60 * The leading dimension of the array AF. LDAF >= max(1,N).
61 *
62 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
63 * The right hand side matrix B.
64 *
65 * LDB (input) INTEGER
66 * The leading dimension of the array B. LDB >= max(1,N).
67 *
68 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
69 * On entry, the solution matrix X, as computed by DPOTRS.
70 * On exit, the improved solution matrix X.
71 *
72 * LDX (input) INTEGER
73 * The leading dimension of the array X. LDX >= max(1,N).
74 *
75 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
76 * The estimated forward error bound for each solution vector
77 * X(j) (the j-th column of the solution matrix X).
78 * If XTRUE is the true solution corresponding to X(j), FERR(j)
79 * is an estimated upper bound for the magnitude of the largest
80 * element in (X(j) - XTRUE) divided by the magnitude of the
81 * largest element in X(j). The estimate is as reliable as
82 * the estimate for RCOND, and is almost always a slight
83 * overestimate of the true error.
84 *
85 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
86 * The componentwise relative backward error of each solution
87 * vector X(j) (i.e., the smallest relative change in
88 * any element of A or B that makes X(j) an exact solution).
89 *
90 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
91 *
92 * IWORK (workspace) INTEGER array, dimension (N)
93 *
94 * INFO (output) INTEGER
95 * = 0: successful exit
96 * < 0: if INFO = -i, the i-th argument had an illegal value
97 *
98 * Internal Parameters
99 * ===================
100 *
101 * ITMAX is the maximum number of steps of iterative refinement.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 INTEGER ITMAX
107 PARAMETER ( ITMAX = 5 )
108 DOUBLE PRECISION ZERO
109 PARAMETER ( ZERO = 0.0D+0 )
110 DOUBLE PRECISION ONE
111 PARAMETER ( ONE = 1.0D+0 )
112 DOUBLE PRECISION TWO
113 PARAMETER ( TWO = 2.0D+0 )
114 DOUBLE PRECISION THREE
115 PARAMETER ( THREE = 3.0D+0 )
116 * ..
117 * .. Local Scalars ..
118 LOGICAL UPPER
119 INTEGER COUNT, I, J, K, KASE, NZ
120 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
121 * ..
122 * .. Local Arrays ..
123 INTEGER ISAVE( 3 )
124 * ..
125 * .. External Subroutines ..
126 EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA
127 * ..
128 * .. Intrinsic Functions ..
129 INTRINSIC ABS, MAX
130 * ..
131 * .. External Functions ..
132 LOGICAL LSAME
133 DOUBLE PRECISION DLAMCH
134 EXTERNAL LSAME, DLAMCH
135 * ..
136 * .. Executable Statements ..
137 *
138 * Test the input parameters.
139 *
140 INFO = 0
141 UPPER = LSAME( UPLO, 'U' )
142 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
143 INFO = -1
144 ELSE IF( N.LT.0 ) THEN
145 INFO = -2
146 ELSE IF( NRHS.LT.0 ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
151 INFO = -7
152 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
153 INFO = -9
154 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
155 INFO = -11
156 END IF
157 IF( INFO.NE.0 ) THEN
158 CALL XERBLA( 'DPORFS', -INFO )
159 RETURN
160 END IF
161 *
162 * Quick return if possible
163 *
164 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
165 DO 10 J = 1, NRHS
166 FERR( J ) = ZERO
167 BERR( J ) = ZERO
168 10 CONTINUE
169 RETURN
170 END IF
171 *
172 * NZ = maximum number of nonzero elements in each row of A, plus 1
173 *
174 NZ = N + 1
175 EPS = DLAMCH( 'Epsilon' )
176 SAFMIN = DLAMCH( 'Safe minimum' )
177 SAFE1 = NZ*SAFMIN
178 SAFE2 = SAFE1 / EPS
179 *
180 * Do for each right hand side
181 *
182 DO 140 J = 1, NRHS
183 *
184 COUNT = 1
185 LSTRES = THREE
186 20 CONTINUE
187 *
188 * Loop until stopping criterion is satisfied.
189 *
190 * Compute residual R = B - A * X
191 *
192 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
193 CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
194 $ WORK( N+1 ), 1 )
195 *
196 * Compute componentwise relative backward error from formula
197 *
198 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
199 *
200 * where abs(Z) is the componentwise absolute value of the matrix
201 * or vector Z. If the i-th component of the denominator is less
202 * than SAFE2, then SAFE1 is added to the i-th components of the
203 * numerator and denominator before dividing.
204 *
205 DO 30 I = 1, N
206 WORK( I ) = ABS( B( I, J ) )
207 30 CONTINUE
208 *
209 * Compute abs(A)*abs(X) + abs(B).
210 *
211 IF( UPPER ) THEN
212 DO 50 K = 1, N
213 S = ZERO
214 XK = ABS( X( K, J ) )
215 DO 40 I = 1, K - 1
216 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
217 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
218 40 CONTINUE
219 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
220 50 CONTINUE
221 ELSE
222 DO 70 K = 1, N
223 S = ZERO
224 XK = ABS( X( K, J ) )
225 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
226 DO 60 I = K + 1, N
227 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
228 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
229 60 CONTINUE
230 WORK( K ) = WORK( K ) + S
231 70 CONTINUE
232 END IF
233 S = ZERO
234 DO 80 I = 1, N
235 IF( WORK( I ).GT.SAFE2 ) THEN
236 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
237 ELSE
238 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
239 $ ( WORK( I )+SAFE1 ) )
240 END IF
241 80 CONTINUE
242 BERR( J ) = S
243 *
244 * Test stopping criterion. Continue iterating if
245 * 1) The residual BERR(J) is larger than machine epsilon, and
246 * 2) BERR(J) decreased by at least a factor of 2 during the
247 * last iteration, and
248 * 3) At most ITMAX iterations tried.
249 *
250 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
251 $ COUNT.LE.ITMAX ) THEN
252 *
253 * Update solution and try again.
254 *
255 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
256 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
257 LSTRES = BERR( J )
258 COUNT = COUNT + 1
259 GO TO 20
260 END IF
261 *
262 * Bound error from formula
263 *
264 * norm(X - XTRUE) / norm(X) .le. FERR =
265 * norm( abs(inv(A))*
266 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
267 *
268 * where
269 * norm(Z) is the magnitude of the largest component of Z
270 * inv(A) is the inverse of A
271 * abs(Z) is the componentwise absolute value of the matrix or
272 * vector Z
273 * NZ is the maximum number of nonzeros in any row of A, plus 1
274 * EPS is machine epsilon
275 *
276 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
277 * is incremented by SAFE1 if the i-th component of
278 * abs(A)*abs(X) + abs(B) is less than SAFE2.
279 *
280 * Use DLACN2 to estimate the infinity-norm of the matrix
281 * inv(A) * diag(W),
282 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
283 *
284 DO 90 I = 1, N
285 IF( WORK( I ).GT.SAFE2 ) THEN
286 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
287 ELSE
288 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
289 END IF
290 90 CONTINUE
291 *
292 KASE = 0
293 100 CONTINUE
294 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
295 $ KASE, ISAVE )
296 IF( KASE.NE.0 ) THEN
297 IF( KASE.EQ.1 ) THEN
298 *
299 * Multiply by diag(W)*inv(A**T).
300 *
301 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
302 DO 110 I = 1, N
303 WORK( N+I ) = WORK( I )*WORK( N+I )
304 110 CONTINUE
305 ELSE IF( KASE.EQ.2 ) THEN
306 *
307 * Multiply by inv(A)*diag(W).
308 *
309 DO 120 I = 1, N
310 WORK( N+I ) = WORK( I )*WORK( N+I )
311 120 CONTINUE
312 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
313 END IF
314 GO TO 100
315 END IF
316 *
317 * Normalize error.
318 *
319 LSTRES = ZERO
320 DO 130 I = 1, N
321 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
322 130 CONTINUE
323 IF( LSTRES.NE.ZERO )
324 $ FERR( J ) = FERR( J ) / LSTRES
325 *
326 140 CONTINUE
327 *
328 RETURN
329 *
330 * End of DPORFS
331 *
332 END
2 $ LDX, FERR, BERR, WORK, IWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.2) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * November 2006
8 *
9 * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
10 *
11 * .. Scalar Arguments ..
12 CHARACTER UPLO
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DPORFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is symmetric positive definite,
26 * and provides error bounds and backward error estimates for the
27 * solution.
28 *
29 * Arguments
30 * =========
31 *
32 * UPLO (input) CHARACTER*1
33 * = 'U': Upper triangle of A is stored;
34 * = 'L': Lower triangle of A is stored.
35 *
36 * N (input) INTEGER
37 * The order of the matrix A. N >= 0.
38 *
39 * NRHS (input) INTEGER
40 * The number of right hand sides, i.e., the number of columns
41 * of the matrices B and X. NRHS >= 0.
42 *
43 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
44 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N
45 * upper triangular part of A contains the upper triangular part
46 * of the matrix A, and the strictly lower triangular part of A
47 * is not referenced. If UPLO = 'L', the leading N-by-N lower
48 * triangular part of A contains the lower triangular part of
49 * the matrix A, and the strictly upper triangular part of A is
50 * not referenced.
51 *
52 * LDA (input) INTEGER
53 * The leading dimension of the array A. LDA >= max(1,N).
54 *
55 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
56 * The triangular factor U or L from the Cholesky factorization
57 * A = U**T*U or A = L*L**T, as computed by DPOTRF.
58 *
59 * LDAF (input) INTEGER
60 * The leading dimension of the array AF. LDAF >= max(1,N).
61 *
62 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
63 * The right hand side matrix B.
64 *
65 * LDB (input) INTEGER
66 * The leading dimension of the array B. LDB >= max(1,N).
67 *
68 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
69 * On entry, the solution matrix X, as computed by DPOTRS.
70 * On exit, the improved solution matrix X.
71 *
72 * LDX (input) INTEGER
73 * The leading dimension of the array X. LDX >= max(1,N).
74 *
75 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
76 * The estimated forward error bound for each solution vector
77 * X(j) (the j-th column of the solution matrix X).
78 * If XTRUE is the true solution corresponding to X(j), FERR(j)
79 * is an estimated upper bound for the magnitude of the largest
80 * element in (X(j) - XTRUE) divided by the magnitude of the
81 * largest element in X(j). The estimate is as reliable as
82 * the estimate for RCOND, and is almost always a slight
83 * overestimate of the true error.
84 *
85 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
86 * The componentwise relative backward error of each solution
87 * vector X(j) (i.e., the smallest relative change in
88 * any element of A or B that makes X(j) an exact solution).
89 *
90 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
91 *
92 * IWORK (workspace) INTEGER array, dimension (N)
93 *
94 * INFO (output) INTEGER
95 * = 0: successful exit
96 * < 0: if INFO = -i, the i-th argument had an illegal value
97 *
98 * Internal Parameters
99 * ===================
100 *
101 * ITMAX is the maximum number of steps of iterative refinement.
102 *
103 * =====================================================================
104 *
105 * .. Parameters ..
106 INTEGER ITMAX
107 PARAMETER ( ITMAX = 5 )
108 DOUBLE PRECISION ZERO
109 PARAMETER ( ZERO = 0.0D+0 )
110 DOUBLE PRECISION ONE
111 PARAMETER ( ONE = 1.0D+0 )
112 DOUBLE PRECISION TWO
113 PARAMETER ( TWO = 2.0D+0 )
114 DOUBLE PRECISION THREE
115 PARAMETER ( THREE = 3.0D+0 )
116 * ..
117 * .. Local Scalars ..
118 LOGICAL UPPER
119 INTEGER COUNT, I, J, K, KASE, NZ
120 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
121 * ..
122 * .. Local Arrays ..
123 INTEGER ISAVE( 3 )
124 * ..
125 * .. External Subroutines ..
126 EXTERNAL DAXPY, DCOPY, DLACN2, DPOTRS, DSYMV, XERBLA
127 * ..
128 * .. Intrinsic Functions ..
129 INTRINSIC ABS, MAX
130 * ..
131 * .. External Functions ..
132 LOGICAL LSAME
133 DOUBLE PRECISION DLAMCH
134 EXTERNAL LSAME, DLAMCH
135 * ..
136 * .. Executable Statements ..
137 *
138 * Test the input parameters.
139 *
140 INFO = 0
141 UPPER = LSAME( UPLO, 'U' )
142 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
143 INFO = -1
144 ELSE IF( N.LT.0 ) THEN
145 INFO = -2
146 ELSE IF( NRHS.LT.0 ) THEN
147 INFO = -3
148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
149 INFO = -5
150 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
151 INFO = -7
152 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
153 INFO = -9
154 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
155 INFO = -11
156 END IF
157 IF( INFO.NE.0 ) THEN
158 CALL XERBLA( 'DPORFS', -INFO )
159 RETURN
160 END IF
161 *
162 * Quick return if possible
163 *
164 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
165 DO 10 J = 1, NRHS
166 FERR( J ) = ZERO
167 BERR( J ) = ZERO
168 10 CONTINUE
169 RETURN
170 END IF
171 *
172 * NZ = maximum number of nonzero elements in each row of A, plus 1
173 *
174 NZ = N + 1
175 EPS = DLAMCH( 'Epsilon' )
176 SAFMIN = DLAMCH( 'Safe minimum' )
177 SAFE1 = NZ*SAFMIN
178 SAFE2 = SAFE1 / EPS
179 *
180 * Do for each right hand side
181 *
182 DO 140 J = 1, NRHS
183 *
184 COUNT = 1
185 LSTRES = THREE
186 20 CONTINUE
187 *
188 * Loop until stopping criterion is satisfied.
189 *
190 * Compute residual R = B - A * X
191 *
192 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
193 CALL DSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
194 $ WORK( N+1 ), 1 )
195 *
196 * Compute componentwise relative backward error from formula
197 *
198 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
199 *
200 * where abs(Z) is the componentwise absolute value of the matrix
201 * or vector Z. If the i-th component of the denominator is less
202 * than SAFE2, then SAFE1 is added to the i-th components of the
203 * numerator and denominator before dividing.
204 *
205 DO 30 I = 1, N
206 WORK( I ) = ABS( B( I, J ) )
207 30 CONTINUE
208 *
209 * Compute abs(A)*abs(X) + abs(B).
210 *
211 IF( UPPER ) THEN
212 DO 50 K = 1, N
213 S = ZERO
214 XK = ABS( X( K, J ) )
215 DO 40 I = 1, K - 1
216 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
217 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
218 40 CONTINUE
219 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK + S
220 50 CONTINUE
221 ELSE
222 DO 70 K = 1, N
223 S = ZERO
224 XK = ABS( X( K, J ) )
225 WORK( K ) = WORK( K ) + ABS( A( K, K ) )*XK
226 DO 60 I = K + 1, N
227 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
228 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
229 60 CONTINUE
230 WORK( K ) = WORK( K ) + S
231 70 CONTINUE
232 END IF
233 S = ZERO
234 DO 80 I = 1, N
235 IF( WORK( I ).GT.SAFE2 ) THEN
236 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
237 ELSE
238 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
239 $ ( WORK( I )+SAFE1 ) )
240 END IF
241 80 CONTINUE
242 BERR( J ) = S
243 *
244 * Test stopping criterion. Continue iterating if
245 * 1) The residual BERR(J) is larger than machine epsilon, and
246 * 2) BERR(J) decreased by at least a factor of 2 during the
247 * last iteration, and
248 * 3) At most ITMAX iterations tried.
249 *
250 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
251 $ COUNT.LE.ITMAX ) THEN
252 *
253 * Update solution and try again.
254 *
255 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
256 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
257 LSTRES = BERR( J )
258 COUNT = COUNT + 1
259 GO TO 20
260 END IF
261 *
262 * Bound error from formula
263 *
264 * norm(X - XTRUE) / norm(X) .le. FERR =
265 * norm( abs(inv(A))*
266 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
267 *
268 * where
269 * norm(Z) is the magnitude of the largest component of Z
270 * inv(A) is the inverse of A
271 * abs(Z) is the componentwise absolute value of the matrix or
272 * vector Z
273 * NZ is the maximum number of nonzeros in any row of A, plus 1
274 * EPS is machine epsilon
275 *
276 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
277 * is incremented by SAFE1 if the i-th component of
278 * abs(A)*abs(X) + abs(B) is less than SAFE2.
279 *
280 * Use DLACN2 to estimate the infinity-norm of the matrix
281 * inv(A) * diag(W),
282 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
283 *
284 DO 90 I = 1, N
285 IF( WORK( I ).GT.SAFE2 ) THEN
286 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
287 ELSE
288 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
289 END IF
290 90 CONTINUE
291 *
292 KASE = 0
293 100 CONTINUE
294 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
295 $ KASE, ISAVE )
296 IF( KASE.NE.0 ) THEN
297 IF( KASE.EQ.1 ) THEN
298 *
299 * Multiply by diag(W)*inv(A**T).
300 *
301 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
302 DO 110 I = 1, N
303 WORK( N+I ) = WORK( I )*WORK( N+I )
304 110 CONTINUE
305 ELSE IF( KASE.EQ.2 ) THEN
306 *
307 * Multiply by inv(A)*diag(W).
308 *
309 DO 120 I = 1, N
310 WORK( N+I ) = WORK( I )*WORK( N+I )
311 120 CONTINUE
312 CALL DPOTRS( UPLO, N, 1, AF, LDAF, WORK( N+1 ), N, INFO )
313 END IF
314 GO TO 100
315 END IF
316 *
317 * Normalize error.
318 *
319 LSTRES = ZERO
320 DO 130 I = 1, N
321 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
322 130 CONTINUE
323 IF( LSTRES.NE.ZERO )
324 $ FERR( J ) = FERR( J ) / LSTRES
325 *
326 140 CONTINUE
327 *
328 RETURN
329 *
330 * End of DPORFS
331 *
332 END