1 SUBROUTINE DGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
2 $ X, 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 TRANS
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DGERFS improves the computed solution to a system of linear
25 * equations and provides error bounds and backward error estimates for
26 * the solution.
27 *
28 * Arguments
29 * =========
30 *
31 * TRANS (input) CHARACTER*1
32 * Specifies the form of the system of equations:
33 * = 'N': A * X = B (No transpose)
34 * = 'T': A**T * X = B (Transpose)
35 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
36 *
37 * N (input) INTEGER
38 * The order of the matrix A. N >= 0.
39 *
40 * NRHS (input) INTEGER
41 * The number of right hand sides, i.e., the number of columns
42 * of the matrices B and X. NRHS >= 0.
43 *
44 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
45 * The original N-by-N matrix A.
46 *
47 * LDA (input) INTEGER
48 * The leading dimension of the array A. LDA >= max(1,N).
49 *
50 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
51 * The factors L and U from the factorization A = P*L*U
52 * as computed by DGETRF.
53 *
54 * LDAF (input) INTEGER
55 * The leading dimension of the array AF. LDAF >= max(1,N).
56 *
57 * IPIV (input) INTEGER array, dimension (N)
58 * The pivot indices from DGETRF; for 1<=i<=N, row i of the
59 * matrix was interchanged with row IPIV(i).
60 *
61 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
62 * The right hand side matrix B.
63 *
64 * LDB (input) INTEGER
65 * The leading dimension of the array B. LDB >= max(1,N).
66 *
67 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
68 * On entry, the solution matrix X, as computed by DGETRS.
69 * On exit, the improved solution matrix X.
70 *
71 * LDX (input) INTEGER
72 * The leading dimension of the array X. LDX >= max(1,N).
73 *
74 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
75 * The estimated forward error bound for each solution vector
76 * X(j) (the j-th column of the solution matrix X).
77 * If XTRUE is the true solution corresponding to X(j), FERR(j)
78 * is an estimated upper bound for the magnitude of the largest
79 * element in (X(j) - XTRUE) divided by the magnitude of the
80 * largest element in X(j). The estimate is as reliable as
81 * the estimate for RCOND, and is almost always a slight
82 * overestimate of the true error.
83 *
84 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
85 * The componentwise relative backward error of each solution
86 * vector X(j) (i.e., the smallest relative change in
87 * any element of A or B that makes X(j) an exact solution).
88 *
89 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
90 *
91 * IWORK (workspace) INTEGER array, dimension (N)
92 *
93 * INFO (output) INTEGER
94 * = 0: successful exit
95 * < 0: if INFO = -i, the i-th argument had an illegal value
96 *
97 * Internal Parameters
98 * ===================
99 *
100 * ITMAX is the maximum number of steps of iterative refinement.
101 *
102 * =====================================================================
103 *
104 * .. Parameters ..
105 INTEGER ITMAX
106 PARAMETER ( ITMAX = 5 )
107 DOUBLE PRECISION ZERO
108 PARAMETER ( ZERO = 0.0D+0 )
109 DOUBLE PRECISION ONE
110 PARAMETER ( ONE = 1.0D+0 )
111 DOUBLE PRECISION TWO
112 PARAMETER ( TWO = 2.0D+0 )
113 DOUBLE PRECISION THREE
114 PARAMETER ( THREE = 3.0D+0 )
115 * ..
116 * .. Local Scalars ..
117 LOGICAL NOTRAN
118 CHARACTER TRANST
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, DGEMV, DGETRS, DLACN2, 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 NOTRAN = LSAME( TRANS, 'N' )
142 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
143 $ LSAME( TRANS, 'C' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( NRHS.LT.0 ) THEN
148 INFO = -3
149 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
150 INFO = -5
151 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
152 INFO = -7
153 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
154 INFO = -10
155 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
156 INFO = -12
157 END IF
158 IF( INFO.NE.0 ) THEN
159 CALL XERBLA( 'DGERFS', -INFO )
160 RETURN
161 END IF
162 *
163 * Quick return if possible
164 *
165 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
166 DO 10 J = 1, NRHS
167 FERR( J ) = ZERO
168 BERR( J ) = ZERO
169 10 CONTINUE
170 RETURN
171 END IF
172 *
173 IF( NOTRAN ) THEN
174 TRANST = 'T'
175 ELSE
176 TRANST = 'N'
177 END IF
178 *
179 * NZ = maximum number of nonzero elements in each row of A, plus 1
180 *
181 NZ = N + 1
182 EPS = DLAMCH( 'Epsilon' )
183 SAFMIN = DLAMCH( 'Safe minimum' )
184 SAFE1 = NZ*SAFMIN
185 SAFE2 = SAFE1 / EPS
186 *
187 * Do for each right hand side
188 *
189 DO 140 J = 1, NRHS
190 *
191 COUNT = 1
192 LSTRES = THREE
193 20 CONTINUE
194 *
195 * Loop until stopping criterion is satisfied.
196 *
197 * Compute residual R = B - op(A) * X,
198 * where op(A) = A, A**T, or A**H, depending on TRANS.
199 *
200 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
201 CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
202 $ WORK( N+1 ), 1 )
203 *
204 * Compute componentwise relative backward error from formula
205 *
206 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
207 *
208 * where abs(Z) is the componentwise absolute value of the matrix
209 * or vector Z. If the i-th component of the denominator is less
210 * than SAFE2, then SAFE1 is added to the i-th components of the
211 * numerator and denominator before dividing.
212 *
213 DO 30 I = 1, N
214 WORK( I ) = ABS( B( I, J ) )
215 30 CONTINUE
216 *
217 * Compute abs(op(A))*abs(X) + abs(B).
218 *
219 IF( NOTRAN ) THEN
220 DO 50 K = 1, N
221 XK = ABS( X( K, J ) )
222 DO 40 I = 1, N
223 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
224 40 CONTINUE
225 50 CONTINUE
226 ELSE
227 DO 70 K = 1, N
228 S = ZERO
229 DO 60 I = 1, N
230 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
231 60 CONTINUE
232 WORK( K ) = WORK( K ) + S
233 70 CONTINUE
234 END IF
235 S = ZERO
236 DO 80 I = 1, N
237 IF( WORK( I ).GT.SAFE2 ) THEN
238 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
239 ELSE
240 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
241 $ ( WORK( I )+SAFE1 ) )
242 END IF
243 80 CONTINUE
244 BERR( J ) = S
245 *
246 * Test stopping criterion. Continue iterating if
247 * 1) The residual BERR(J) is larger than machine epsilon, and
248 * 2) BERR(J) decreased by at least a factor of 2 during the
249 * last iteration, and
250 * 3) At most ITMAX iterations tried.
251 *
252 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
253 $ COUNT.LE.ITMAX ) THEN
254 *
255 * Update solution and try again.
256 *
257 CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
258 $ INFO )
259 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
260 LSTRES = BERR( J )
261 COUNT = COUNT + 1
262 GO TO 20
263 END IF
264 *
265 * Bound error from formula
266 *
267 * norm(X - XTRUE) / norm(X) .le. FERR =
268 * norm( abs(inv(op(A)))*
269 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
270 *
271 * where
272 * norm(Z) is the magnitude of the largest component of Z
273 * inv(op(A)) is the inverse of op(A)
274 * abs(Z) is the componentwise absolute value of the matrix or
275 * vector Z
276 * NZ is the maximum number of nonzeros in any row of A, plus 1
277 * EPS is machine epsilon
278 *
279 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
280 * is incremented by SAFE1 if the i-th component of
281 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
282 *
283 * Use DLACN2 to estimate the infinity-norm of the matrix
284 * inv(op(A)) * diag(W),
285 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
286 *
287 DO 90 I = 1, N
288 IF( WORK( I ).GT.SAFE2 ) THEN
289 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
290 ELSE
291 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
292 END IF
293 90 CONTINUE
294 *
295 KASE = 0
296 100 CONTINUE
297 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
298 $ KASE, ISAVE )
299 IF( KASE.NE.0 ) THEN
300 IF( KASE.EQ.1 ) THEN
301 *
302 * Multiply by diag(W)*inv(op(A)**T).
303 *
304 CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
305 $ N, INFO )
306 DO 110 I = 1, N
307 WORK( N+I ) = WORK( I )*WORK( N+I )
308 110 CONTINUE
309 ELSE
310 *
311 * Multiply by inv(op(A))*diag(W).
312 *
313 DO 120 I = 1, N
314 WORK( N+I ) = WORK( I )*WORK( N+I )
315 120 CONTINUE
316 CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
317 $ INFO )
318 END IF
319 GO TO 100
320 END IF
321 *
322 * Normalize error.
323 *
324 LSTRES = ZERO
325 DO 130 I = 1, N
326 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
327 130 CONTINUE
328 IF( LSTRES.NE.ZERO )
329 $ FERR( J ) = FERR( J ) / LSTRES
330 *
331 140 CONTINUE
332 *
333 RETURN
334 *
335 * End of DGERFS
336 *
337 END
2 $ X, 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 TRANS
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DGERFS improves the computed solution to a system of linear
25 * equations and provides error bounds and backward error estimates for
26 * the solution.
27 *
28 * Arguments
29 * =========
30 *
31 * TRANS (input) CHARACTER*1
32 * Specifies the form of the system of equations:
33 * = 'N': A * X = B (No transpose)
34 * = 'T': A**T * X = B (Transpose)
35 * = 'C': A**H * X = B (Conjugate transpose = Transpose)
36 *
37 * N (input) INTEGER
38 * The order of the matrix A. N >= 0.
39 *
40 * NRHS (input) INTEGER
41 * The number of right hand sides, i.e., the number of columns
42 * of the matrices B and X. NRHS >= 0.
43 *
44 * A (input) DOUBLE PRECISION array, dimension (LDA,N)
45 * The original N-by-N matrix A.
46 *
47 * LDA (input) INTEGER
48 * The leading dimension of the array A. LDA >= max(1,N).
49 *
50 * AF (input) DOUBLE PRECISION array, dimension (LDAF,N)
51 * The factors L and U from the factorization A = P*L*U
52 * as computed by DGETRF.
53 *
54 * LDAF (input) INTEGER
55 * The leading dimension of the array AF. LDAF >= max(1,N).
56 *
57 * IPIV (input) INTEGER array, dimension (N)
58 * The pivot indices from DGETRF; for 1<=i<=N, row i of the
59 * matrix was interchanged with row IPIV(i).
60 *
61 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
62 * The right hand side matrix B.
63 *
64 * LDB (input) INTEGER
65 * The leading dimension of the array B. LDB >= max(1,N).
66 *
67 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
68 * On entry, the solution matrix X, as computed by DGETRS.
69 * On exit, the improved solution matrix X.
70 *
71 * LDX (input) INTEGER
72 * The leading dimension of the array X. LDX >= max(1,N).
73 *
74 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
75 * The estimated forward error bound for each solution vector
76 * X(j) (the j-th column of the solution matrix X).
77 * If XTRUE is the true solution corresponding to X(j), FERR(j)
78 * is an estimated upper bound for the magnitude of the largest
79 * element in (X(j) - XTRUE) divided by the magnitude of the
80 * largest element in X(j). The estimate is as reliable as
81 * the estimate for RCOND, and is almost always a slight
82 * overestimate of the true error.
83 *
84 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
85 * The componentwise relative backward error of each solution
86 * vector X(j) (i.e., the smallest relative change in
87 * any element of A or B that makes X(j) an exact solution).
88 *
89 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
90 *
91 * IWORK (workspace) INTEGER array, dimension (N)
92 *
93 * INFO (output) INTEGER
94 * = 0: successful exit
95 * < 0: if INFO = -i, the i-th argument had an illegal value
96 *
97 * Internal Parameters
98 * ===================
99 *
100 * ITMAX is the maximum number of steps of iterative refinement.
101 *
102 * =====================================================================
103 *
104 * .. Parameters ..
105 INTEGER ITMAX
106 PARAMETER ( ITMAX = 5 )
107 DOUBLE PRECISION ZERO
108 PARAMETER ( ZERO = 0.0D+0 )
109 DOUBLE PRECISION ONE
110 PARAMETER ( ONE = 1.0D+0 )
111 DOUBLE PRECISION TWO
112 PARAMETER ( TWO = 2.0D+0 )
113 DOUBLE PRECISION THREE
114 PARAMETER ( THREE = 3.0D+0 )
115 * ..
116 * .. Local Scalars ..
117 LOGICAL NOTRAN
118 CHARACTER TRANST
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, DGEMV, DGETRS, DLACN2, 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 NOTRAN = LSAME( TRANS, 'N' )
142 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
143 $ LSAME( TRANS, 'C' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( NRHS.LT.0 ) THEN
148 INFO = -3
149 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
150 INFO = -5
151 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
152 INFO = -7
153 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
154 INFO = -10
155 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
156 INFO = -12
157 END IF
158 IF( INFO.NE.0 ) THEN
159 CALL XERBLA( 'DGERFS', -INFO )
160 RETURN
161 END IF
162 *
163 * Quick return if possible
164 *
165 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
166 DO 10 J = 1, NRHS
167 FERR( J ) = ZERO
168 BERR( J ) = ZERO
169 10 CONTINUE
170 RETURN
171 END IF
172 *
173 IF( NOTRAN ) THEN
174 TRANST = 'T'
175 ELSE
176 TRANST = 'N'
177 END IF
178 *
179 * NZ = maximum number of nonzero elements in each row of A, plus 1
180 *
181 NZ = N + 1
182 EPS = DLAMCH( 'Epsilon' )
183 SAFMIN = DLAMCH( 'Safe minimum' )
184 SAFE1 = NZ*SAFMIN
185 SAFE2 = SAFE1 / EPS
186 *
187 * Do for each right hand side
188 *
189 DO 140 J = 1, NRHS
190 *
191 COUNT = 1
192 LSTRES = THREE
193 20 CONTINUE
194 *
195 * Loop until stopping criterion is satisfied.
196 *
197 * Compute residual R = B - op(A) * X,
198 * where op(A) = A, A**T, or A**H, depending on TRANS.
199 *
200 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
201 CALL DGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE,
202 $ WORK( N+1 ), 1 )
203 *
204 * Compute componentwise relative backward error from formula
205 *
206 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
207 *
208 * where abs(Z) is the componentwise absolute value of the matrix
209 * or vector Z. If the i-th component of the denominator is less
210 * than SAFE2, then SAFE1 is added to the i-th components of the
211 * numerator and denominator before dividing.
212 *
213 DO 30 I = 1, N
214 WORK( I ) = ABS( B( I, J ) )
215 30 CONTINUE
216 *
217 * Compute abs(op(A))*abs(X) + abs(B).
218 *
219 IF( NOTRAN ) THEN
220 DO 50 K = 1, N
221 XK = ABS( X( K, J ) )
222 DO 40 I = 1, N
223 WORK( I ) = WORK( I ) + ABS( A( I, K ) )*XK
224 40 CONTINUE
225 50 CONTINUE
226 ELSE
227 DO 70 K = 1, N
228 S = ZERO
229 DO 60 I = 1, N
230 S = S + ABS( A( I, K ) )*ABS( X( I, J ) )
231 60 CONTINUE
232 WORK( K ) = WORK( K ) + S
233 70 CONTINUE
234 END IF
235 S = ZERO
236 DO 80 I = 1, N
237 IF( WORK( I ).GT.SAFE2 ) THEN
238 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
239 ELSE
240 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
241 $ ( WORK( I )+SAFE1 ) )
242 END IF
243 80 CONTINUE
244 BERR( J ) = S
245 *
246 * Test stopping criterion. Continue iterating if
247 * 1) The residual BERR(J) is larger than machine epsilon, and
248 * 2) BERR(J) decreased by at least a factor of 2 during the
249 * last iteration, and
250 * 3) At most ITMAX iterations tried.
251 *
252 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
253 $ COUNT.LE.ITMAX ) THEN
254 *
255 * Update solution and try again.
256 *
257 CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
258 $ INFO )
259 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
260 LSTRES = BERR( J )
261 COUNT = COUNT + 1
262 GO TO 20
263 END IF
264 *
265 * Bound error from formula
266 *
267 * norm(X - XTRUE) / norm(X) .le. FERR =
268 * norm( abs(inv(op(A)))*
269 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
270 *
271 * where
272 * norm(Z) is the magnitude of the largest component of Z
273 * inv(op(A)) is the inverse of op(A)
274 * abs(Z) is the componentwise absolute value of the matrix or
275 * vector Z
276 * NZ is the maximum number of nonzeros in any row of A, plus 1
277 * EPS is machine epsilon
278 *
279 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
280 * is incremented by SAFE1 if the i-th component of
281 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
282 *
283 * Use DLACN2 to estimate the infinity-norm of the matrix
284 * inv(op(A)) * diag(W),
285 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
286 *
287 DO 90 I = 1, N
288 IF( WORK( I ).GT.SAFE2 ) THEN
289 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
290 ELSE
291 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
292 END IF
293 90 CONTINUE
294 *
295 KASE = 0
296 100 CONTINUE
297 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
298 $ KASE, ISAVE )
299 IF( KASE.NE.0 ) THEN
300 IF( KASE.EQ.1 ) THEN
301 *
302 * Multiply by diag(W)*inv(op(A)**T).
303 *
304 CALL DGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK( N+1 ),
305 $ N, INFO )
306 DO 110 I = 1, N
307 WORK( N+I ) = WORK( I )*WORK( N+I )
308 110 CONTINUE
309 ELSE
310 *
311 * Multiply by inv(op(A))*diag(W).
312 *
313 DO 120 I = 1, N
314 WORK( N+I ) = WORK( I )*WORK( N+I )
315 120 CONTINUE
316 CALL DGETRS( TRANS, N, 1, AF, LDAF, IPIV, WORK( N+1 ), N,
317 $ INFO )
318 END IF
319 GO TO 100
320 END IF
321 *
322 * Normalize error.
323 *
324 LSTRES = ZERO
325 DO 130 I = 1, N
326 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
327 130 CONTINUE
328 IF( LSTRES.NE.ZERO )
329 $ FERR( J ) = FERR( J ) / LSTRES
330 *
331 140 CONTINUE
332 *
333 RETURN
334 *
335 * End of DGERFS
336 *
337 END