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