1 SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
2 $ 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, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
18 $ FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DSPRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is symmetric indefinite
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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
49 *
50 * AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
51 * The factored form of the matrix A. AFP contains the block
52 * diagonal matrix D and the multipliers used to obtain the
53 * factor U or L from the factorization A = U*D*U**T or
54 * A = L*D*L**T as computed by DSPTRF, stored as a packed
55 * triangular matrix.
56 *
57 * IPIV (input) INTEGER array, dimension (N)
58 * Details of the interchanges and the block structure of D
59 * as determined by DSPTRF.
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 DSPTRS.
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 UPPER
118 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
119 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
120 * ..
121 * .. Local Arrays ..
122 INTEGER ISAVE( 3 )
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, MAX
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 DOUBLE PRECISION DLAMCH
133 EXTERNAL LSAME, DLAMCH
134 * ..
135 * .. Executable Statements ..
136 *
137 * Test the input parameters.
138 *
139 INFO = 0
140 UPPER = LSAME( UPLO, 'U' )
141 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
142 INFO = -1
143 ELSE IF( N.LT.0 ) THEN
144 INFO = -2
145 ELSE IF( NRHS.LT.0 ) THEN
146 INFO = -3
147 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
148 INFO = -8
149 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
150 INFO = -10
151 END IF
152 IF( INFO.NE.0 ) THEN
153 CALL XERBLA( 'DSPRFS', -INFO )
154 RETURN
155 END IF
156 *
157 * Quick return if possible
158 *
159 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
160 DO 10 J = 1, NRHS
161 FERR( J ) = ZERO
162 BERR( J ) = ZERO
163 10 CONTINUE
164 RETURN
165 END IF
166 *
167 * NZ = maximum number of nonzero elements in each row of A, plus 1
168 *
169 NZ = N + 1
170 EPS = DLAMCH( 'Epsilon' )
171 SAFMIN = DLAMCH( 'Safe minimum' )
172 SAFE1 = NZ*SAFMIN
173 SAFE2 = SAFE1 / EPS
174 *
175 * Do for each right hand side
176 *
177 DO 140 J = 1, NRHS
178 *
179 COUNT = 1
180 LSTRES = THREE
181 20 CONTINUE
182 *
183 * Loop until stopping criterion is satisfied.
184 *
185 * Compute residual R = B - A * X
186 *
187 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
188 CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
189 $ 1 )
190 *
191 * Compute componentwise relative backward error from formula
192 *
193 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
194 *
195 * where abs(Z) is the componentwise absolute value of the matrix
196 * or vector Z. If the i-th component of the denominator is less
197 * than SAFE2, then SAFE1 is added to the i-th components of the
198 * numerator and denominator before dividing.
199 *
200 DO 30 I = 1, N
201 WORK( I ) = ABS( B( I, J ) )
202 30 CONTINUE
203 *
204 * Compute abs(A)*abs(X) + abs(B).
205 *
206 KK = 1
207 IF( UPPER ) THEN
208 DO 50 K = 1, N
209 S = ZERO
210 XK = ABS( X( K, J ) )
211 IK = KK
212 DO 40 I = 1, K - 1
213 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
214 S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
215 IK = IK + 1
216 40 CONTINUE
217 WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
218 KK = KK + K
219 50 CONTINUE
220 ELSE
221 DO 70 K = 1, N
222 S = ZERO
223 XK = ABS( X( K, J ) )
224 WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
225 IK = KK + 1
226 DO 60 I = K + 1, N
227 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
228 S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
229 IK = IK + 1
230 60 CONTINUE
231 WORK( K ) = WORK( K ) + S
232 KK = KK + ( N-K+1 )
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 DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
258 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
259 LSTRES = BERR( J )
260 COUNT = COUNT + 1
261 GO TO 20
262 END IF
263 *
264 * Bound error from formula
265 *
266 * norm(X - XTRUE) / norm(X) .le. FERR =
267 * norm( abs(inv(A))*
268 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
269 *
270 * where
271 * norm(Z) is the magnitude of the largest component of Z
272 * inv(A) is the inverse of A
273 * abs(Z) is the componentwise absolute value of the matrix or
274 * vector Z
275 * NZ is the maximum number of nonzeros in any row of A, plus 1
276 * EPS is machine epsilon
277 *
278 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
279 * is incremented by SAFE1 if the i-th component of
280 * abs(A)*abs(X) + abs(B) is less than SAFE2.
281 *
282 * Use DLACN2 to estimate the infinity-norm of the matrix
283 * inv(A) * diag(W),
284 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
285 *
286 DO 90 I = 1, N
287 IF( WORK( I ).GT.SAFE2 ) THEN
288 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
289 ELSE
290 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
291 END IF
292 90 CONTINUE
293 *
294 KASE = 0
295 100 CONTINUE
296 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
297 $ KASE, ISAVE )
298 IF( KASE.NE.0 ) THEN
299 IF( KASE.EQ.1 ) THEN
300 *
301 * Multiply by diag(W)*inv(A**T).
302 *
303 CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
304 $ INFO )
305 DO 110 I = 1, N
306 WORK( N+I ) = WORK( I )*WORK( N+I )
307 110 CONTINUE
308 ELSE IF( KASE.EQ.2 ) THEN
309 *
310 * Multiply by inv(A)*diag(W).
311 *
312 DO 120 I = 1, N
313 WORK( N+I ) = WORK( I )*WORK( N+I )
314 120 CONTINUE
315 CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
316 $ INFO )
317 END IF
318 GO TO 100
319 END IF
320 *
321 * Normalize error.
322 *
323 LSTRES = ZERO
324 DO 130 I = 1, N
325 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
326 130 CONTINUE
327 IF( LSTRES.NE.ZERO )
328 $ FERR( J ) = FERR( J ) / LSTRES
329 *
330 140 CONTINUE
331 *
332 RETURN
333 *
334 * End of DSPRFS
335 *
336 END
2 $ 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, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * ), IWORK( * )
17 DOUBLE PRECISION AFP( * ), AP( * ), B( LDB, * ), BERR( * ),
18 $ FERR( * ), WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * DSPRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is symmetric indefinite
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)*(2*n-j)/2) = A(i,j) for j<=i<=n.
49 *
50 * AFP (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
51 * The factored form of the matrix A. AFP contains the block
52 * diagonal matrix D and the multipliers used to obtain the
53 * factor U or L from the factorization A = U*D*U**T or
54 * A = L*D*L**T as computed by DSPTRF, stored as a packed
55 * triangular matrix.
56 *
57 * IPIV (input) INTEGER array, dimension (N)
58 * Details of the interchanges and the block structure of D
59 * as determined by DSPTRF.
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 DSPTRS.
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 UPPER
118 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
119 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
120 * ..
121 * .. Local Arrays ..
122 INTEGER ISAVE( 3 )
123 * ..
124 * .. External Subroutines ..
125 EXTERNAL DAXPY, DCOPY, DLACN2, DSPMV, DSPTRS, XERBLA
126 * ..
127 * .. Intrinsic Functions ..
128 INTRINSIC ABS, MAX
129 * ..
130 * .. External Functions ..
131 LOGICAL LSAME
132 DOUBLE PRECISION DLAMCH
133 EXTERNAL LSAME, DLAMCH
134 * ..
135 * .. Executable Statements ..
136 *
137 * Test the input parameters.
138 *
139 INFO = 0
140 UPPER = LSAME( UPLO, 'U' )
141 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
142 INFO = -1
143 ELSE IF( N.LT.0 ) THEN
144 INFO = -2
145 ELSE IF( NRHS.LT.0 ) THEN
146 INFO = -3
147 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
148 INFO = -8
149 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
150 INFO = -10
151 END IF
152 IF( INFO.NE.0 ) THEN
153 CALL XERBLA( 'DSPRFS', -INFO )
154 RETURN
155 END IF
156 *
157 * Quick return if possible
158 *
159 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
160 DO 10 J = 1, NRHS
161 FERR( J ) = ZERO
162 BERR( J ) = ZERO
163 10 CONTINUE
164 RETURN
165 END IF
166 *
167 * NZ = maximum number of nonzero elements in each row of A, plus 1
168 *
169 NZ = N + 1
170 EPS = DLAMCH( 'Epsilon' )
171 SAFMIN = DLAMCH( 'Safe minimum' )
172 SAFE1 = NZ*SAFMIN
173 SAFE2 = SAFE1 / EPS
174 *
175 * Do for each right hand side
176 *
177 DO 140 J = 1, NRHS
178 *
179 COUNT = 1
180 LSTRES = THREE
181 20 CONTINUE
182 *
183 * Loop until stopping criterion is satisfied.
184 *
185 * Compute residual R = B - A * X
186 *
187 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
188 CALL DSPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK( N+1 ),
189 $ 1 )
190 *
191 * Compute componentwise relative backward error from formula
192 *
193 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
194 *
195 * where abs(Z) is the componentwise absolute value of the matrix
196 * or vector Z. If the i-th component of the denominator is less
197 * than SAFE2, then SAFE1 is added to the i-th components of the
198 * numerator and denominator before dividing.
199 *
200 DO 30 I = 1, N
201 WORK( I ) = ABS( B( I, J ) )
202 30 CONTINUE
203 *
204 * Compute abs(A)*abs(X) + abs(B).
205 *
206 KK = 1
207 IF( UPPER ) THEN
208 DO 50 K = 1, N
209 S = ZERO
210 XK = ABS( X( K, J ) )
211 IK = KK
212 DO 40 I = 1, K - 1
213 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
214 S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
215 IK = IK + 1
216 40 CONTINUE
217 WORK( K ) = WORK( K ) + ABS( AP( KK+K-1 ) )*XK + S
218 KK = KK + K
219 50 CONTINUE
220 ELSE
221 DO 70 K = 1, N
222 S = ZERO
223 XK = ABS( X( K, J ) )
224 WORK( K ) = WORK( K ) + ABS( AP( KK ) )*XK
225 IK = KK + 1
226 DO 60 I = K + 1, N
227 WORK( I ) = WORK( I ) + ABS( AP( IK ) )*XK
228 S = S + ABS( AP( IK ) )*ABS( X( I, J ) )
229 IK = IK + 1
230 60 CONTINUE
231 WORK( K ) = WORK( K ) + S
232 KK = KK + ( N-K+1 )
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 DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N, INFO )
258 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
259 LSTRES = BERR( J )
260 COUNT = COUNT + 1
261 GO TO 20
262 END IF
263 *
264 * Bound error from formula
265 *
266 * norm(X - XTRUE) / norm(X) .le. FERR =
267 * norm( abs(inv(A))*
268 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
269 *
270 * where
271 * norm(Z) is the magnitude of the largest component of Z
272 * inv(A) is the inverse of A
273 * abs(Z) is the componentwise absolute value of the matrix or
274 * vector Z
275 * NZ is the maximum number of nonzeros in any row of A, plus 1
276 * EPS is machine epsilon
277 *
278 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
279 * is incremented by SAFE1 if the i-th component of
280 * abs(A)*abs(X) + abs(B) is less than SAFE2.
281 *
282 * Use DLACN2 to estimate the infinity-norm of the matrix
283 * inv(A) * diag(W),
284 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
285 *
286 DO 90 I = 1, N
287 IF( WORK( I ).GT.SAFE2 ) THEN
288 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
289 ELSE
290 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
291 END IF
292 90 CONTINUE
293 *
294 KASE = 0
295 100 CONTINUE
296 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
297 $ KASE, ISAVE )
298 IF( KASE.NE.0 ) THEN
299 IF( KASE.EQ.1 ) THEN
300 *
301 * Multiply by diag(W)*inv(A**T).
302 *
303 CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
304 $ INFO )
305 DO 110 I = 1, N
306 WORK( N+I ) = WORK( I )*WORK( N+I )
307 110 CONTINUE
308 ELSE IF( KASE.EQ.2 ) THEN
309 *
310 * Multiply by inv(A)*diag(W).
311 *
312 DO 120 I = 1, N
313 WORK( N+I ) = WORK( I )*WORK( N+I )
314 120 CONTINUE
315 CALL DSPTRS( UPLO, N, 1, AFP, IPIV, WORK( N+1 ), N,
316 $ INFO )
317 END IF
318 GO TO 100
319 END IF
320 *
321 * Normalize error.
322 *
323 LSTRES = ZERO
324 DO 130 I = 1, N
325 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
326 130 CONTINUE
327 IF( LSTRES.NE.ZERO )
328 $ FERR( J ) = FERR( J ) / LSTRES
329 *
330 140 CONTINUE
331 *
332 RETURN
333 *
334 * End of DSPRFS
335 *
336 END