1 SUBROUTINE ZPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
2 $ LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
10 *
11 * .. Scalar Arguments ..
12 CHARACTER UPLO
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPORFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian 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) COMPLEX*16 array, dimension (LDA,N)
44 * The Hermitian 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) COMPLEX*16 array, dimension (LDAF,N)
56 * The triangular factor U or L from the Cholesky factorization
57 * A = U**H*U or A = L*L**H, as computed by ZPOTRF.
58 *
59 * LDAF (input) INTEGER
60 * The leading dimension of the array AF. LDAF >= max(1,N).
61 *
62 * B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
69 * On entry, the solution matrix X, as computed by ZPOTRS.
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) COMPLEX*16 array, dimension (2*N)
91 *
92 * RWORK (workspace) DOUBLE PRECISION 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 COMPLEX*16 ONE
111 PARAMETER ( ONE = ( 1.0D+0, 0.0D+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 COMPLEX*16 ZDUM
122 * ..
123 * .. Local Arrays ..
124 INTEGER ISAVE( 3 )
125 * ..
126 * .. External Subroutines ..
127 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
128 * ..
129 * .. Intrinsic Functions ..
130 INTRINSIC ABS, DBLE, DIMAG, MAX
131 * ..
132 * .. External Functions ..
133 LOGICAL LSAME
134 DOUBLE PRECISION DLAMCH
135 EXTERNAL LSAME, DLAMCH
136 * ..
137 * .. Statement Functions ..
138 DOUBLE PRECISION CABS1
139 * ..
140 * .. Statement Function definitions ..
141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
142 * ..
143 * .. Executable Statements ..
144 *
145 * Test the input parameters.
146 *
147 INFO = 0
148 UPPER = LSAME( UPLO, 'U' )
149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
150 INFO = -1
151 ELSE IF( N.LT.0 ) THEN
152 INFO = -2
153 ELSE IF( NRHS.LT.0 ) THEN
154 INFO = -3
155 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
156 INFO = -5
157 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
158 INFO = -7
159 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
160 INFO = -9
161 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
162 INFO = -11
163 END IF
164 IF( INFO.NE.0 ) THEN
165 CALL XERBLA( 'ZPORFS', -INFO )
166 RETURN
167 END IF
168 *
169 * Quick return if possible
170 *
171 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
172 DO 10 J = 1, NRHS
173 FERR( J ) = ZERO
174 BERR( J ) = ZERO
175 10 CONTINUE
176 RETURN
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 - A * X
198 *
199 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
200 CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
201 *
202 * Compute componentwise relative backward error from formula
203 *
204 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
205 *
206 * where abs(Z) is the componentwise absolute value of the matrix
207 * or vector Z. If the i-th component of the denominator is less
208 * than SAFE2, then SAFE1 is added to the i-th components of the
209 * numerator and denominator before dividing.
210 *
211 DO 30 I = 1, N
212 RWORK( I ) = CABS1( B( I, J ) )
213 30 CONTINUE
214 *
215 * Compute abs(A)*abs(X) + abs(B).
216 *
217 IF( UPPER ) THEN
218 DO 50 K = 1, N
219 S = ZERO
220 XK = CABS1( X( K, J ) )
221 DO 40 I = 1, K - 1
222 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
223 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
224 40 CONTINUE
225 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
226 50 CONTINUE
227 ELSE
228 DO 70 K = 1, N
229 S = ZERO
230 XK = CABS1( X( K, J ) )
231 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
232 DO 60 I = K + 1, N
233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
234 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
235 60 CONTINUE
236 RWORK( K ) = RWORK( K ) + S
237 70 CONTINUE
238 END IF
239 S = ZERO
240 DO 80 I = 1, N
241 IF( RWORK( I ).GT.SAFE2 ) THEN
242 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
243 ELSE
244 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
245 $ ( RWORK( I )+SAFE1 ) )
246 END IF
247 80 CONTINUE
248 BERR( J ) = S
249 *
250 * Test stopping criterion. Continue iterating if
251 * 1) The residual BERR(J) is larger than machine epsilon, and
252 * 2) BERR(J) decreased by at least a factor of 2 during the
253 * last iteration, and
254 * 3) At most ITMAX iterations tried.
255 *
256 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
257 $ COUNT.LE.ITMAX ) THEN
258 *
259 * Update solution and try again.
260 *
261 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
262 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
263 LSTRES = BERR( J )
264 COUNT = COUNT + 1
265 GO TO 20
266 END IF
267 *
268 * Bound error from formula
269 *
270 * norm(X - XTRUE) / norm(X) .le. FERR =
271 * norm( abs(inv(A))*
272 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
273 *
274 * where
275 * norm(Z) is the magnitude of the largest component of Z
276 * inv(A) is the inverse of A
277 * abs(Z) is the componentwise absolute value of the matrix or
278 * vector Z
279 * NZ is the maximum number of nonzeros in any row of A, plus 1
280 * EPS is machine epsilon
281 *
282 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
283 * is incremented by SAFE1 if the i-th component of
284 * abs(A)*abs(X) + abs(B) is less than SAFE2.
285 *
286 * Use ZLACN2 to estimate the infinity-norm of the matrix
287 * inv(A) * diag(W),
288 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
289 *
290 DO 90 I = 1, N
291 IF( RWORK( I ).GT.SAFE2 ) THEN
292 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
293 ELSE
294 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
295 $ SAFE1
296 END IF
297 90 CONTINUE
298 *
299 KASE = 0
300 100 CONTINUE
301 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
302 IF( KASE.NE.0 ) THEN
303 IF( KASE.EQ.1 ) THEN
304 *
305 * Multiply by diag(W)*inv(A**H).
306 *
307 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
308 DO 110 I = 1, N
309 WORK( I ) = RWORK( I )*WORK( I )
310 110 CONTINUE
311 ELSE IF( KASE.EQ.2 ) THEN
312 *
313 * Multiply by inv(A)*diag(W).
314 *
315 DO 120 I = 1, N
316 WORK( I ) = RWORK( I )*WORK( I )
317 120 CONTINUE
318 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
319 END IF
320 GO TO 100
321 END IF
322 *
323 * Normalize error.
324 *
325 LSTRES = ZERO
326 DO 130 I = 1, N
327 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
328 130 CONTINUE
329 IF( LSTRES.NE.ZERO )
330 $ FERR( J ) = FERR( J ) / LSTRES
331 *
332 140 CONTINUE
333 *
334 RETURN
335 *
336 * End of ZPORFS
337 *
338 END
2 $ LDX, FERR, BERR, WORK, RWORK, 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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
10 *
11 * .. Scalar Arguments ..
12 CHARACTER UPLO
13 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPORFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian 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) COMPLEX*16 array, dimension (LDA,N)
44 * The Hermitian 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) COMPLEX*16 array, dimension (LDAF,N)
56 * The triangular factor U or L from the Cholesky factorization
57 * A = U**H*U or A = L*L**H, as computed by ZPOTRF.
58 *
59 * LDAF (input) INTEGER
60 * The leading dimension of the array AF. LDAF >= max(1,N).
61 *
62 * B (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
69 * On entry, the solution matrix X, as computed by ZPOTRS.
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) COMPLEX*16 array, dimension (2*N)
91 *
92 * RWORK (workspace) DOUBLE PRECISION 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 COMPLEX*16 ONE
111 PARAMETER ( ONE = ( 1.0D+0, 0.0D+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 COMPLEX*16 ZDUM
122 * ..
123 * .. Local Arrays ..
124 INTEGER ISAVE( 3 )
125 * ..
126 * .. External Subroutines ..
127 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHEMV, ZLACN2, ZPOTRS
128 * ..
129 * .. Intrinsic Functions ..
130 INTRINSIC ABS, DBLE, DIMAG, MAX
131 * ..
132 * .. External Functions ..
133 LOGICAL LSAME
134 DOUBLE PRECISION DLAMCH
135 EXTERNAL LSAME, DLAMCH
136 * ..
137 * .. Statement Functions ..
138 DOUBLE PRECISION CABS1
139 * ..
140 * .. Statement Function definitions ..
141 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
142 * ..
143 * .. Executable Statements ..
144 *
145 * Test the input parameters.
146 *
147 INFO = 0
148 UPPER = LSAME( UPLO, 'U' )
149 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
150 INFO = -1
151 ELSE IF( N.LT.0 ) THEN
152 INFO = -2
153 ELSE IF( NRHS.LT.0 ) THEN
154 INFO = -3
155 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
156 INFO = -5
157 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
158 INFO = -7
159 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
160 INFO = -9
161 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
162 INFO = -11
163 END IF
164 IF( INFO.NE.0 ) THEN
165 CALL XERBLA( 'ZPORFS', -INFO )
166 RETURN
167 END IF
168 *
169 * Quick return if possible
170 *
171 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
172 DO 10 J = 1, NRHS
173 FERR( J ) = ZERO
174 BERR( J ) = ZERO
175 10 CONTINUE
176 RETURN
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 - A * X
198 *
199 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
200 CALL ZHEMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
201 *
202 * Compute componentwise relative backward error from formula
203 *
204 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
205 *
206 * where abs(Z) is the componentwise absolute value of the matrix
207 * or vector Z. If the i-th component of the denominator is less
208 * than SAFE2, then SAFE1 is added to the i-th components of the
209 * numerator and denominator before dividing.
210 *
211 DO 30 I = 1, N
212 RWORK( I ) = CABS1( B( I, J ) )
213 30 CONTINUE
214 *
215 * Compute abs(A)*abs(X) + abs(B).
216 *
217 IF( UPPER ) THEN
218 DO 50 K = 1, N
219 S = ZERO
220 XK = CABS1( X( K, J ) )
221 DO 40 I = 1, K - 1
222 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
223 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
224 40 CONTINUE
225 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK + S
226 50 CONTINUE
227 ELSE
228 DO 70 K = 1, N
229 S = ZERO
230 XK = CABS1( X( K, J ) )
231 RWORK( K ) = RWORK( K ) + ABS( DBLE( A( K, K ) ) )*XK
232 DO 60 I = K + 1, N
233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
234 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
235 60 CONTINUE
236 RWORK( K ) = RWORK( K ) + S
237 70 CONTINUE
238 END IF
239 S = ZERO
240 DO 80 I = 1, N
241 IF( RWORK( I ).GT.SAFE2 ) THEN
242 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
243 ELSE
244 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
245 $ ( RWORK( I )+SAFE1 ) )
246 END IF
247 80 CONTINUE
248 BERR( J ) = S
249 *
250 * Test stopping criterion. Continue iterating if
251 * 1) The residual BERR(J) is larger than machine epsilon, and
252 * 2) BERR(J) decreased by at least a factor of 2 during the
253 * last iteration, and
254 * 3) At most ITMAX iterations tried.
255 *
256 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
257 $ COUNT.LE.ITMAX ) THEN
258 *
259 * Update solution and try again.
260 *
261 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
262 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
263 LSTRES = BERR( J )
264 COUNT = COUNT + 1
265 GO TO 20
266 END IF
267 *
268 * Bound error from formula
269 *
270 * norm(X - XTRUE) / norm(X) .le. FERR =
271 * norm( abs(inv(A))*
272 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
273 *
274 * where
275 * norm(Z) is the magnitude of the largest component of Z
276 * inv(A) is the inverse of A
277 * abs(Z) is the componentwise absolute value of the matrix or
278 * vector Z
279 * NZ is the maximum number of nonzeros in any row of A, plus 1
280 * EPS is machine epsilon
281 *
282 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
283 * is incremented by SAFE1 if the i-th component of
284 * abs(A)*abs(X) + abs(B) is less than SAFE2.
285 *
286 * Use ZLACN2 to estimate the infinity-norm of the matrix
287 * inv(A) * diag(W),
288 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
289 *
290 DO 90 I = 1, N
291 IF( RWORK( I ).GT.SAFE2 ) THEN
292 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
293 ELSE
294 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
295 $ SAFE1
296 END IF
297 90 CONTINUE
298 *
299 KASE = 0
300 100 CONTINUE
301 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
302 IF( KASE.NE.0 ) THEN
303 IF( KASE.EQ.1 ) THEN
304 *
305 * Multiply by diag(W)*inv(A**H).
306 *
307 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
308 DO 110 I = 1, N
309 WORK( I ) = RWORK( I )*WORK( I )
310 110 CONTINUE
311 ELSE IF( KASE.EQ.2 ) THEN
312 *
313 * Multiply by inv(A)*diag(W).
314 *
315 DO 120 I = 1, N
316 WORK( I ) = RWORK( I )*WORK( I )
317 120 CONTINUE
318 CALL ZPOTRS( UPLO, N, 1, AF, LDAF, WORK, N, INFO )
319 END IF
320 GO TO 100
321 END IF
322 *
323 * Normalize error.
324 *
325 LSTRES = ZERO
326 DO 130 I = 1, N
327 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
328 130 CONTINUE
329 IF( LSTRES.NE.ZERO )
330 $ FERR( J ) = FERR( J ) / LSTRES
331 *
332 140 CONTINUE
333 *
334 RETURN
335 *
336 * End of ZPORFS
337 *
338 END