1 SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
2 $ X, 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 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
19 $ WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZSYRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is symmetric indefinite, and
27 * provides error bounds and backward error estimates 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 * A (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDAF,N)
56 * The factored form of the matrix A. AF contains the block
57 * diagonal matrix D and the multipliers used to obtain the
58 * factor U or L from the factorization A = U*D*U**T or
59 * A = L*D*L**T as computed by ZSYTRF.
60 *
61 * LDAF (input) INTEGER
62 * The leading dimension of the array AF. LDAF >= max(1,N).
63 *
64 * IPIV (input) INTEGER array, dimension (N)
65 * Details of the interchanges and the block structure of D
66 * as determined by ZSYTRF.
67 *
68 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
69 * The right hand side matrix B.
70 *
71 * LDB (input) INTEGER
72 * The leading dimension of the array B. LDB >= max(1,N).
73 *
74 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
75 * On entry, the solution matrix X, as computed by ZSYTRS.
76 * On exit, the improved solution matrix X.
77 *
78 * LDX (input) INTEGER
79 * The leading dimension of the array X. LDX >= max(1,N).
80 *
81 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
82 * The estimated forward error bound for each solution vector
83 * X(j) (the j-th column of the solution matrix X).
84 * If XTRUE is the true solution corresponding to X(j), FERR(j)
85 * is an estimated upper bound for the magnitude of the largest
86 * element in (X(j) - XTRUE) divided by the magnitude of the
87 * largest element in X(j). The estimate is as reliable as
88 * the estimate for RCOND, and is almost always a slight
89 * overestimate of the true error.
90 *
91 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
92 * The componentwise relative backward error of each solution
93 * vector X(j) (i.e., the smallest relative change in
94 * any element of A or B that makes X(j) an exact solution).
95 *
96 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
97 *
98 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
99 *
100 * INFO (output) INTEGER
101 * = 0: successful exit
102 * < 0: if INFO = -i, the i-th argument had an illegal value
103 *
104 * Internal Parameters
105 * ===================
106 *
107 * ITMAX is the maximum number of steps of iterative refinement.
108 *
109 * =====================================================================
110 *
111 * .. Parameters ..
112 INTEGER ITMAX
113 PARAMETER ( ITMAX = 5 )
114 DOUBLE PRECISION ZERO
115 PARAMETER ( ZERO = 0.0D+0 )
116 COMPLEX*16 ONE
117 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
118 DOUBLE PRECISION TWO
119 PARAMETER ( TWO = 2.0D+0 )
120 DOUBLE PRECISION THREE
121 PARAMETER ( THREE = 3.0D+0 )
122 * ..
123 * .. Local Scalars ..
124 LOGICAL UPPER
125 INTEGER COUNT, I, J, K, KASE, NZ
126 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
127 COMPLEX*16 ZDUM
128 * ..
129 * .. Local Arrays ..
130 INTEGER ISAVE( 3 )
131 * ..
132 * .. External Subroutines ..
133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSYMV, ZSYTRS
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC ABS, DBLE, DIMAG, MAX
137 * ..
138 * .. External Functions ..
139 LOGICAL LSAME
140 DOUBLE PRECISION DLAMCH
141 EXTERNAL LSAME, DLAMCH
142 * ..
143 * .. Statement Functions ..
144 DOUBLE PRECISION CABS1
145 * ..
146 * .. Statement Function definitions ..
147 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
148 * ..
149 * .. Executable Statements ..
150 *
151 * Test the input parameters.
152 *
153 INFO = 0
154 UPPER = LSAME( UPLO, 'U' )
155 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
156 INFO = -1
157 ELSE IF( N.LT.0 ) THEN
158 INFO = -2
159 ELSE IF( NRHS.LT.0 ) THEN
160 INFO = -3
161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162 INFO = -5
163 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
164 INFO = -7
165 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
166 INFO = -10
167 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
168 INFO = -12
169 END IF
170 IF( INFO.NE.0 ) THEN
171 CALL XERBLA( 'ZSYRFS', -INFO )
172 RETURN
173 END IF
174 *
175 * Quick return if possible
176 *
177 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
178 DO 10 J = 1, NRHS
179 FERR( J ) = ZERO
180 BERR( J ) = ZERO
181 10 CONTINUE
182 RETURN
183 END IF
184 *
185 * NZ = maximum number of nonzero elements in each row of A, plus 1
186 *
187 NZ = N + 1
188 EPS = DLAMCH( 'Epsilon' )
189 SAFMIN = DLAMCH( 'Safe minimum' )
190 SAFE1 = NZ*SAFMIN
191 SAFE2 = SAFE1 / EPS
192 *
193 * Do for each right hand side
194 *
195 DO 140 J = 1, NRHS
196 *
197 COUNT = 1
198 LSTRES = THREE
199 20 CONTINUE
200 *
201 * Loop until stopping criterion is satisfied.
202 *
203 * Compute residual R = B - A * X
204 *
205 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
206 CALL ZSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
207 *
208 * Compute componentwise relative backward error from formula
209 *
210 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
211 *
212 * where abs(Z) is the componentwise absolute value of the matrix
213 * or vector Z. If the i-th component of the denominator is less
214 * than SAFE2, then SAFE1 is added to the i-th components of the
215 * numerator and denominator before dividing.
216 *
217 DO 30 I = 1, N
218 RWORK( I ) = CABS1( B( I, J ) )
219 30 CONTINUE
220 *
221 * Compute abs(A)*abs(X) + abs(B).
222 *
223 IF( UPPER ) THEN
224 DO 50 K = 1, N
225 S = ZERO
226 XK = CABS1( X( K, J ) )
227 DO 40 I = 1, K - 1
228 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
229 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
230 40 CONTINUE
231 RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK + S
232 50 CONTINUE
233 ELSE
234 DO 70 K = 1, N
235 S = ZERO
236 XK = CABS1( X( K, J ) )
237 RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK
238 DO 60 I = K + 1, N
239 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
240 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
241 60 CONTINUE
242 RWORK( K ) = RWORK( K ) + S
243 70 CONTINUE
244 END IF
245 S = ZERO
246 DO 80 I = 1, N
247 IF( RWORK( I ).GT.SAFE2 ) THEN
248 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
249 ELSE
250 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
251 $ ( RWORK( I )+SAFE1 ) )
252 END IF
253 80 CONTINUE
254 BERR( J ) = S
255 *
256 * Test stopping criterion. Continue iterating if
257 * 1) The residual BERR(J) is larger than machine epsilon, and
258 * 2) BERR(J) decreased by at least a factor of 2 during the
259 * last iteration, and
260 * 3) At most ITMAX iterations tried.
261 *
262 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
263 $ COUNT.LE.ITMAX ) THEN
264 *
265 * Update solution and try again.
266 *
267 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
268 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
269 LSTRES = BERR( J )
270 COUNT = COUNT + 1
271 GO TO 20
272 END IF
273 *
274 * Bound error from formula
275 *
276 * norm(X - XTRUE) / norm(X) .le. FERR =
277 * norm( abs(inv(A))*
278 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
279 *
280 * where
281 * norm(Z) is the magnitude of the largest component of Z
282 * inv(A) is the inverse of A
283 * abs(Z) is the componentwise absolute value of the matrix or
284 * vector Z
285 * NZ is the maximum number of nonzeros in any row of A, plus 1
286 * EPS is machine epsilon
287 *
288 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
289 * is incremented by SAFE1 if the i-th component of
290 * abs(A)*abs(X) + abs(B) is less than SAFE2.
291 *
292 * Use ZLACN2 to estimate the infinity-norm of the matrix
293 * inv(A) * diag(W),
294 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
295 *
296 DO 90 I = 1, N
297 IF( RWORK( I ).GT.SAFE2 ) THEN
298 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
299 ELSE
300 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
301 $ SAFE1
302 END IF
303 90 CONTINUE
304 *
305 KASE = 0
306 100 CONTINUE
307 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
308 IF( KASE.NE.0 ) THEN
309 IF( KASE.EQ.1 ) THEN
310 *
311 * Multiply by diag(W)*inv(A**T).
312 *
313 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
314 DO 110 I = 1, N
315 WORK( I ) = RWORK( I )*WORK( I )
316 110 CONTINUE
317 ELSE IF( KASE.EQ.2 ) THEN
318 *
319 * Multiply by inv(A)*diag(W).
320 *
321 DO 120 I = 1, N
322 WORK( I ) = RWORK( I )*WORK( I )
323 120 CONTINUE
324 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
325 END IF
326 GO TO 100
327 END IF
328 *
329 * Normalize error.
330 *
331 LSTRES = ZERO
332 DO 130 I = 1, N
333 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
334 130 CONTINUE
335 IF( LSTRES.NE.ZERO )
336 $ FERR( J ) = FERR( J ) / LSTRES
337 *
338 140 CONTINUE
339 *
340 RETURN
341 *
342 * End of ZSYRFS
343 *
344 END
2 $ X, 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 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
19 $ WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZSYRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is symmetric indefinite, and
27 * provides error bounds and backward error estimates 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 * A (input) COMPLEX*16 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) COMPLEX*16 array, dimension (LDAF,N)
56 * The factored form of the matrix A. AF contains the block
57 * diagonal matrix D and the multipliers used to obtain the
58 * factor U or L from the factorization A = U*D*U**T or
59 * A = L*D*L**T as computed by ZSYTRF.
60 *
61 * LDAF (input) INTEGER
62 * The leading dimension of the array AF. LDAF >= max(1,N).
63 *
64 * IPIV (input) INTEGER array, dimension (N)
65 * Details of the interchanges and the block structure of D
66 * as determined by ZSYTRF.
67 *
68 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
69 * The right hand side matrix B.
70 *
71 * LDB (input) INTEGER
72 * The leading dimension of the array B. LDB >= max(1,N).
73 *
74 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
75 * On entry, the solution matrix X, as computed by ZSYTRS.
76 * On exit, the improved solution matrix X.
77 *
78 * LDX (input) INTEGER
79 * The leading dimension of the array X. LDX >= max(1,N).
80 *
81 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
82 * The estimated forward error bound for each solution vector
83 * X(j) (the j-th column of the solution matrix X).
84 * If XTRUE is the true solution corresponding to X(j), FERR(j)
85 * is an estimated upper bound for the magnitude of the largest
86 * element in (X(j) - XTRUE) divided by the magnitude of the
87 * largest element in X(j). The estimate is as reliable as
88 * the estimate for RCOND, and is almost always a slight
89 * overestimate of the true error.
90 *
91 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
92 * The componentwise relative backward error of each solution
93 * vector X(j) (i.e., the smallest relative change in
94 * any element of A or B that makes X(j) an exact solution).
95 *
96 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
97 *
98 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
99 *
100 * INFO (output) INTEGER
101 * = 0: successful exit
102 * < 0: if INFO = -i, the i-th argument had an illegal value
103 *
104 * Internal Parameters
105 * ===================
106 *
107 * ITMAX is the maximum number of steps of iterative refinement.
108 *
109 * =====================================================================
110 *
111 * .. Parameters ..
112 INTEGER ITMAX
113 PARAMETER ( ITMAX = 5 )
114 DOUBLE PRECISION ZERO
115 PARAMETER ( ZERO = 0.0D+0 )
116 COMPLEX*16 ONE
117 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
118 DOUBLE PRECISION TWO
119 PARAMETER ( TWO = 2.0D+0 )
120 DOUBLE PRECISION THREE
121 PARAMETER ( THREE = 3.0D+0 )
122 * ..
123 * .. Local Scalars ..
124 LOGICAL UPPER
125 INTEGER COUNT, I, J, K, KASE, NZ
126 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
127 COMPLEX*16 ZDUM
128 * ..
129 * .. Local Arrays ..
130 INTEGER ISAVE( 3 )
131 * ..
132 * .. External Subroutines ..
133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZLACN2, ZSYMV, ZSYTRS
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC ABS, DBLE, DIMAG, MAX
137 * ..
138 * .. External Functions ..
139 LOGICAL LSAME
140 DOUBLE PRECISION DLAMCH
141 EXTERNAL LSAME, DLAMCH
142 * ..
143 * .. Statement Functions ..
144 DOUBLE PRECISION CABS1
145 * ..
146 * .. Statement Function definitions ..
147 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
148 * ..
149 * .. Executable Statements ..
150 *
151 * Test the input parameters.
152 *
153 INFO = 0
154 UPPER = LSAME( UPLO, 'U' )
155 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
156 INFO = -1
157 ELSE IF( N.LT.0 ) THEN
158 INFO = -2
159 ELSE IF( NRHS.LT.0 ) THEN
160 INFO = -3
161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
162 INFO = -5
163 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
164 INFO = -7
165 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
166 INFO = -10
167 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
168 INFO = -12
169 END IF
170 IF( INFO.NE.0 ) THEN
171 CALL XERBLA( 'ZSYRFS', -INFO )
172 RETURN
173 END IF
174 *
175 * Quick return if possible
176 *
177 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
178 DO 10 J = 1, NRHS
179 FERR( J ) = ZERO
180 BERR( J ) = ZERO
181 10 CONTINUE
182 RETURN
183 END IF
184 *
185 * NZ = maximum number of nonzero elements in each row of A, plus 1
186 *
187 NZ = N + 1
188 EPS = DLAMCH( 'Epsilon' )
189 SAFMIN = DLAMCH( 'Safe minimum' )
190 SAFE1 = NZ*SAFMIN
191 SAFE2 = SAFE1 / EPS
192 *
193 * Do for each right hand side
194 *
195 DO 140 J = 1, NRHS
196 *
197 COUNT = 1
198 LSTRES = THREE
199 20 CONTINUE
200 *
201 * Loop until stopping criterion is satisfied.
202 *
203 * Compute residual R = B - A * X
204 *
205 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
206 CALL ZSYMV( UPLO, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK, 1 )
207 *
208 * Compute componentwise relative backward error from formula
209 *
210 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
211 *
212 * where abs(Z) is the componentwise absolute value of the matrix
213 * or vector Z. If the i-th component of the denominator is less
214 * than SAFE2, then SAFE1 is added to the i-th components of the
215 * numerator and denominator before dividing.
216 *
217 DO 30 I = 1, N
218 RWORK( I ) = CABS1( B( I, J ) )
219 30 CONTINUE
220 *
221 * Compute abs(A)*abs(X) + abs(B).
222 *
223 IF( UPPER ) THEN
224 DO 50 K = 1, N
225 S = ZERO
226 XK = CABS1( X( K, J ) )
227 DO 40 I = 1, K - 1
228 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
229 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
230 40 CONTINUE
231 RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK + S
232 50 CONTINUE
233 ELSE
234 DO 70 K = 1, N
235 S = ZERO
236 XK = CABS1( X( K, J ) )
237 RWORK( K ) = RWORK( K ) + CABS1( A( K, K ) )*XK
238 DO 60 I = K + 1, N
239 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
240 S = S + CABS1( A( I, K ) )*CABS1( X( I, J ) )
241 60 CONTINUE
242 RWORK( K ) = RWORK( K ) + S
243 70 CONTINUE
244 END IF
245 S = ZERO
246 DO 80 I = 1, N
247 IF( RWORK( I ).GT.SAFE2 ) THEN
248 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
249 ELSE
250 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
251 $ ( RWORK( I )+SAFE1 ) )
252 END IF
253 80 CONTINUE
254 BERR( J ) = S
255 *
256 * Test stopping criterion. Continue iterating if
257 * 1) The residual BERR(J) is larger than machine epsilon, and
258 * 2) BERR(J) decreased by at least a factor of 2 during the
259 * last iteration, and
260 * 3) At most ITMAX iterations tried.
261 *
262 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
263 $ COUNT.LE.ITMAX ) THEN
264 *
265 * Update solution and try again.
266 *
267 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
268 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
269 LSTRES = BERR( J )
270 COUNT = COUNT + 1
271 GO TO 20
272 END IF
273 *
274 * Bound error from formula
275 *
276 * norm(X - XTRUE) / norm(X) .le. FERR =
277 * norm( abs(inv(A))*
278 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
279 *
280 * where
281 * norm(Z) is the magnitude of the largest component of Z
282 * inv(A) is the inverse of A
283 * abs(Z) is the componentwise absolute value of the matrix or
284 * vector Z
285 * NZ is the maximum number of nonzeros in any row of A, plus 1
286 * EPS is machine epsilon
287 *
288 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
289 * is incremented by SAFE1 if the i-th component of
290 * abs(A)*abs(X) + abs(B) is less than SAFE2.
291 *
292 * Use ZLACN2 to estimate the infinity-norm of the matrix
293 * inv(A) * diag(W),
294 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
295 *
296 DO 90 I = 1, N
297 IF( RWORK( I ).GT.SAFE2 ) THEN
298 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
299 ELSE
300 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
301 $ SAFE1
302 END IF
303 90 CONTINUE
304 *
305 KASE = 0
306 100 CONTINUE
307 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
308 IF( KASE.NE.0 ) THEN
309 IF( KASE.EQ.1 ) THEN
310 *
311 * Multiply by diag(W)*inv(A**T).
312 *
313 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
314 DO 110 I = 1, N
315 WORK( I ) = RWORK( I )*WORK( I )
316 110 CONTINUE
317 ELSE IF( KASE.EQ.2 ) THEN
318 *
319 * Multiply by inv(A)*diag(W).
320 *
321 DO 120 I = 1, N
322 WORK( I ) = RWORK( I )*WORK( I )
323 120 CONTINUE
324 CALL ZSYTRS( UPLO, N, 1, AF, LDAF, IPIV, WORK, N, INFO )
325 END IF
326 GO TO 100
327 END IF
328 *
329 * Normalize error.
330 *
331 LSTRES = ZERO
332 DO 130 I = 1, N
333 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
334 130 CONTINUE
335 IF( LSTRES.NE.ZERO )
336 $ FERR( J ) = FERR( J ) / LSTRES
337 *
338 140 CONTINUE
339 *
340 RETURN
341 *
342 * End of ZSYRFS
343 *
344 END