1 SUBROUTINE ZHPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX,
2 $ 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, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
19 $ X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHPRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is Hermitian indefinite
27 * and packed, and provides error bounds and backward error estimates
28 * for the solution.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * = 'U': Upper triangle of A is stored;
35 * = 'L': Lower triangle of A is stored.
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 * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
45 * The upper or lower triangle of the Hermitian matrix A, packed
46 * columnwise in a linear array. The j-th column of A is stored
47 * in the array AP as follows:
48 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
49 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
50 *
51 * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
52 * The factored form of the matrix A. AFP contains the block
53 * diagonal matrix D and the multipliers used to obtain the
54 * factor U or L from the factorization A = U*D*U**H or
55 * A = L*D*L**H as computed by ZHPTRF, stored as a packed
56 * triangular matrix.
57 *
58 * IPIV (input) INTEGER array, dimension (N)
59 * Details of the interchanges and the block structure of D
60 * as determined by ZHPTRF.
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 ZHPTRS.
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, IK, J, K, KASE, KK, 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, ZHPMV, ZHPTRS, ZLACN2
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( LDB.LT.MAX( 1, N ) ) THEN
156 INFO = -8
157 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
158 INFO = -10
159 END IF
160 IF( INFO.NE.0 ) THEN
161 CALL XERBLA( 'ZHPRFS', -INFO )
162 RETURN
163 END IF
164 *
165 * Quick return if possible
166 *
167 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
168 DO 10 J = 1, NRHS
169 FERR( J ) = ZERO
170 BERR( J ) = ZERO
171 10 CONTINUE
172 RETURN
173 END IF
174 *
175 * NZ = maximum number of nonzero elements in each row of A, plus 1
176 *
177 NZ = N + 1
178 EPS = DLAMCH( 'Epsilon' )
179 SAFMIN = DLAMCH( 'Safe minimum' )
180 SAFE1 = NZ*SAFMIN
181 SAFE2 = SAFE1 / EPS
182 *
183 * Do for each right hand side
184 *
185 DO 140 J = 1, NRHS
186 *
187 COUNT = 1
188 LSTRES = THREE
189 20 CONTINUE
190 *
191 * Loop until stopping criterion is satisfied.
192 *
193 * Compute residual R = B - A * X
194 *
195 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
196 CALL ZHPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
197 *
198 * Compute componentwise relative backward error from formula
199 *
200 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
201 *
202 * where abs(Z) is the componentwise absolute value of the matrix
203 * or vector Z. If the i-th component of the denominator is less
204 * than SAFE2, then SAFE1 is added to the i-th components of the
205 * numerator and denominator before dividing.
206 *
207 DO 30 I = 1, N
208 RWORK( I ) = CABS1( B( I, J ) )
209 30 CONTINUE
210 *
211 * Compute abs(A)*abs(X) + abs(B).
212 *
213 KK = 1
214 IF( UPPER ) THEN
215 DO 50 K = 1, N
216 S = ZERO
217 XK = CABS1( X( K, J ) )
218 IK = KK
219 DO 40 I = 1, K - 1
220 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
221 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
222 IK = IK + 1
223 40 CONTINUE
224 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
225 $ XK + S
226 KK = KK + K
227 50 CONTINUE
228 ELSE
229 DO 70 K = 1, N
230 S = ZERO
231 XK = CABS1( X( K, J ) )
232 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
233 IK = KK + 1
234 DO 60 I = K + 1, N
235 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
236 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
237 IK = IK + 1
238 60 CONTINUE
239 RWORK( K ) = RWORK( K ) + S
240 KK = KK + ( N-K+1 )
241 70 CONTINUE
242 END IF
243 S = ZERO
244 DO 80 I = 1, N
245 IF( RWORK( I ).GT.SAFE2 ) THEN
246 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
247 ELSE
248 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
249 $ ( RWORK( I )+SAFE1 ) )
250 END IF
251 80 CONTINUE
252 BERR( J ) = S
253 *
254 * Test stopping criterion. Continue iterating if
255 * 1) The residual BERR(J) is larger than machine epsilon, and
256 * 2) BERR(J) decreased by at least a factor of 2 during the
257 * last iteration, and
258 * 3) At most ITMAX iterations tried.
259 *
260 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
261 $ COUNT.LE.ITMAX ) THEN
262 *
263 * Update solution and try again.
264 *
265 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
266 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
267 LSTRES = BERR( J )
268 COUNT = COUNT + 1
269 GO TO 20
270 END IF
271 *
272 * Bound error from formula
273 *
274 * norm(X - XTRUE) / norm(X) .le. FERR =
275 * norm( abs(inv(A))*
276 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
277 *
278 * where
279 * norm(Z) is the magnitude of the largest component of Z
280 * inv(A) is the inverse of A
281 * abs(Z) is the componentwise absolute value of the matrix or
282 * vector Z
283 * NZ is the maximum number of nonzeros in any row of A, plus 1
284 * EPS is machine epsilon
285 *
286 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
287 * is incremented by SAFE1 if the i-th component of
288 * abs(A)*abs(X) + abs(B) is less than SAFE2.
289 *
290 * Use ZLACN2 to estimate the infinity-norm of the matrix
291 * inv(A) * diag(W),
292 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
293 *
294 DO 90 I = 1, N
295 IF( RWORK( I ).GT.SAFE2 ) THEN
296 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
297 ELSE
298 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
299 $ SAFE1
300 END IF
301 90 CONTINUE
302 *
303 KASE = 0
304 100 CONTINUE
305 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
306 IF( KASE.NE.0 ) THEN
307 IF( KASE.EQ.1 ) THEN
308 *
309 * Multiply by diag(W)*inv(A**H).
310 *
311 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
312 DO 110 I = 1, N
313 WORK( I ) = RWORK( I )*WORK( I )
314 110 CONTINUE
315 ELSE IF( KASE.EQ.2 ) THEN
316 *
317 * Multiply by inv(A)*diag(W).
318 *
319 DO 120 I = 1, N
320 WORK( I ) = RWORK( I )*WORK( I )
321 120 CONTINUE
322 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
323 END IF
324 GO TO 100
325 END IF
326 *
327 * Normalize error.
328 *
329 LSTRES = ZERO
330 DO 130 I = 1, N
331 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
332 130 CONTINUE
333 IF( LSTRES.NE.ZERO )
334 $ FERR( J ) = FERR( J ) / LSTRES
335 *
336 140 CONTINUE
337 *
338 RETURN
339 *
340 * End of ZHPRFS
341 *
342 END
2 $ 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, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 INTEGER IPIV( * )
17 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
18 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
19 $ X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * ZHPRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is Hermitian indefinite
27 * and packed, and provides error bounds and backward error estimates
28 * for the solution.
29 *
30 * Arguments
31 * =========
32 *
33 * UPLO (input) CHARACTER*1
34 * = 'U': Upper triangle of A is stored;
35 * = 'L': Lower triangle of A is stored.
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 * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
45 * The upper or lower triangle of the Hermitian matrix A, packed
46 * columnwise in a linear array. The j-th column of A is stored
47 * in the array AP as follows:
48 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
49 * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
50 *
51 * AFP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
52 * The factored form of the matrix A. AFP contains the block
53 * diagonal matrix D and the multipliers used to obtain the
54 * factor U or L from the factorization A = U*D*U**H or
55 * A = L*D*L**H as computed by ZHPTRF, stored as a packed
56 * triangular matrix.
57 *
58 * IPIV (input) INTEGER array, dimension (N)
59 * Details of the interchanges and the block structure of D
60 * as determined by ZHPTRF.
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 ZHPTRS.
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, IK, J, K, KASE, KK, 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, ZHPMV, ZHPTRS, ZLACN2
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( LDB.LT.MAX( 1, N ) ) THEN
156 INFO = -8
157 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
158 INFO = -10
159 END IF
160 IF( INFO.NE.0 ) THEN
161 CALL XERBLA( 'ZHPRFS', -INFO )
162 RETURN
163 END IF
164 *
165 * Quick return if possible
166 *
167 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
168 DO 10 J = 1, NRHS
169 FERR( J ) = ZERO
170 BERR( J ) = ZERO
171 10 CONTINUE
172 RETURN
173 END IF
174 *
175 * NZ = maximum number of nonzero elements in each row of A, plus 1
176 *
177 NZ = N + 1
178 EPS = DLAMCH( 'Epsilon' )
179 SAFMIN = DLAMCH( 'Safe minimum' )
180 SAFE1 = NZ*SAFMIN
181 SAFE2 = SAFE1 / EPS
182 *
183 * Do for each right hand side
184 *
185 DO 140 J = 1, NRHS
186 *
187 COUNT = 1
188 LSTRES = THREE
189 20 CONTINUE
190 *
191 * Loop until stopping criterion is satisfied.
192 *
193 * Compute residual R = B - A * X
194 *
195 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
196 CALL ZHPMV( UPLO, N, -ONE, AP, X( 1, J ), 1, ONE, WORK, 1 )
197 *
198 * Compute componentwise relative backward error from formula
199 *
200 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
201 *
202 * where abs(Z) is the componentwise absolute value of the matrix
203 * or vector Z. If the i-th component of the denominator is less
204 * than SAFE2, then SAFE1 is added to the i-th components of the
205 * numerator and denominator before dividing.
206 *
207 DO 30 I = 1, N
208 RWORK( I ) = CABS1( B( I, J ) )
209 30 CONTINUE
210 *
211 * Compute abs(A)*abs(X) + abs(B).
212 *
213 KK = 1
214 IF( UPPER ) THEN
215 DO 50 K = 1, N
216 S = ZERO
217 XK = CABS1( X( K, J ) )
218 IK = KK
219 DO 40 I = 1, K - 1
220 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
221 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
222 IK = IK + 1
223 40 CONTINUE
224 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
225 $ XK + S
226 KK = KK + K
227 50 CONTINUE
228 ELSE
229 DO 70 K = 1, N
230 S = ZERO
231 XK = CABS1( X( K, J ) )
232 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
233 IK = KK + 1
234 DO 60 I = K + 1, N
235 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
236 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
237 IK = IK + 1
238 60 CONTINUE
239 RWORK( K ) = RWORK( K ) + S
240 KK = KK + ( N-K+1 )
241 70 CONTINUE
242 END IF
243 S = ZERO
244 DO 80 I = 1, N
245 IF( RWORK( I ).GT.SAFE2 ) THEN
246 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
247 ELSE
248 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
249 $ ( RWORK( I )+SAFE1 ) )
250 END IF
251 80 CONTINUE
252 BERR( J ) = S
253 *
254 * Test stopping criterion. Continue iterating if
255 * 1) The residual BERR(J) is larger than machine epsilon, and
256 * 2) BERR(J) decreased by at least a factor of 2 during the
257 * last iteration, and
258 * 3) At most ITMAX iterations tried.
259 *
260 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
261 $ COUNT.LE.ITMAX ) THEN
262 *
263 * Update solution and try again.
264 *
265 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
266 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
267 LSTRES = BERR( J )
268 COUNT = COUNT + 1
269 GO TO 20
270 END IF
271 *
272 * Bound error from formula
273 *
274 * norm(X - XTRUE) / norm(X) .le. FERR =
275 * norm( abs(inv(A))*
276 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
277 *
278 * where
279 * norm(Z) is the magnitude of the largest component of Z
280 * inv(A) is the inverse of A
281 * abs(Z) is the componentwise absolute value of the matrix or
282 * vector Z
283 * NZ is the maximum number of nonzeros in any row of A, plus 1
284 * EPS is machine epsilon
285 *
286 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
287 * is incremented by SAFE1 if the i-th component of
288 * abs(A)*abs(X) + abs(B) is less than SAFE2.
289 *
290 * Use ZLACN2 to estimate the infinity-norm of the matrix
291 * inv(A) * diag(W),
292 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
293 *
294 DO 90 I = 1, N
295 IF( RWORK( I ).GT.SAFE2 ) THEN
296 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
297 ELSE
298 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
299 $ SAFE1
300 END IF
301 90 CONTINUE
302 *
303 KASE = 0
304 100 CONTINUE
305 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
306 IF( KASE.NE.0 ) THEN
307 IF( KASE.EQ.1 ) THEN
308 *
309 * Multiply by diag(W)*inv(A**H).
310 *
311 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
312 DO 110 I = 1, N
313 WORK( I ) = RWORK( I )*WORK( I )
314 110 CONTINUE
315 ELSE IF( KASE.EQ.2 ) THEN
316 *
317 * Multiply by inv(A)*diag(W).
318 *
319 DO 120 I = 1, N
320 WORK( I ) = RWORK( I )*WORK( I )
321 120 CONTINUE
322 CALL ZHPTRS( UPLO, N, 1, AFP, IPIV, WORK, N, INFO )
323 END IF
324 GO TO 100
325 END IF
326 *
327 * Normalize error.
328 *
329 LSTRES = ZERO
330 DO 130 I = 1, N
331 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
332 130 CONTINUE
333 IF( LSTRES.NE.ZERO )
334 $ FERR( J ) = FERR( J ) / LSTRES
335 *
336 140 CONTINUE
337 *
338 RETURN
339 *
340 * End of ZHPRFS
341 *
342 END