1 SUBROUTINE ZPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
2 $ 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 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
18 $ X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPPRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian 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) COMPLEX*16 array, dimension (N*(N+1)/2)
44 * The upper or lower triangle of the Hermitian 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) COMPLEX*16 array, dimension (N*(N+1)/2)
51 * The triangular factor U or L from the Cholesky factorization
52 * A = U**H*U or A = L*L**H, 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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
63 * On entry, the solution matrix X, as computed by ZPPTRS.
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) COMPLEX*16 array, dimension (2*N)
85 *
86 * RWORK (workspace) DOUBLE PRECISION 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 COMPLEX*16 CONE
105 PARAMETER ( CONE = ( 1.0D+0, 0.0D+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 COMPLEX*16 ZDUM
116 * ..
117 * .. Local Arrays ..
118 INTEGER ISAVE( 3 )
119 * ..
120 * .. External Subroutines ..
121 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, DBLE, DIMAG, MAX
125 * ..
126 * .. External Functions ..
127 LOGICAL LSAME
128 DOUBLE PRECISION DLAMCH
129 EXTERNAL LSAME, DLAMCH
130 * ..
131 * .. Statement Functions ..
132 DOUBLE PRECISION CABS1
133 * ..
134 * .. Statement Function definitions ..
135 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
136 * ..
137 * .. Executable Statements ..
138 *
139 * Test the input parameters.
140 *
141 INFO = 0
142 UPPER = LSAME( UPLO, 'U' )
143 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( NRHS.LT.0 ) THEN
148 INFO = -3
149 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
150 INFO = -7
151 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
152 INFO = -9
153 END IF
154 IF( INFO.NE.0 ) THEN
155 CALL XERBLA( 'ZPPRFS', -INFO )
156 RETURN
157 END IF
158 *
159 * Quick return if possible
160 *
161 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
162 DO 10 J = 1, NRHS
163 FERR( J ) = ZERO
164 BERR( J ) = ZERO
165 10 CONTINUE
166 RETURN
167 END IF
168 *
169 * NZ = maximum number of nonzero elements in each row of A, plus 1
170 *
171 NZ = N + 1
172 EPS = DLAMCH( 'Epsilon' )
173 SAFMIN = DLAMCH( 'Safe minimum' )
174 SAFE1 = NZ*SAFMIN
175 SAFE2 = SAFE1 / EPS
176 *
177 * Do for each right hand side
178 *
179 DO 140 J = 1, NRHS
180 *
181 COUNT = 1
182 LSTRES = THREE
183 20 CONTINUE
184 *
185 * Loop until stopping criterion is satisfied.
186 *
187 * Compute residual R = B - A * X
188 *
189 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
190 CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
191 *
192 * Compute componentwise relative backward error from formula
193 *
194 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
195 *
196 * where abs(Z) is the componentwise absolute value of the matrix
197 * or vector Z. If the i-th component of the denominator is less
198 * than SAFE2, then SAFE1 is added to the i-th components of the
199 * numerator and denominator before dividing.
200 *
201 DO 30 I = 1, N
202 RWORK( I ) = CABS1( B( I, J ) )
203 30 CONTINUE
204 *
205 * Compute abs(A)*abs(X) + abs(B).
206 *
207 KK = 1
208 IF( UPPER ) THEN
209 DO 50 K = 1, N
210 S = ZERO
211 XK = CABS1( X( K, J ) )
212 IK = KK
213 DO 40 I = 1, K - 1
214 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
215 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
216 IK = IK + 1
217 40 CONTINUE
218 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
219 $ XK + S
220 KK = KK + K
221 50 CONTINUE
222 ELSE
223 DO 70 K = 1, N
224 S = ZERO
225 XK = CABS1( X( K, J ) )
226 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
227 IK = KK + 1
228 DO 60 I = K + 1, N
229 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
230 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
231 IK = IK + 1
232 60 CONTINUE
233 RWORK( K ) = RWORK( K ) + S
234 KK = KK + ( N-K+1 )
235 70 CONTINUE
236 END IF
237 S = ZERO
238 DO 80 I = 1, N
239 IF( RWORK( I ).GT.SAFE2 ) THEN
240 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
241 ELSE
242 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
243 $ ( RWORK( I )+SAFE1 ) )
244 END IF
245 80 CONTINUE
246 BERR( J ) = S
247 *
248 * Test stopping criterion. Continue iterating if
249 * 1) The residual BERR(J) is larger than machine epsilon, and
250 * 2) BERR(J) decreased by at least a factor of 2 during the
251 * last iteration, and
252 * 3) At most ITMAX iterations tried.
253 *
254 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
255 $ COUNT.LE.ITMAX ) THEN
256 *
257 * Update solution and try again.
258 *
259 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
260 CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
261 LSTRES = BERR( J )
262 COUNT = COUNT + 1
263 GO TO 20
264 END IF
265 *
266 * Bound error from formula
267 *
268 * norm(X - XTRUE) / norm(X) .le. FERR =
269 * norm( abs(inv(A))*
270 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
271 *
272 * where
273 * norm(Z) is the magnitude of the largest component of Z
274 * inv(A) is the inverse of A
275 * abs(Z) is the componentwise absolute value of the matrix or
276 * vector Z
277 * NZ is the maximum number of nonzeros in any row of A, plus 1
278 * EPS is machine epsilon
279 *
280 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
281 * is incremented by SAFE1 if the i-th component of
282 * abs(A)*abs(X) + abs(B) is less than SAFE2.
283 *
284 * Use ZLACN2 to estimate the infinity-norm of the matrix
285 * inv(A) * diag(W),
286 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
287 *
288 DO 90 I = 1, N
289 IF( RWORK( I ).GT.SAFE2 ) THEN
290 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
291 ELSE
292 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
293 $ SAFE1
294 END IF
295 90 CONTINUE
296 *
297 KASE = 0
298 100 CONTINUE
299 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
300 IF( KASE.NE.0 ) THEN
301 IF( KASE.EQ.1 ) THEN
302 *
303 * Multiply by diag(W)*inv(A**H).
304 *
305 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
306 DO 110 I = 1, N
307 WORK( I ) = RWORK( I )*WORK( I )
308 110 CONTINUE
309 ELSE IF( KASE.EQ.2 ) THEN
310 *
311 * Multiply by inv(A)*diag(W).
312 *
313 DO 120 I = 1, N
314 WORK( I ) = RWORK( I )*WORK( I )
315 120 CONTINUE
316 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, 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, CABS1( 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 ZPPRFS
335 *
336 END
2 $ 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 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
18 $ X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPPRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian 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) COMPLEX*16 array, dimension (N*(N+1)/2)
44 * The upper or lower triangle of the Hermitian 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) COMPLEX*16 array, dimension (N*(N+1)/2)
51 * The triangular factor U or L from the Cholesky factorization
52 * A = U**H*U or A = L*L**H, 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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDX,NRHS)
63 * On entry, the solution matrix X, as computed by ZPPTRS.
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) COMPLEX*16 array, dimension (2*N)
85 *
86 * RWORK (workspace) DOUBLE PRECISION 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 COMPLEX*16 CONE
105 PARAMETER ( CONE = ( 1.0D+0, 0.0D+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 COMPLEX*16 ZDUM
116 * ..
117 * .. Local Arrays ..
118 INTEGER ISAVE( 3 )
119 * ..
120 * .. External Subroutines ..
121 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHPMV, ZLACN2, ZPPTRS
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, DBLE, DIMAG, MAX
125 * ..
126 * .. External Functions ..
127 LOGICAL LSAME
128 DOUBLE PRECISION DLAMCH
129 EXTERNAL LSAME, DLAMCH
130 * ..
131 * .. Statement Functions ..
132 DOUBLE PRECISION CABS1
133 * ..
134 * .. Statement Function definitions ..
135 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
136 * ..
137 * .. Executable Statements ..
138 *
139 * Test the input parameters.
140 *
141 INFO = 0
142 UPPER = LSAME( UPLO, 'U' )
143 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
144 INFO = -1
145 ELSE IF( N.LT.0 ) THEN
146 INFO = -2
147 ELSE IF( NRHS.LT.0 ) THEN
148 INFO = -3
149 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
150 INFO = -7
151 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
152 INFO = -9
153 END IF
154 IF( INFO.NE.0 ) THEN
155 CALL XERBLA( 'ZPPRFS', -INFO )
156 RETURN
157 END IF
158 *
159 * Quick return if possible
160 *
161 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
162 DO 10 J = 1, NRHS
163 FERR( J ) = ZERO
164 BERR( J ) = ZERO
165 10 CONTINUE
166 RETURN
167 END IF
168 *
169 * NZ = maximum number of nonzero elements in each row of A, plus 1
170 *
171 NZ = N + 1
172 EPS = DLAMCH( 'Epsilon' )
173 SAFMIN = DLAMCH( 'Safe minimum' )
174 SAFE1 = NZ*SAFMIN
175 SAFE2 = SAFE1 / EPS
176 *
177 * Do for each right hand side
178 *
179 DO 140 J = 1, NRHS
180 *
181 COUNT = 1
182 LSTRES = THREE
183 20 CONTINUE
184 *
185 * Loop until stopping criterion is satisfied.
186 *
187 * Compute residual R = B - A * X
188 *
189 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
190 CALL ZHPMV( UPLO, N, -CONE, AP, X( 1, J ), 1, CONE, WORK, 1 )
191 *
192 * Compute componentwise relative backward error from formula
193 *
194 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
195 *
196 * where abs(Z) is the componentwise absolute value of the matrix
197 * or vector Z. If the i-th component of the denominator is less
198 * than SAFE2, then SAFE1 is added to the i-th components of the
199 * numerator and denominator before dividing.
200 *
201 DO 30 I = 1, N
202 RWORK( I ) = CABS1( B( I, J ) )
203 30 CONTINUE
204 *
205 * Compute abs(A)*abs(X) + abs(B).
206 *
207 KK = 1
208 IF( UPPER ) THEN
209 DO 50 K = 1, N
210 S = ZERO
211 XK = CABS1( X( K, J ) )
212 IK = KK
213 DO 40 I = 1, K - 1
214 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
215 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
216 IK = IK + 1
217 40 CONTINUE
218 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK+K-1 ) ) )*
219 $ XK + S
220 KK = KK + K
221 50 CONTINUE
222 ELSE
223 DO 70 K = 1, N
224 S = ZERO
225 XK = CABS1( X( K, J ) )
226 RWORK( K ) = RWORK( K ) + ABS( DBLE( AP( KK ) ) )*XK
227 IK = KK + 1
228 DO 60 I = K + 1, N
229 RWORK( I ) = RWORK( I ) + CABS1( AP( IK ) )*XK
230 S = S + CABS1( AP( IK ) )*CABS1( X( I, J ) )
231 IK = IK + 1
232 60 CONTINUE
233 RWORK( K ) = RWORK( K ) + S
234 KK = KK + ( N-K+1 )
235 70 CONTINUE
236 END IF
237 S = ZERO
238 DO 80 I = 1, N
239 IF( RWORK( I ).GT.SAFE2 ) THEN
240 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
241 ELSE
242 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
243 $ ( RWORK( I )+SAFE1 ) )
244 END IF
245 80 CONTINUE
246 BERR( J ) = S
247 *
248 * Test stopping criterion. Continue iterating if
249 * 1) The residual BERR(J) is larger than machine epsilon, and
250 * 2) BERR(J) decreased by at least a factor of 2 during the
251 * last iteration, and
252 * 3) At most ITMAX iterations tried.
253 *
254 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
255 $ COUNT.LE.ITMAX ) THEN
256 *
257 * Update solution and try again.
258 *
259 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
260 CALL ZAXPY( N, CONE, WORK, 1, X( 1, J ), 1 )
261 LSTRES = BERR( J )
262 COUNT = COUNT + 1
263 GO TO 20
264 END IF
265 *
266 * Bound error from formula
267 *
268 * norm(X - XTRUE) / norm(X) .le. FERR =
269 * norm( abs(inv(A))*
270 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
271 *
272 * where
273 * norm(Z) is the magnitude of the largest component of Z
274 * inv(A) is the inverse of A
275 * abs(Z) is the componentwise absolute value of the matrix or
276 * vector Z
277 * NZ is the maximum number of nonzeros in any row of A, plus 1
278 * EPS is machine epsilon
279 *
280 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
281 * is incremented by SAFE1 if the i-th component of
282 * abs(A)*abs(X) + abs(B) is less than SAFE2.
283 *
284 * Use ZLACN2 to estimate the infinity-norm of the matrix
285 * inv(A) * diag(W),
286 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
287 *
288 DO 90 I = 1, N
289 IF( RWORK( I ).GT.SAFE2 ) THEN
290 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
291 ELSE
292 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
293 $ SAFE1
294 END IF
295 90 CONTINUE
296 *
297 KASE = 0
298 100 CONTINUE
299 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
300 IF( KASE.NE.0 ) THEN
301 IF( KASE.EQ.1 ) THEN
302 *
303 * Multiply by diag(W)*inv(A**H).
304 *
305 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, INFO )
306 DO 110 I = 1, N
307 WORK( I ) = RWORK( I )*WORK( I )
308 110 CONTINUE
309 ELSE IF( KASE.EQ.2 ) THEN
310 *
311 * Multiply by inv(A)*diag(W).
312 *
313 DO 120 I = 1, N
314 WORK( I ) = RWORK( I )*WORK( I )
315 120 CONTINUE
316 CALL ZPPTRS( UPLO, N, 1, AFP, WORK, N, 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, CABS1( 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 ZPPRFS
335 *
336 END