1 SUBROUTINE ZGERFS( TRANS, 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 TRANS
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 * ZGERFS improves the computed solution to a system of linear
26 * equations and provides error bounds and backward error estimates for
27 * the solution.
28 *
29 * Arguments
30 * =========
31 *
32 * TRANS (input) CHARACTER*1
33 * Specifies the form of the system of equations:
34 * = 'N': A * X = B (No transpose)
35 * = 'T': A**T * X = B (Transpose)
36 * = 'C': A**H * X = B (Conjugate transpose)
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * NRHS (input) INTEGER
42 * The number of right hand sides, i.e., the number of columns
43 * of the matrices B and X. NRHS >= 0.
44 *
45 * A (input) COMPLEX*16 array, dimension (LDA,N)
46 * The original N-by-N matrix A.
47 *
48 * LDA (input) INTEGER
49 * The leading dimension of the array A. LDA >= max(1,N).
50 *
51 * AF (input) COMPLEX*16 array, dimension (LDAF,N)
52 * The factors L and U from the factorization A = P*L*U
53 * as computed by ZGETRF.
54 *
55 * LDAF (input) INTEGER
56 * The leading dimension of the array AF. LDAF >= max(1,N).
57 *
58 * IPIV (input) INTEGER array, dimension (N)
59 * The pivot indices from ZGETRF; for 1<=i<=N, row i of the
60 * matrix was interchanged with row IPIV(i).
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 ZGETRS.
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 NOTRAN
119 CHARACTER TRANSN, TRANST
120 INTEGER COUNT, I, J, K, KASE, NZ
121 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
122 COMPLEX*16 ZDUM
123 * ..
124 * .. Local Arrays ..
125 INTEGER ISAVE( 3 )
126 * ..
127 * .. External Functions ..
128 LOGICAL LSAME
129 DOUBLE PRECISION DLAMCH
130 EXTERNAL LSAME, DLAMCH
131 * ..
132 * .. External Subroutines ..
133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC ABS, DBLE, DIMAG, MAX
137 * ..
138 * .. Statement Functions ..
139 DOUBLE PRECISION CABS1
140 * ..
141 * .. Statement Function definitions ..
142 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
143 * ..
144 * .. Executable Statements ..
145 *
146 * Test the input parameters.
147 *
148 INFO = 0
149 NOTRAN = LSAME( TRANS, 'N' )
150 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
151 $ LSAME( TRANS, 'C' ) ) THEN
152 INFO = -1
153 ELSE IF( N.LT.0 ) THEN
154 INFO = -2
155 ELSE IF( NRHS.LT.0 ) THEN
156 INFO = -3
157 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
158 INFO = -5
159 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
160 INFO = -7
161 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
162 INFO = -10
163 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
164 INFO = -12
165 END IF
166 IF( INFO.NE.0 ) THEN
167 CALL XERBLA( 'ZGERFS', -INFO )
168 RETURN
169 END IF
170 *
171 * Quick return if possible
172 *
173 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
174 DO 10 J = 1, NRHS
175 FERR( J ) = ZERO
176 BERR( J ) = ZERO
177 10 CONTINUE
178 RETURN
179 END IF
180 *
181 IF( NOTRAN ) THEN
182 TRANSN = 'N'
183 TRANST = 'C'
184 ELSE
185 TRANSN = 'C'
186 TRANST = 'N'
187 END IF
188 *
189 * NZ = maximum number of nonzero elements in each row of A, plus 1
190 *
191 NZ = N + 1
192 EPS = DLAMCH( 'Epsilon' )
193 SAFMIN = DLAMCH( 'Safe minimum' )
194 SAFE1 = NZ*SAFMIN
195 SAFE2 = SAFE1 / EPS
196 *
197 * Do for each right hand side
198 *
199 DO 140 J = 1, NRHS
200 *
201 COUNT = 1
202 LSTRES = THREE
203 20 CONTINUE
204 *
205 * Loop until stopping criterion is satisfied.
206 *
207 * Compute residual R = B - op(A) * X,
208 * where op(A) = A, A**T, or A**H, depending on TRANS.
209 *
210 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
211 CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK,
212 $ 1 )
213 *
214 * Compute componentwise relative backward error from formula
215 *
216 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
217 *
218 * where abs(Z) is the componentwise absolute value of the matrix
219 * or vector Z. If the i-th component of the denominator is less
220 * than SAFE2, then SAFE1 is added to the i-th components of the
221 * numerator and denominator before dividing.
222 *
223 DO 30 I = 1, N
224 RWORK( I ) = CABS1( B( I, J ) )
225 30 CONTINUE
226 *
227 * Compute abs(op(A))*abs(X) + abs(B).
228 *
229 IF( NOTRAN ) THEN
230 DO 50 K = 1, N
231 XK = CABS1( X( K, J ) )
232 DO 40 I = 1, N
233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
234 40 CONTINUE
235 50 CONTINUE
236 ELSE
237 DO 70 K = 1, N
238 S = ZERO
239 DO 60 I = 1, N
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 ZGETRS( TRANS, 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(op(A)))*
278 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
279 *
280 * where
281 * norm(Z) is the magnitude of the largest component of Z
282 * inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B))
289 * is incremented by SAFE1 if the i-th component of
290 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
291 *
292 * Use ZLACN2 to estimate the infinity-norm of the matrix
293 * inv(op(A)) * diag(W),
294 * where W = abs(R) + NZ*EPS*( abs(op(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(op(A)**H).
312 *
313 CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N,
314 $ INFO )
315 DO 110 I = 1, N
316 WORK( I ) = RWORK( I )*WORK( I )
317 110 CONTINUE
318 ELSE
319 *
320 * Multiply by inv(op(A))*diag(W).
321 *
322 DO 120 I = 1, N
323 WORK( I ) = RWORK( I )*WORK( I )
324 120 CONTINUE
325 CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N,
326 $ INFO )
327 END IF
328 GO TO 100
329 END IF
330 *
331 * Normalize error.
332 *
333 LSTRES = ZERO
334 DO 130 I = 1, N
335 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
336 130 CONTINUE
337 IF( LSTRES.NE.ZERO )
338 $ FERR( J ) = FERR( J ) / LSTRES
339 *
340 140 CONTINUE
341 *
342 RETURN
343 *
344 * End of ZGERFS
345 *
346 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 TRANS
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 * ZGERFS improves the computed solution to a system of linear
26 * equations and provides error bounds and backward error estimates for
27 * the solution.
28 *
29 * Arguments
30 * =========
31 *
32 * TRANS (input) CHARACTER*1
33 * Specifies the form of the system of equations:
34 * = 'N': A * X = B (No transpose)
35 * = 'T': A**T * X = B (Transpose)
36 * = 'C': A**H * X = B (Conjugate transpose)
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * NRHS (input) INTEGER
42 * The number of right hand sides, i.e., the number of columns
43 * of the matrices B and X. NRHS >= 0.
44 *
45 * A (input) COMPLEX*16 array, dimension (LDA,N)
46 * The original N-by-N matrix A.
47 *
48 * LDA (input) INTEGER
49 * The leading dimension of the array A. LDA >= max(1,N).
50 *
51 * AF (input) COMPLEX*16 array, dimension (LDAF,N)
52 * The factors L and U from the factorization A = P*L*U
53 * as computed by ZGETRF.
54 *
55 * LDAF (input) INTEGER
56 * The leading dimension of the array AF. LDAF >= max(1,N).
57 *
58 * IPIV (input) INTEGER array, dimension (N)
59 * The pivot indices from ZGETRF; for 1<=i<=N, row i of the
60 * matrix was interchanged with row IPIV(i).
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 ZGETRS.
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 NOTRAN
119 CHARACTER TRANSN, TRANST
120 INTEGER COUNT, I, J, K, KASE, NZ
121 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
122 COMPLEX*16 ZDUM
123 * ..
124 * .. Local Arrays ..
125 INTEGER ISAVE( 3 )
126 * ..
127 * .. External Functions ..
128 LOGICAL LSAME
129 DOUBLE PRECISION DLAMCH
130 EXTERNAL LSAME, DLAMCH
131 * ..
132 * .. External Subroutines ..
133 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGETRS, ZLACN2
134 * ..
135 * .. Intrinsic Functions ..
136 INTRINSIC ABS, DBLE, DIMAG, MAX
137 * ..
138 * .. Statement Functions ..
139 DOUBLE PRECISION CABS1
140 * ..
141 * .. Statement Function definitions ..
142 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
143 * ..
144 * .. Executable Statements ..
145 *
146 * Test the input parameters.
147 *
148 INFO = 0
149 NOTRAN = LSAME( TRANS, 'N' )
150 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
151 $ LSAME( TRANS, 'C' ) ) THEN
152 INFO = -1
153 ELSE IF( N.LT.0 ) THEN
154 INFO = -2
155 ELSE IF( NRHS.LT.0 ) THEN
156 INFO = -3
157 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
158 INFO = -5
159 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
160 INFO = -7
161 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
162 INFO = -10
163 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
164 INFO = -12
165 END IF
166 IF( INFO.NE.0 ) THEN
167 CALL XERBLA( 'ZGERFS', -INFO )
168 RETURN
169 END IF
170 *
171 * Quick return if possible
172 *
173 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
174 DO 10 J = 1, NRHS
175 FERR( J ) = ZERO
176 BERR( J ) = ZERO
177 10 CONTINUE
178 RETURN
179 END IF
180 *
181 IF( NOTRAN ) THEN
182 TRANSN = 'N'
183 TRANST = 'C'
184 ELSE
185 TRANSN = 'C'
186 TRANST = 'N'
187 END IF
188 *
189 * NZ = maximum number of nonzero elements in each row of A, plus 1
190 *
191 NZ = N + 1
192 EPS = DLAMCH( 'Epsilon' )
193 SAFMIN = DLAMCH( 'Safe minimum' )
194 SAFE1 = NZ*SAFMIN
195 SAFE2 = SAFE1 / EPS
196 *
197 * Do for each right hand side
198 *
199 DO 140 J = 1, NRHS
200 *
201 COUNT = 1
202 LSTRES = THREE
203 20 CONTINUE
204 *
205 * Loop until stopping criterion is satisfied.
206 *
207 * Compute residual R = B - op(A) * X,
208 * where op(A) = A, A**T, or A**H, depending on TRANS.
209 *
210 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
211 CALL ZGEMV( TRANS, N, N, -ONE, A, LDA, X( 1, J ), 1, ONE, WORK,
212 $ 1 )
213 *
214 * Compute componentwise relative backward error from formula
215 *
216 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
217 *
218 * where abs(Z) is the componentwise absolute value of the matrix
219 * or vector Z. If the i-th component of the denominator is less
220 * than SAFE2, then SAFE1 is added to the i-th components of the
221 * numerator and denominator before dividing.
222 *
223 DO 30 I = 1, N
224 RWORK( I ) = CABS1( B( I, J ) )
225 30 CONTINUE
226 *
227 * Compute abs(op(A))*abs(X) + abs(B).
228 *
229 IF( NOTRAN ) THEN
230 DO 50 K = 1, N
231 XK = CABS1( X( K, J ) )
232 DO 40 I = 1, N
233 RWORK( I ) = RWORK( I ) + CABS1( A( I, K ) )*XK
234 40 CONTINUE
235 50 CONTINUE
236 ELSE
237 DO 70 K = 1, N
238 S = ZERO
239 DO 60 I = 1, N
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 ZGETRS( TRANS, 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(op(A)))*
278 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
279 *
280 * where
281 * norm(Z) is the magnitude of the largest component of Z
282 * inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B))
289 * is incremented by SAFE1 if the i-th component of
290 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
291 *
292 * Use ZLACN2 to estimate the infinity-norm of the matrix
293 * inv(op(A)) * diag(W),
294 * where W = abs(R) + NZ*EPS*( abs(op(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(op(A)**H).
312 *
313 CALL ZGETRS( TRANST, N, 1, AF, LDAF, IPIV, WORK, N,
314 $ INFO )
315 DO 110 I = 1, N
316 WORK( I ) = RWORK( I )*WORK( I )
317 110 CONTINUE
318 ELSE
319 *
320 * Multiply by inv(op(A))*diag(W).
321 *
322 DO 120 I = 1, N
323 WORK( I ) = RWORK( I )*WORK( I )
324 120 CONTINUE
325 CALL ZGETRS( TRANSN, N, 1, AF, LDAF, IPIV, WORK, N,
326 $ INFO )
327 END IF
328 GO TO 100
329 END IF
330 *
331 * Normalize error.
332 *
333 LSTRES = ZERO
334 DO 130 I = 1, N
335 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
336 130 CONTINUE
337 IF( LSTRES.NE.ZERO )
338 $ FERR( J ) = FERR( J ) / LSTRES
339 *
340 140 CONTINUE
341 *
342 RETURN
343 *
344 * End of ZGERFS
345 *
346 END