1 SUBROUTINE ZPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
2 $ LDB, 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, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPBRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian positive definite
26 * and banded, 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 * KD (input) INTEGER
40 * The number of superdiagonals of the matrix A if UPLO = 'U',
41 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
42 *
43 * NRHS (input) INTEGER
44 * The number of right hand sides, i.e., the number of columns
45 * of the matrices B and X. NRHS >= 0.
46 *
47 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
48 * The upper or lower triangle of the Hermitian band matrix A,
49 * stored in the first KD+1 rows of the array. The j-th column
50 * of A is stored in the j-th column of the array AB as follows:
51 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
52 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
53 *
54 * LDAB (input) INTEGER
55 * The leading dimension of the array AB. LDAB >= KD+1.
56 *
57 * AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
58 * The triangular factor U or L from the Cholesky factorization
59 * A = U**H*U or A = L*L**H of the band matrix A as computed by
60 * ZPBTRF, in the same storage format as A (see AB).
61 *
62 * LDAFB (input) INTEGER
63 * The leading dimension of the array AFB. LDAFB >= KD+1.
64 *
65 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
66 * The right hand side matrix B.
67 *
68 * LDB (input) INTEGER
69 * The leading dimension of the array B. LDB >= max(1,N).
70 *
71 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
72 * On entry, the solution matrix X, as computed by ZPBTRS.
73 * On exit, the improved solution matrix X.
74 *
75 * LDX (input) INTEGER
76 * The leading dimension of the array X. LDX >= max(1,N).
77 *
78 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
79 * The estimated forward error bound for each solution vector
80 * X(j) (the j-th column of the solution matrix X).
81 * If XTRUE is the true solution corresponding to X(j), FERR(j)
82 * is an estimated upper bound for the magnitude of the largest
83 * element in (X(j) - XTRUE) divided by the magnitude of the
84 * largest element in X(j). The estimate is as reliable as
85 * the estimate for RCOND, and is almost always a slight
86 * overestimate of the true error.
87 *
88 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
89 * The componentwise relative backward error of each solution
90 * vector X(j) (i.e., the smallest relative change in
91 * any element of A or B that makes X(j) an exact solution).
92 *
93 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
94 *
95 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
96 *
97 * INFO (output) INTEGER
98 * = 0: successful exit
99 * < 0: if INFO = -i, the i-th argument had an illegal value
100 *
101 * Internal Parameters
102 * ===================
103 *
104 * ITMAX is the maximum number of steps of iterative refinement.
105 *
106 * =====================================================================
107 *
108 * .. Parameters ..
109 INTEGER ITMAX
110 PARAMETER ( ITMAX = 5 )
111 DOUBLE PRECISION ZERO
112 PARAMETER ( ZERO = 0.0D+0 )
113 COMPLEX*16 ONE
114 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
115 DOUBLE PRECISION TWO
116 PARAMETER ( TWO = 2.0D+0 )
117 DOUBLE PRECISION THREE
118 PARAMETER ( THREE = 3.0D+0 )
119 * ..
120 * .. Local Scalars ..
121 LOGICAL UPPER
122 INTEGER COUNT, I, J, K, KASE, L, NZ
123 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
124 COMPLEX*16 ZDUM
125 * ..
126 * .. Local Arrays ..
127 INTEGER ISAVE( 3 )
128 * ..
129 * .. External Subroutines ..
130 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHBMV, ZLACN2, ZPBTRS
131 * ..
132 * .. Intrinsic Functions ..
133 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
134 * ..
135 * .. External Functions ..
136 LOGICAL LSAME
137 DOUBLE PRECISION DLAMCH
138 EXTERNAL LSAME, DLAMCH
139 * ..
140 * .. Statement Functions ..
141 DOUBLE PRECISION CABS1
142 * ..
143 * .. Statement Function definitions ..
144 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
145 * ..
146 * .. Executable Statements ..
147 *
148 * Test the input parameters.
149 *
150 INFO = 0
151 UPPER = LSAME( UPLO, 'U' )
152 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
153 INFO = -1
154 ELSE IF( N.LT.0 ) THEN
155 INFO = -2
156 ELSE IF( KD.LT.0 ) THEN
157 INFO = -3
158 ELSE IF( NRHS.LT.0 ) THEN
159 INFO = -4
160 ELSE IF( LDAB.LT.KD+1 ) THEN
161 INFO = -6
162 ELSE IF( LDAFB.LT.KD+1 ) THEN
163 INFO = -8
164 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
165 INFO = -10
166 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
167 INFO = -12
168 END IF
169 IF( INFO.NE.0 ) THEN
170 CALL XERBLA( 'ZPBRFS', -INFO )
171 RETURN
172 END IF
173 *
174 * Quick return if possible
175 *
176 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
177 DO 10 J = 1, NRHS
178 FERR( J ) = ZERO
179 BERR( J ) = ZERO
180 10 CONTINUE
181 RETURN
182 END IF
183 *
184 * NZ = maximum number of nonzero elements in each row of A, plus 1
185 *
186 NZ = MIN( N+1, 2*KD+2 )
187 EPS = DLAMCH( 'Epsilon' )
188 SAFMIN = DLAMCH( 'Safe minimum' )
189 SAFE1 = NZ*SAFMIN
190 SAFE2 = SAFE1 / EPS
191 *
192 * Do for each right hand side
193 *
194 DO 140 J = 1, NRHS
195 *
196 COUNT = 1
197 LSTRES = THREE
198 20 CONTINUE
199 *
200 * Loop until stopping criterion is satisfied.
201 *
202 * Compute residual R = B - A * X
203 *
204 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
205 CALL ZHBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
206 $ 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 L = KD + 1 - K
228 DO 40 I = MAX( 1, K-KD ), K - 1
229 RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
230 S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
231 40 CONTINUE
232 RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( KD+1, K ) ) )*
233 $ XK + S
234 50 CONTINUE
235 ELSE
236 DO 70 K = 1, N
237 S = ZERO
238 XK = CABS1( X( K, J ) )
239 RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( 1, K ) ) )*XK
240 L = 1 - K
241 DO 60 I = K + 1, MIN( N, K+KD )
242 RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
243 S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
244 60 CONTINUE
245 RWORK( K ) = RWORK( K ) + S
246 70 CONTINUE
247 END IF
248 S = ZERO
249 DO 80 I = 1, N
250 IF( RWORK( I ).GT.SAFE2 ) THEN
251 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
252 ELSE
253 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
254 $ ( RWORK( I )+SAFE1 ) )
255 END IF
256 80 CONTINUE
257 BERR( J ) = S
258 *
259 * Test stopping criterion. Continue iterating if
260 * 1) The residual BERR(J) is larger than machine epsilon, and
261 * 2) BERR(J) decreased by at least a factor of 2 during the
262 * last iteration, and
263 * 3) At most ITMAX iterations tried.
264 *
265 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
266 $ COUNT.LE.ITMAX ) THEN
267 *
268 * Update solution and try again.
269 *
270 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
271 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
272 LSTRES = BERR( J )
273 COUNT = COUNT + 1
274 GO TO 20
275 END IF
276 *
277 * Bound error from formula
278 *
279 * norm(X - XTRUE) / norm(X) .le. FERR =
280 * norm( abs(inv(A))*
281 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
282 *
283 * where
284 * norm(Z) is the magnitude of the largest component of Z
285 * inv(A) is the inverse of A
286 * abs(Z) is the componentwise absolute value of the matrix or
287 * vector Z
288 * NZ is the maximum number of nonzeros in any row of A, plus 1
289 * EPS is machine epsilon
290 *
291 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
292 * is incremented by SAFE1 if the i-th component of
293 * abs(A)*abs(X) + abs(B) is less than SAFE2.
294 *
295 * Use ZLACN2 to estimate the infinity-norm of the matrix
296 * inv(A) * diag(W),
297 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
298 *
299 DO 90 I = 1, N
300 IF( RWORK( I ).GT.SAFE2 ) THEN
301 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
302 ELSE
303 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
304 $ SAFE1
305 END IF
306 90 CONTINUE
307 *
308 KASE = 0
309 100 CONTINUE
310 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
311 IF( KASE.NE.0 ) THEN
312 IF( KASE.EQ.1 ) THEN
313 *
314 * Multiply by diag(W)*inv(A**H).
315 *
316 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
317 DO 110 I = 1, N
318 WORK( I ) = RWORK( I )*WORK( I )
319 110 CONTINUE
320 ELSE IF( KASE.EQ.2 ) THEN
321 *
322 * Multiply by inv(A)*diag(W).
323 *
324 DO 120 I = 1, N
325 WORK( I ) = RWORK( I )*WORK( I )
326 120 CONTINUE
327 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
328 END IF
329 GO TO 100
330 END IF
331 *
332 * Normalize error.
333 *
334 LSTRES = ZERO
335 DO 130 I = 1, N
336 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
337 130 CONTINUE
338 IF( LSTRES.NE.ZERO )
339 $ FERR( J ) = FERR( J ) / LSTRES
340 *
341 140 CONTINUE
342 *
343 RETURN
344 *
345 * End of ZPBRFS
346 *
347 END
2 $ LDB, 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, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
14 * ..
15 * .. Array Arguments ..
16 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
17 COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
18 $ WORK( * ), X( LDX, * )
19 * ..
20 *
21 * Purpose
22 * =======
23 *
24 * ZPBRFS improves the computed solution to a system of linear
25 * equations when the coefficient matrix is Hermitian positive definite
26 * and banded, 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 * KD (input) INTEGER
40 * The number of superdiagonals of the matrix A if UPLO = 'U',
41 * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
42 *
43 * NRHS (input) INTEGER
44 * The number of right hand sides, i.e., the number of columns
45 * of the matrices B and X. NRHS >= 0.
46 *
47 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
48 * The upper or lower triangle of the Hermitian band matrix A,
49 * stored in the first KD+1 rows of the array. The j-th column
50 * of A is stored in the j-th column of the array AB as follows:
51 * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
52 * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
53 *
54 * LDAB (input) INTEGER
55 * The leading dimension of the array AB. LDAB >= KD+1.
56 *
57 * AFB (input) COMPLEX*16 array, dimension (LDAFB,N)
58 * The triangular factor U or L from the Cholesky factorization
59 * A = U**H*U or A = L*L**H of the band matrix A as computed by
60 * ZPBTRF, in the same storage format as A (see AB).
61 *
62 * LDAFB (input) INTEGER
63 * The leading dimension of the array AFB. LDAFB >= KD+1.
64 *
65 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
66 * The right hand side matrix B.
67 *
68 * LDB (input) INTEGER
69 * The leading dimension of the array B. LDB >= max(1,N).
70 *
71 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
72 * On entry, the solution matrix X, as computed by ZPBTRS.
73 * On exit, the improved solution matrix X.
74 *
75 * LDX (input) INTEGER
76 * The leading dimension of the array X. LDX >= max(1,N).
77 *
78 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
79 * The estimated forward error bound for each solution vector
80 * X(j) (the j-th column of the solution matrix X).
81 * If XTRUE is the true solution corresponding to X(j), FERR(j)
82 * is an estimated upper bound for the magnitude of the largest
83 * element in (X(j) - XTRUE) divided by the magnitude of the
84 * largest element in X(j). The estimate is as reliable as
85 * the estimate for RCOND, and is almost always a slight
86 * overestimate of the true error.
87 *
88 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
89 * The componentwise relative backward error of each solution
90 * vector X(j) (i.e., the smallest relative change in
91 * any element of A or B that makes X(j) an exact solution).
92 *
93 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
94 *
95 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
96 *
97 * INFO (output) INTEGER
98 * = 0: successful exit
99 * < 0: if INFO = -i, the i-th argument had an illegal value
100 *
101 * Internal Parameters
102 * ===================
103 *
104 * ITMAX is the maximum number of steps of iterative refinement.
105 *
106 * =====================================================================
107 *
108 * .. Parameters ..
109 INTEGER ITMAX
110 PARAMETER ( ITMAX = 5 )
111 DOUBLE PRECISION ZERO
112 PARAMETER ( ZERO = 0.0D+0 )
113 COMPLEX*16 ONE
114 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) )
115 DOUBLE PRECISION TWO
116 PARAMETER ( TWO = 2.0D+0 )
117 DOUBLE PRECISION THREE
118 PARAMETER ( THREE = 3.0D+0 )
119 * ..
120 * .. Local Scalars ..
121 LOGICAL UPPER
122 INTEGER COUNT, I, J, K, KASE, L, NZ
123 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
124 COMPLEX*16 ZDUM
125 * ..
126 * .. Local Arrays ..
127 INTEGER ISAVE( 3 )
128 * ..
129 * .. External Subroutines ..
130 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZHBMV, ZLACN2, ZPBTRS
131 * ..
132 * .. Intrinsic Functions ..
133 INTRINSIC ABS, DBLE, DIMAG, MAX, MIN
134 * ..
135 * .. External Functions ..
136 LOGICAL LSAME
137 DOUBLE PRECISION DLAMCH
138 EXTERNAL LSAME, DLAMCH
139 * ..
140 * .. Statement Functions ..
141 DOUBLE PRECISION CABS1
142 * ..
143 * .. Statement Function definitions ..
144 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
145 * ..
146 * .. Executable Statements ..
147 *
148 * Test the input parameters.
149 *
150 INFO = 0
151 UPPER = LSAME( UPLO, 'U' )
152 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
153 INFO = -1
154 ELSE IF( N.LT.0 ) THEN
155 INFO = -2
156 ELSE IF( KD.LT.0 ) THEN
157 INFO = -3
158 ELSE IF( NRHS.LT.0 ) THEN
159 INFO = -4
160 ELSE IF( LDAB.LT.KD+1 ) THEN
161 INFO = -6
162 ELSE IF( LDAFB.LT.KD+1 ) THEN
163 INFO = -8
164 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
165 INFO = -10
166 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
167 INFO = -12
168 END IF
169 IF( INFO.NE.0 ) THEN
170 CALL XERBLA( 'ZPBRFS', -INFO )
171 RETURN
172 END IF
173 *
174 * Quick return if possible
175 *
176 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
177 DO 10 J = 1, NRHS
178 FERR( J ) = ZERO
179 BERR( J ) = ZERO
180 10 CONTINUE
181 RETURN
182 END IF
183 *
184 * NZ = maximum number of nonzero elements in each row of A, plus 1
185 *
186 NZ = MIN( N+1, 2*KD+2 )
187 EPS = DLAMCH( 'Epsilon' )
188 SAFMIN = DLAMCH( 'Safe minimum' )
189 SAFE1 = NZ*SAFMIN
190 SAFE2 = SAFE1 / EPS
191 *
192 * Do for each right hand side
193 *
194 DO 140 J = 1, NRHS
195 *
196 COUNT = 1
197 LSTRES = THREE
198 20 CONTINUE
199 *
200 * Loop until stopping criterion is satisfied.
201 *
202 * Compute residual R = B - A * X
203 *
204 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
205 CALL ZHBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
206 $ 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 L = KD + 1 - K
228 DO 40 I = MAX( 1, K-KD ), K - 1
229 RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
230 S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
231 40 CONTINUE
232 RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( KD+1, K ) ) )*
233 $ XK + S
234 50 CONTINUE
235 ELSE
236 DO 70 K = 1, N
237 S = ZERO
238 XK = CABS1( X( K, J ) )
239 RWORK( K ) = RWORK( K ) + ABS( DBLE( AB( 1, K ) ) )*XK
240 L = 1 - K
241 DO 60 I = K + 1, MIN( N, K+KD )
242 RWORK( I ) = RWORK( I ) + CABS1( AB( L+I, K ) )*XK
243 S = S + CABS1( AB( L+I, K ) )*CABS1( X( I, J ) )
244 60 CONTINUE
245 RWORK( K ) = RWORK( K ) + S
246 70 CONTINUE
247 END IF
248 S = ZERO
249 DO 80 I = 1, N
250 IF( RWORK( I ).GT.SAFE2 ) THEN
251 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
252 ELSE
253 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
254 $ ( RWORK( I )+SAFE1 ) )
255 END IF
256 80 CONTINUE
257 BERR( J ) = S
258 *
259 * Test stopping criterion. Continue iterating if
260 * 1) The residual BERR(J) is larger than machine epsilon, and
261 * 2) BERR(J) decreased by at least a factor of 2 during the
262 * last iteration, and
263 * 3) At most ITMAX iterations tried.
264 *
265 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
266 $ COUNT.LE.ITMAX ) THEN
267 *
268 * Update solution and try again.
269 *
270 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
271 CALL ZAXPY( N, ONE, WORK, 1, X( 1, J ), 1 )
272 LSTRES = BERR( J )
273 COUNT = COUNT + 1
274 GO TO 20
275 END IF
276 *
277 * Bound error from formula
278 *
279 * norm(X - XTRUE) / norm(X) .le. FERR =
280 * norm( abs(inv(A))*
281 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
282 *
283 * where
284 * norm(Z) is the magnitude of the largest component of Z
285 * inv(A) is the inverse of A
286 * abs(Z) is the componentwise absolute value of the matrix or
287 * vector Z
288 * NZ is the maximum number of nonzeros in any row of A, plus 1
289 * EPS is machine epsilon
290 *
291 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
292 * is incremented by SAFE1 if the i-th component of
293 * abs(A)*abs(X) + abs(B) is less than SAFE2.
294 *
295 * Use ZLACN2 to estimate the infinity-norm of the matrix
296 * inv(A) * diag(W),
297 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
298 *
299 DO 90 I = 1, N
300 IF( RWORK( I ).GT.SAFE2 ) THEN
301 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
302 ELSE
303 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
304 $ SAFE1
305 END IF
306 90 CONTINUE
307 *
308 KASE = 0
309 100 CONTINUE
310 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
311 IF( KASE.NE.0 ) THEN
312 IF( KASE.EQ.1 ) THEN
313 *
314 * Multiply by diag(W)*inv(A**H).
315 *
316 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
317 DO 110 I = 1, N
318 WORK( I ) = RWORK( I )*WORK( I )
319 110 CONTINUE
320 ELSE IF( KASE.EQ.2 ) THEN
321 *
322 * Multiply by inv(A)*diag(W).
323 *
324 DO 120 I = 1, N
325 WORK( I ) = RWORK( I )*WORK( I )
326 120 CONTINUE
327 CALL ZPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK, N, INFO )
328 END IF
329 GO TO 100
330 END IF
331 *
332 * Normalize error.
333 *
334 LSTRES = ZERO
335 DO 130 I = 1, N
336 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
337 130 CONTINUE
338 IF( LSTRES.NE.ZERO )
339 $ FERR( J ) = FERR( J ) / LSTRES
340 *
341 140 CONTINUE
342 *
343 RETURN
344 *
345 * End of ZPBRFS
346 *
347 END