1 SUBROUTINE DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
2 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
3 $ INFO )
4 *
5 * -- LAPACK routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
11 *
12 * .. Scalar Arguments ..
13 CHARACTER TRANS
14 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
15 * ..
16 * .. Array Arguments ..
17 INTEGER IPIV( * ), IWORK( * )
18 DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
19 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DGBRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is banded, and provides
27 * error bounds and backward error estimates for 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 = Transpose)
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * KL (input) INTEGER
42 * The number of subdiagonals within the band of A. KL >= 0.
43 *
44 * KU (input) INTEGER
45 * The number of superdiagonals within the band of A. KU >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of right hand sides, i.e., the number of columns
49 * of the matrices B and X. NRHS >= 0.
50 *
51 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
52 * The original band matrix A, stored in rows 1 to KL+KU+1.
53 * The j-th column of A is stored in the j-th column of the
54 * array AB as follows:
55 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
56 *
57 * LDAB (input) INTEGER
58 * The leading dimension of the array AB. LDAB >= KL+KU+1.
59 *
60 * AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
61 * Details of the LU factorization of the band matrix A, as
62 * computed by DGBTRF. U is stored as an upper triangular band
63 * matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
64 * the multipliers used during the factorization are stored in
65 * rows KL+KU+2 to 2*KL+KU+1.
66 *
67 * LDAFB (input) INTEGER
68 * The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
69 *
70 * IPIV (input) INTEGER array, dimension (N)
71 * The pivot indices from DGBTRF; for 1<=i<=N, row i of the
72 * matrix was interchanged with row IPIV(i).
73 *
74 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
75 * The right hand side matrix B.
76 *
77 * LDB (input) INTEGER
78 * The leading dimension of the array B. LDB >= max(1,N).
79 *
80 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
81 * On entry, the solution matrix X, as computed by DGBTRS.
82 * On exit, the improved solution matrix X.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
88 * The estimated forward error bound for each solution vector
89 * X(j) (the j-th column of the solution matrix X).
90 * If XTRUE is the true solution corresponding to X(j), FERR(j)
91 * is an estimated upper bound for the magnitude of the largest
92 * element in (X(j) - XTRUE) divided by the magnitude of the
93 * largest element in X(j). The estimate is as reliable as
94 * the estimate for RCOND, and is almost always a slight
95 * overestimate of the true error.
96 *
97 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
98 * The componentwise relative backward error of each solution
99 * vector X(j) (i.e., the smallest relative change in
100 * any element of A or B that makes X(j) an exact solution).
101 *
102 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
103 *
104 * IWORK (workspace) INTEGER array, dimension (N)
105 *
106 * INFO (output) INTEGER
107 * = 0: successful exit
108 * < 0: if INFO = -i, the i-th argument had an illegal value
109 *
110 * Internal Parameters
111 * ===================
112 *
113 * ITMAX is the maximum number of steps of iterative refinement.
114 *
115 * =====================================================================
116 *
117 * .. Parameters ..
118 INTEGER ITMAX
119 PARAMETER ( ITMAX = 5 )
120 DOUBLE PRECISION ZERO
121 PARAMETER ( ZERO = 0.0D+0 )
122 DOUBLE PRECISION ONE
123 PARAMETER ( ONE = 1.0D+0 )
124 DOUBLE PRECISION TWO
125 PARAMETER ( TWO = 2.0D+0 )
126 DOUBLE PRECISION THREE
127 PARAMETER ( THREE = 3.0D+0 )
128 * ..
129 * .. Local Scalars ..
130 LOGICAL NOTRAN
131 CHARACTER TRANST
132 INTEGER COUNT, I, J, K, KASE, KK, NZ
133 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
134 * ..
135 * .. Local Arrays ..
136 INTEGER ISAVE( 3 )
137 * ..
138 * .. External Subroutines ..
139 EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2, XERBLA
140 * ..
141 * .. Intrinsic Functions ..
142 INTRINSIC ABS, MAX, MIN
143 * ..
144 * .. External Functions ..
145 LOGICAL LSAME
146 DOUBLE PRECISION DLAMCH
147 EXTERNAL LSAME, DLAMCH
148 * ..
149 * .. Executable Statements ..
150 *
151 * Test the input parameters.
152 *
153 INFO = 0
154 NOTRAN = LSAME( TRANS, 'N' )
155 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
156 $ LSAME( TRANS, 'C' ) ) THEN
157 INFO = -1
158 ELSE IF( N.LT.0 ) THEN
159 INFO = -2
160 ELSE IF( KL.LT.0 ) THEN
161 INFO = -3
162 ELSE IF( KU.LT.0 ) THEN
163 INFO = -4
164 ELSE IF( NRHS.LT.0 ) THEN
165 INFO = -5
166 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
167 INFO = -7
168 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
169 INFO = -9
170 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
171 INFO = -12
172 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
173 INFO = -14
174 END IF
175 IF( INFO.NE.0 ) THEN
176 CALL XERBLA( 'DGBRFS', -INFO )
177 RETURN
178 END IF
179 *
180 * Quick return if possible
181 *
182 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
183 DO 10 J = 1, NRHS
184 FERR( J ) = ZERO
185 BERR( J ) = ZERO
186 10 CONTINUE
187 RETURN
188 END IF
189 *
190 IF( NOTRAN ) THEN
191 TRANST = 'T'
192 ELSE
193 TRANST = 'N'
194 END IF
195 *
196 * NZ = maximum number of nonzero elements in each row of A, plus 1
197 *
198 NZ = MIN( KL+KU+2, N+1 )
199 EPS = DLAMCH( 'Epsilon' )
200 SAFMIN = DLAMCH( 'Safe minimum' )
201 SAFE1 = NZ*SAFMIN
202 SAFE2 = SAFE1 / EPS
203 *
204 * Do for each right hand side
205 *
206 DO 140 J = 1, NRHS
207 *
208 COUNT = 1
209 LSTRES = THREE
210 20 CONTINUE
211 *
212 * Loop until stopping criterion is satisfied.
213 *
214 * Compute residual R = B - op(A) * X,
215 * where op(A) = A, A**T, or A**H, depending on TRANS.
216 *
217 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
218 CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
219 $ ONE, WORK( N+1 ), 1 )
220 *
221 * Compute componentwise relative backward error from formula
222 *
223 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
224 *
225 * where abs(Z) is the componentwise absolute value of the matrix
226 * or vector Z. If the i-th component of the denominator is less
227 * than SAFE2, then SAFE1 is added to the i-th components of the
228 * numerator and denominator before dividing.
229 *
230 DO 30 I = 1, N
231 WORK( I ) = ABS( B( I, J ) )
232 30 CONTINUE
233 *
234 * Compute abs(op(A))*abs(X) + abs(B).
235 *
236 IF( NOTRAN ) THEN
237 DO 50 K = 1, N
238 KK = KU + 1 - K
239 XK = ABS( X( K, J ) )
240 DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
241 WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
242 40 CONTINUE
243 50 CONTINUE
244 ELSE
245 DO 70 K = 1, N
246 S = ZERO
247 KK = KU + 1 - K
248 DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
249 S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
250 60 CONTINUE
251 WORK( K ) = WORK( K ) + S
252 70 CONTINUE
253 END IF
254 S = ZERO
255 DO 80 I = 1, N
256 IF( WORK( I ).GT.SAFE2 ) THEN
257 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
258 ELSE
259 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
260 $ ( WORK( I )+SAFE1 ) )
261 END IF
262 80 CONTINUE
263 BERR( J ) = S
264 *
265 * Test stopping criterion. Continue iterating if
266 * 1) The residual BERR(J) is larger than machine epsilon, and
267 * 2) BERR(J) decreased by at least a factor of 2 during the
268 * last iteration, and
269 * 3) At most ITMAX iterations tried.
270 *
271 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
272 $ COUNT.LE.ITMAX ) THEN
273 *
274 * Update solution and try again.
275 *
276 CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
277 $ WORK( N+1 ), N, INFO )
278 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
279 LSTRES = BERR( J )
280 COUNT = COUNT + 1
281 GO TO 20
282 END IF
283 *
284 * Bound error from formula
285 *
286 * norm(X - XTRUE) / norm(X) .le. FERR =
287 * norm( abs(inv(op(A)))*
288 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
289 *
290 * where
291 * norm(Z) is the magnitude of the largest component of Z
292 * inv(op(A)) is the inverse of op(A)
293 * abs(Z) is the componentwise absolute value of the matrix or
294 * vector Z
295 * NZ is the maximum number of nonzeros in any row of A, plus 1
296 * EPS is machine epsilon
297 *
298 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
299 * is incremented by SAFE1 if the i-th component of
300 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
301 *
302 * Use DLACN2 to estimate the infinity-norm of the matrix
303 * inv(op(A)) * diag(W),
304 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
305 *
306 DO 90 I = 1, N
307 IF( WORK( I ).GT.SAFE2 ) THEN
308 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
309 ELSE
310 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
311 END IF
312 90 CONTINUE
313 *
314 KASE = 0
315 100 CONTINUE
316 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
317 $ KASE, ISAVE )
318 IF( KASE.NE.0 ) THEN
319 IF( KASE.EQ.1 ) THEN
320 *
321 * Multiply by diag(W)*inv(op(A)**T).
322 *
323 CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
324 $ WORK( N+1 ), N, INFO )
325 DO 110 I = 1, N
326 WORK( N+I ) = WORK( N+I )*WORK( I )
327 110 CONTINUE
328 ELSE
329 *
330 * Multiply by inv(op(A))*diag(W).
331 *
332 DO 120 I = 1, N
333 WORK( N+I ) = WORK( N+I )*WORK( I )
334 120 CONTINUE
335 CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
336 $ WORK( N+1 ), N, INFO )
337 END IF
338 GO TO 100
339 END IF
340 *
341 * Normalize error.
342 *
343 LSTRES = ZERO
344 DO 130 I = 1, N
345 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
346 130 CONTINUE
347 IF( LSTRES.NE.ZERO )
348 $ FERR( J ) = FERR( J ) / LSTRES
349 *
350 140 CONTINUE
351 *
352 RETURN
353 *
354 * End of DGBRFS
355 *
356 END
2 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK,
3 $ INFO )
4 *
5 * -- LAPACK routine (version 3.2) --
6 * -- LAPACK is a software package provided by Univ. of Tennessee, --
7 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8 * November 2006
9 *
10 * Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH.
11 *
12 * .. Scalar Arguments ..
13 CHARACTER TRANS
14 INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
15 * ..
16 * .. Array Arguments ..
17 INTEGER IPIV( * ), IWORK( * )
18 DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
19 $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
20 * ..
21 *
22 * Purpose
23 * =======
24 *
25 * DGBRFS improves the computed solution to a system of linear
26 * equations when the coefficient matrix is banded, and provides
27 * error bounds and backward error estimates for 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 = Transpose)
37 *
38 * N (input) INTEGER
39 * The order of the matrix A. N >= 0.
40 *
41 * KL (input) INTEGER
42 * The number of subdiagonals within the band of A. KL >= 0.
43 *
44 * KU (input) INTEGER
45 * The number of superdiagonals within the band of A. KU >= 0.
46 *
47 * NRHS (input) INTEGER
48 * The number of right hand sides, i.e., the number of columns
49 * of the matrices B and X. NRHS >= 0.
50 *
51 * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
52 * The original band matrix A, stored in rows 1 to KL+KU+1.
53 * The j-th column of A is stored in the j-th column of the
54 * array AB as follows:
55 * AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
56 *
57 * LDAB (input) INTEGER
58 * The leading dimension of the array AB. LDAB >= KL+KU+1.
59 *
60 * AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N)
61 * Details of the LU factorization of the band matrix A, as
62 * computed by DGBTRF. U is stored as an upper triangular band
63 * matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
64 * the multipliers used during the factorization are stored in
65 * rows KL+KU+2 to 2*KL+KU+1.
66 *
67 * LDAFB (input) INTEGER
68 * The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
69 *
70 * IPIV (input) INTEGER array, dimension (N)
71 * The pivot indices from DGBTRF; for 1<=i<=N, row i of the
72 * matrix was interchanged with row IPIV(i).
73 *
74 * B (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
75 * The right hand side matrix B.
76 *
77 * LDB (input) INTEGER
78 * The leading dimension of the array B. LDB >= max(1,N).
79 *
80 * X (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
81 * On entry, the solution matrix X, as computed by DGBTRS.
82 * On exit, the improved solution matrix X.
83 *
84 * LDX (input) INTEGER
85 * The leading dimension of the array X. LDX >= max(1,N).
86 *
87 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
88 * The estimated forward error bound for each solution vector
89 * X(j) (the j-th column of the solution matrix X).
90 * If XTRUE is the true solution corresponding to X(j), FERR(j)
91 * is an estimated upper bound for the magnitude of the largest
92 * element in (X(j) - XTRUE) divided by the magnitude of the
93 * largest element in X(j). The estimate is as reliable as
94 * the estimate for RCOND, and is almost always a slight
95 * overestimate of the true error.
96 *
97 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
98 * The componentwise relative backward error of each solution
99 * vector X(j) (i.e., the smallest relative change in
100 * any element of A or B that makes X(j) an exact solution).
101 *
102 * WORK (workspace) DOUBLE PRECISION array, dimension (3*N)
103 *
104 * IWORK (workspace) INTEGER array, dimension (N)
105 *
106 * INFO (output) INTEGER
107 * = 0: successful exit
108 * < 0: if INFO = -i, the i-th argument had an illegal value
109 *
110 * Internal Parameters
111 * ===================
112 *
113 * ITMAX is the maximum number of steps of iterative refinement.
114 *
115 * =====================================================================
116 *
117 * .. Parameters ..
118 INTEGER ITMAX
119 PARAMETER ( ITMAX = 5 )
120 DOUBLE PRECISION ZERO
121 PARAMETER ( ZERO = 0.0D+0 )
122 DOUBLE PRECISION ONE
123 PARAMETER ( ONE = 1.0D+0 )
124 DOUBLE PRECISION TWO
125 PARAMETER ( TWO = 2.0D+0 )
126 DOUBLE PRECISION THREE
127 PARAMETER ( THREE = 3.0D+0 )
128 * ..
129 * .. Local Scalars ..
130 LOGICAL NOTRAN
131 CHARACTER TRANST
132 INTEGER COUNT, I, J, K, KASE, KK, NZ
133 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
134 * ..
135 * .. Local Arrays ..
136 INTEGER ISAVE( 3 )
137 * ..
138 * .. External Subroutines ..
139 EXTERNAL DAXPY, DCOPY, DGBMV, DGBTRS, DLACN2, XERBLA
140 * ..
141 * .. Intrinsic Functions ..
142 INTRINSIC ABS, MAX, MIN
143 * ..
144 * .. External Functions ..
145 LOGICAL LSAME
146 DOUBLE PRECISION DLAMCH
147 EXTERNAL LSAME, DLAMCH
148 * ..
149 * .. Executable Statements ..
150 *
151 * Test the input parameters.
152 *
153 INFO = 0
154 NOTRAN = LSAME( TRANS, 'N' )
155 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
156 $ LSAME( TRANS, 'C' ) ) THEN
157 INFO = -1
158 ELSE IF( N.LT.0 ) THEN
159 INFO = -2
160 ELSE IF( KL.LT.0 ) THEN
161 INFO = -3
162 ELSE IF( KU.LT.0 ) THEN
163 INFO = -4
164 ELSE IF( NRHS.LT.0 ) THEN
165 INFO = -5
166 ELSE IF( LDAB.LT.KL+KU+1 ) THEN
167 INFO = -7
168 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
169 INFO = -9
170 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
171 INFO = -12
172 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
173 INFO = -14
174 END IF
175 IF( INFO.NE.0 ) THEN
176 CALL XERBLA( 'DGBRFS', -INFO )
177 RETURN
178 END IF
179 *
180 * Quick return if possible
181 *
182 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
183 DO 10 J = 1, NRHS
184 FERR( J ) = ZERO
185 BERR( J ) = ZERO
186 10 CONTINUE
187 RETURN
188 END IF
189 *
190 IF( NOTRAN ) THEN
191 TRANST = 'T'
192 ELSE
193 TRANST = 'N'
194 END IF
195 *
196 * NZ = maximum number of nonzero elements in each row of A, plus 1
197 *
198 NZ = MIN( KL+KU+2, N+1 )
199 EPS = DLAMCH( 'Epsilon' )
200 SAFMIN = DLAMCH( 'Safe minimum' )
201 SAFE1 = NZ*SAFMIN
202 SAFE2 = SAFE1 / EPS
203 *
204 * Do for each right hand side
205 *
206 DO 140 J = 1, NRHS
207 *
208 COUNT = 1
209 LSTRES = THREE
210 20 CONTINUE
211 *
212 * Loop until stopping criterion is satisfied.
213 *
214 * Compute residual R = B - op(A) * X,
215 * where op(A) = A, A**T, or A**H, depending on TRANS.
216 *
217 CALL DCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
218 CALL DGBMV( TRANS, N, N, KL, KU, -ONE, AB, LDAB, X( 1, J ), 1,
219 $ ONE, WORK( N+1 ), 1 )
220 *
221 * Compute componentwise relative backward error from formula
222 *
223 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
224 *
225 * where abs(Z) is the componentwise absolute value of the matrix
226 * or vector Z. If the i-th component of the denominator is less
227 * than SAFE2, then SAFE1 is added to the i-th components of the
228 * numerator and denominator before dividing.
229 *
230 DO 30 I = 1, N
231 WORK( I ) = ABS( B( I, J ) )
232 30 CONTINUE
233 *
234 * Compute abs(op(A))*abs(X) + abs(B).
235 *
236 IF( NOTRAN ) THEN
237 DO 50 K = 1, N
238 KK = KU + 1 - K
239 XK = ABS( X( K, J ) )
240 DO 40 I = MAX( 1, K-KU ), MIN( N, K+KL )
241 WORK( I ) = WORK( I ) + ABS( AB( KK+I, K ) )*XK
242 40 CONTINUE
243 50 CONTINUE
244 ELSE
245 DO 70 K = 1, N
246 S = ZERO
247 KK = KU + 1 - K
248 DO 60 I = MAX( 1, K-KU ), MIN( N, K+KL )
249 S = S + ABS( AB( KK+I, K ) )*ABS( X( I, J ) )
250 60 CONTINUE
251 WORK( K ) = WORK( K ) + S
252 70 CONTINUE
253 END IF
254 S = ZERO
255 DO 80 I = 1, N
256 IF( WORK( I ).GT.SAFE2 ) THEN
257 S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
258 ELSE
259 S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
260 $ ( WORK( I )+SAFE1 ) )
261 END IF
262 80 CONTINUE
263 BERR( J ) = S
264 *
265 * Test stopping criterion. Continue iterating if
266 * 1) The residual BERR(J) is larger than machine epsilon, and
267 * 2) BERR(J) decreased by at least a factor of 2 during the
268 * last iteration, and
269 * 3) At most ITMAX iterations tried.
270 *
271 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
272 $ COUNT.LE.ITMAX ) THEN
273 *
274 * Update solution and try again.
275 *
276 CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
277 $ WORK( N+1 ), N, INFO )
278 CALL DAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
279 LSTRES = BERR( J )
280 COUNT = COUNT + 1
281 GO TO 20
282 END IF
283 *
284 * Bound error from formula
285 *
286 * norm(X - XTRUE) / norm(X) .le. FERR =
287 * norm( abs(inv(op(A)))*
288 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
289 *
290 * where
291 * norm(Z) is the magnitude of the largest component of Z
292 * inv(op(A)) is the inverse of op(A)
293 * abs(Z) is the componentwise absolute value of the matrix or
294 * vector Z
295 * NZ is the maximum number of nonzeros in any row of A, plus 1
296 * EPS is machine epsilon
297 *
298 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
299 * is incremented by SAFE1 if the i-th component of
300 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
301 *
302 * Use DLACN2 to estimate the infinity-norm of the matrix
303 * inv(op(A)) * diag(W),
304 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
305 *
306 DO 90 I = 1, N
307 IF( WORK( I ).GT.SAFE2 ) THEN
308 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
309 ELSE
310 WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
311 END IF
312 90 CONTINUE
313 *
314 KASE = 0
315 100 CONTINUE
316 CALL DLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
317 $ KASE, ISAVE )
318 IF( KASE.NE.0 ) THEN
319 IF( KASE.EQ.1 ) THEN
320 *
321 * Multiply by diag(W)*inv(op(A)**T).
322 *
323 CALL DGBTRS( TRANST, N, KL, KU, 1, AFB, LDAFB, IPIV,
324 $ WORK( N+1 ), N, INFO )
325 DO 110 I = 1, N
326 WORK( N+I ) = WORK( N+I )*WORK( I )
327 110 CONTINUE
328 ELSE
329 *
330 * Multiply by inv(op(A))*diag(W).
331 *
332 DO 120 I = 1, N
333 WORK( N+I ) = WORK( N+I )*WORK( I )
334 120 CONTINUE
335 CALL DGBTRS( TRANS, N, KL, KU, 1, AFB, LDAFB, IPIV,
336 $ WORK( N+1 ), N, INFO )
337 END IF
338 GO TO 100
339 END IF
340 *
341 * Normalize error.
342 *
343 LSTRES = ZERO
344 DO 130 I = 1, N
345 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
346 130 CONTINUE
347 IF( LSTRES.NE.ZERO )
348 $ FERR( J ) = FERR( J ) / LSTRES
349 *
350 140 CONTINUE
351 *
352 RETURN
353 *
354 * End of DGBRFS
355 *
356 END