1 SUBROUTINE ZGTRFS( TRANS, N, NRHS, DL, D, DU, DLF, DF, DUF, DU2,
2 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
11 *
12 * .. Scalar Arguments ..
13 CHARACTER TRANS
14 INTEGER INFO, LDB, LDX, N, NRHS
15 * ..
16 * .. Array Arguments ..
17 INTEGER IPIV( * )
18 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
19 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
20 $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
21 $ WORK( * ), X( LDX, * )
22 * ..
23 *
24 * Purpose
25 * =======
26 *
27 * ZGTRFS improves the computed solution to a system of linear
28 * equations when the coefficient matrix is tridiagonal, and provides
29 * error bounds and backward error estimates for the solution.
30 *
31 * Arguments
32 * =========
33 *
34 * TRANS (input) CHARACTER*1
35 * Specifies the form of the system of equations:
36 * = 'N': A * X = B (No transpose)
37 * = 'T': A**T * X = B (Transpose)
38 * = 'C': A**H * X = B (Conjugate transpose)
39 *
40 * N (input) INTEGER
41 * The order of the matrix A. N >= 0.
42 *
43 * NRHS (input) INTEGER
44 * The number of right hand sides, i.e., the number of columns
45 * of the matrix B. NRHS >= 0.
46 *
47 * DL (input) COMPLEX*16 array, dimension (N-1)
48 * The (n-1) subdiagonal elements of A.
49 *
50 * D (input) COMPLEX*16 array, dimension (N)
51 * The diagonal elements of A.
52 *
53 * DU (input) COMPLEX*16 array, dimension (N-1)
54 * The (n-1) superdiagonal elements of A.
55 *
56 * DLF (input) COMPLEX*16 array, dimension (N-1)
57 * The (n-1) multipliers that define the matrix L from the
58 * LU factorization of A as computed by ZGTTRF.
59 *
60 * DF (input) COMPLEX*16 array, dimension (N)
61 * The n diagonal elements of the upper triangular matrix U from
62 * the LU factorization of A.
63 *
64 * DUF (input) COMPLEX*16 array, dimension (N-1)
65 * The (n-1) elements of the first superdiagonal of U.
66 *
67 * DU2 (input) COMPLEX*16 array, dimension (N-2)
68 * The (n-2) elements of the second superdiagonal of U.
69 *
70 * IPIV (input) INTEGER array, dimension (N)
71 * The pivot indices; for 1 <= i <= n, row i of the matrix was
72 * interchanged with row IPIV(i). IPIV(i) will always be either
73 * i or i+1; IPIV(i) = i indicates a row interchange was not
74 * required.
75 *
76 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
77 * The right hand side matrix B.
78 *
79 * LDB (input) INTEGER
80 * The leading dimension of the array B. LDB >= max(1,N).
81 *
82 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
83 * On entry, the solution matrix X, as computed by ZGTTRS.
84 * On exit, the improved solution matrix X.
85 *
86 * LDX (input) INTEGER
87 * The leading dimension of the array X. LDX >= max(1,N).
88 *
89 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
90 * The estimated forward error bound for each solution vector
91 * X(j) (the j-th column of the solution matrix X).
92 * If XTRUE is the true solution corresponding to X(j), FERR(j)
93 * is an estimated upper bound for the magnitude of the largest
94 * element in (X(j) - XTRUE) divided by the magnitude of the
95 * largest element in X(j). The estimate is as reliable as
96 * the estimate for RCOND, and is almost always a slight
97 * overestimate of the true error.
98 *
99 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
100 * The componentwise relative backward error of each solution
101 * vector X(j) (i.e., the smallest relative change in
102 * any element of A or B that makes X(j) an exact solution).
103 *
104 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
105 *
106 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
107 *
108 * INFO (output) INTEGER
109 * = 0: successful exit
110 * < 0: if INFO = -i, the i-th argument had an illegal value
111 *
112 * Internal Parameters
113 * ===================
114 *
115 * ITMAX is the maximum number of steps of iterative refinement.
116 *
117 * =====================================================================
118 *
119 * .. Parameters ..
120 INTEGER ITMAX
121 PARAMETER ( ITMAX = 5 )
122 DOUBLE PRECISION ZERO, ONE
123 PARAMETER ( ZERO = 0.0D+0, 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 TRANSN, TRANST
132 INTEGER COUNT, I, J, KASE, NZ
133 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
134 COMPLEX*16 ZDUM
135 * ..
136 * .. Local Arrays ..
137 INTEGER ISAVE( 3 )
138 * ..
139 * .. External Subroutines ..
140 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGTTRS, ZLACN2, ZLAGTM
141 * ..
142 * .. Intrinsic Functions ..
143 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, MAX
144 * ..
145 * .. External Functions ..
146 LOGICAL LSAME
147 DOUBLE PRECISION DLAMCH
148 EXTERNAL LSAME, DLAMCH
149 * ..
150 * .. Statement Functions ..
151 DOUBLE PRECISION CABS1
152 * ..
153 * .. Statement Function definitions ..
154 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
155 * ..
156 * .. Executable Statements ..
157 *
158 * Test the input parameters.
159 *
160 INFO = 0
161 NOTRAN = LSAME( TRANS, 'N' )
162 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
163 $ LSAME( TRANS, 'C' ) ) THEN
164 INFO = -1
165 ELSE IF( N.LT.0 ) THEN
166 INFO = -2
167 ELSE IF( NRHS.LT.0 ) THEN
168 INFO = -3
169 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
170 INFO = -13
171 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
172 INFO = -15
173 END IF
174 IF( INFO.NE.0 ) THEN
175 CALL XERBLA( 'ZGTRFS', -INFO )
176 RETURN
177 END IF
178 *
179 * Quick return if possible
180 *
181 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
182 DO 10 J = 1, NRHS
183 FERR( J ) = ZERO
184 BERR( J ) = ZERO
185 10 CONTINUE
186 RETURN
187 END IF
188 *
189 IF( NOTRAN ) THEN
190 TRANSN = 'N'
191 TRANST = 'C'
192 ELSE
193 TRANSN = 'C'
194 TRANST = 'N'
195 END IF
196 *
197 * NZ = maximum number of nonzero elements in each row of A, plus 1
198 *
199 NZ = 4
200 EPS = DLAMCH( 'Epsilon' )
201 SAFMIN = DLAMCH( 'Safe minimum' )
202 SAFE1 = NZ*SAFMIN
203 SAFE2 = SAFE1 / EPS
204 *
205 * Do for each right hand side
206 *
207 DO 110 J = 1, NRHS
208 *
209 COUNT = 1
210 LSTRES = THREE
211 20 CONTINUE
212 *
213 * Loop until stopping criterion is satisfied.
214 *
215 * Compute residual R = B - op(A) * X,
216 * where op(A) = A, A**T, or A**H, depending on TRANS.
217 *
218 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
219 CALL ZLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
220 $ WORK, N )
221 *
222 * Compute abs(op(A))*abs(x) + abs(b) for use in the backward
223 * error bound.
224 *
225 IF( NOTRAN ) THEN
226 IF( N.EQ.1 ) THEN
227 RWORK( 1 ) = CABS1( B( 1, J ) ) +
228 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
229 ELSE
230 RWORK( 1 ) = CABS1( B( 1, J ) ) +
231 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
232 $ CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
233 DO 30 I = 2, N - 1
234 RWORK( I ) = CABS1( B( I, J ) ) +
235 $ CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
236 $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
237 $ CABS1( DU( I ) )*CABS1( X( I+1, J ) )
238 30 CONTINUE
239 RWORK( N ) = CABS1( B( N, J ) ) +
240 $ CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
241 $ CABS1( D( N ) )*CABS1( X( N, J ) )
242 END IF
243 ELSE
244 IF( N.EQ.1 ) THEN
245 RWORK( 1 ) = CABS1( B( 1, J ) ) +
246 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
247 ELSE
248 RWORK( 1 ) = CABS1( B( 1, J ) ) +
249 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
250 $ CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
251 DO 40 I = 2, N - 1
252 RWORK( I ) = CABS1( B( I, J ) ) +
253 $ CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
254 $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
255 $ CABS1( DL( I ) )*CABS1( X( I+1, J ) )
256 40 CONTINUE
257 RWORK( N ) = CABS1( B( N, J ) ) +
258 $ CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
259 $ CABS1( D( N ) )*CABS1( X( N, J ) )
260 END IF
261 END IF
262 *
263 * Compute componentwise relative backward error from formula
264 *
265 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
266 *
267 * where abs(Z) is the componentwise absolute value of the matrix
268 * or vector Z. If the i-th component of the denominator is less
269 * than SAFE2, then SAFE1 is added to the i-th components of the
270 * numerator and denominator before dividing.
271 *
272 S = ZERO
273 DO 50 I = 1, N
274 IF( RWORK( I ).GT.SAFE2 ) THEN
275 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
276 ELSE
277 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
278 $ ( RWORK( I )+SAFE1 ) )
279 END IF
280 50 CONTINUE
281 BERR( J ) = S
282 *
283 * Test stopping criterion. Continue iterating if
284 * 1) The residual BERR(J) is larger than machine epsilon, and
285 * 2) BERR(J) decreased by at least a factor of 2 during the
286 * last iteration, and
287 * 3) At most ITMAX iterations tried.
288 *
289 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
290 $ COUNT.LE.ITMAX ) THEN
291 *
292 * Update solution and try again.
293 *
294 CALL ZGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
295 $ INFO )
296 CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
297 LSTRES = BERR( J )
298 COUNT = COUNT + 1
299 GO TO 20
300 END IF
301 *
302 * Bound error from formula
303 *
304 * norm(X - XTRUE) / norm(X) .le. FERR =
305 * norm( abs(inv(op(A)))*
306 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
307 *
308 * where
309 * norm(Z) is the magnitude of the largest component of Z
310 * inv(op(A)) is the inverse of op(A)
311 * abs(Z) is the componentwise absolute value of the matrix or
312 * vector Z
313 * NZ is the maximum number of nonzeros in any row of A, plus 1
314 * EPS is machine epsilon
315 *
316 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
317 * is incremented by SAFE1 if the i-th component of
318 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
319 *
320 * Use ZLACN2 to estimate the infinity-norm of the matrix
321 * inv(op(A)) * diag(W),
322 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
323 *
324 DO 60 I = 1, N
325 IF( RWORK( I ).GT.SAFE2 ) THEN
326 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
327 ELSE
328 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
329 $ SAFE1
330 END IF
331 60 CONTINUE
332 *
333 KASE = 0
334 70 CONTINUE
335 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
336 IF( KASE.NE.0 ) THEN
337 IF( KASE.EQ.1 ) THEN
338 *
339 * Multiply by diag(W)*inv(op(A)**H).
340 *
341 CALL ZGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
342 $ N, INFO )
343 DO 80 I = 1, N
344 WORK( I ) = RWORK( I )*WORK( I )
345 80 CONTINUE
346 ELSE
347 *
348 * Multiply by inv(op(A))*diag(W).
349 *
350 DO 90 I = 1, N
351 WORK( I ) = RWORK( I )*WORK( I )
352 90 CONTINUE
353 CALL ZGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
354 $ N, INFO )
355 END IF
356 GO TO 70
357 END IF
358 *
359 * Normalize error.
360 *
361 LSTRES = ZERO
362 DO 100 I = 1, N
363 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
364 100 CONTINUE
365 IF( LSTRES.NE.ZERO )
366 $ FERR( J ) = FERR( J ) / LSTRES
367 *
368 110 CONTINUE
369 *
370 RETURN
371 *
372 * End of ZGTRFS
373 *
374 END
2 $ IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
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 ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
11 *
12 * .. Scalar Arguments ..
13 CHARACTER TRANS
14 INTEGER INFO, LDB, LDX, N, NRHS
15 * ..
16 * .. Array Arguments ..
17 INTEGER IPIV( * )
18 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
19 COMPLEX*16 B( LDB, * ), D( * ), DF( * ), DL( * ),
20 $ DLF( * ), DU( * ), DU2( * ), DUF( * ),
21 $ WORK( * ), X( LDX, * )
22 * ..
23 *
24 * Purpose
25 * =======
26 *
27 * ZGTRFS improves the computed solution to a system of linear
28 * equations when the coefficient matrix is tridiagonal, and provides
29 * error bounds and backward error estimates for the solution.
30 *
31 * Arguments
32 * =========
33 *
34 * TRANS (input) CHARACTER*1
35 * Specifies the form of the system of equations:
36 * = 'N': A * X = B (No transpose)
37 * = 'T': A**T * X = B (Transpose)
38 * = 'C': A**H * X = B (Conjugate transpose)
39 *
40 * N (input) INTEGER
41 * The order of the matrix A. N >= 0.
42 *
43 * NRHS (input) INTEGER
44 * The number of right hand sides, i.e., the number of columns
45 * of the matrix B. NRHS >= 0.
46 *
47 * DL (input) COMPLEX*16 array, dimension (N-1)
48 * The (n-1) subdiagonal elements of A.
49 *
50 * D (input) COMPLEX*16 array, dimension (N)
51 * The diagonal elements of A.
52 *
53 * DU (input) COMPLEX*16 array, dimension (N-1)
54 * The (n-1) superdiagonal elements of A.
55 *
56 * DLF (input) COMPLEX*16 array, dimension (N-1)
57 * The (n-1) multipliers that define the matrix L from the
58 * LU factorization of A as computed by ZGTTRF.
59 *
60 * DF (input) COMPLEX*16 array, dimension (N)
61 * The n diagonal elements of the upper triangular matrix U from
62 * the LU factorization of A.
63 *
64 * DUF (input) COMPLEX*16 array, dimension (N-1)
65 * The (n-1) elements of the first superdiagonal of U.
66 *
67 * DU2 (input) COMPLEX*16 array, dimension (N-2)
68 * The (n-2) elements of the second superdiagonal of U.
69 *
70 * IPIV (input) INTEGER array, dimension (N)
71 * The pivot indices; for 1 <= i <= n, row i of the matrix was
72 * interchanged with row IPIV(i). IPIV(i) will always be either
73 * i or i+1; IPIV(i) = i indicates a row interchange was not
74 * required.
75 *
76 * B (input) COMPLEX*16 array, dimension (LDB,NRHS)
77 * The right hand side matrix B.
78 *
79 * LDB (input) INTEGER
80 * The leading dimension of the array B. LDB >= max(1,N).
81 *
82 * X (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
83 * On entry, the solution matrix X, as computed by ZGTTRS.
84 * On exit, the improved solution matrix X.
85 *
86 * LDX (input) INTEGER
87 * The leading dimension of the array X. LDX >= max(1,N).
88 *
89 * FERR (output) DOUBLE PRECISION array, dimension (NRHS)
90 * The estimated forward error bound for each solution vector
91 * X(j) (the j-th column of the solution matrix X).
92 * If XTRUE is the true solution corresponding to X(j), FERR(j)
93 * is an estimated upper bound for the magnitude of the largest
94 * element in (X(j) - XTRUE) divided by the magnitude of the
95 * largest element in X(j). The estimate is as reliable as
96 * the estimate for RCOND, and is almost always a slight
97 * overestimate of the true error.
98 *
99 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
100 * The componentwise relative backward error of each solution
101 * vector X(j) (i.e., the smallest relative change in
102 * any element of A or B that makes X(j) an exact solution).
103 *
104 * WORK (workspace) COMPLEX*16 array, dimension (2*N)
105 *
106 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
107 *
108 * INFO (output) INTEGER
109 * = 0: successful exit
110 * < 0: if INFO = -i, the i-th argument had an illegal value
111 *
112 * Internal Parameters
113 * ===================
114 *
115 * ITMAX is the maximum number of steps of iterative refinement.
116 *
117 * =====================================================================
118 *
119 * .. Parameters ..
120 INTEGER ITMAX
121 PARAMETER ( ITMAX = 5 )
122 DOUBLE PRECISION ZERO, ONE
123 PARAMETER ( ZERO = 0.0D+0, 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 TRANSN, TRANST
132 INTEGER COUNT, I, J, KASE, NZ
133 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
134 COMPLEX*16 ZDUM
135 * ..
136 * .. Local Arrays ..
137 INTEGER ISAVE( 3 )
138 * ..
139 * .. External Subroutines ..
140 EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGTTRS, ZLACN2, ZLAGTM
141 * ..
142 * .. Intrinsic Functions ..
143 INTRINSIC ABS, DBLE, DCMPLX, DIMAG, MAX
144 * ..
145 * .. External Functions ..
146 LOGICAL LSAME
147 DOUBLE PRECISION DLAMCH
148 EXTERNAL LSAME, DLAMCH
149 * ..
150 * .. Statement Functions ..
151 DOUBLE PRECISION CABS1
152 * ..
153 * .. Statement Function definitions ..
154 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
155 * ..
156 * .. Executable Statements ..
157 *
158 * Test the input parameters.
159 *
160 INFO = 0
161 NOTRAN = LSAME( TRANS, 'N' )
162 IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
163 $ LSAME( TRANS, 'C' ) ) THEN
164 INFO = -1
165 ELSE IF( N.LT.0 ) THEN
166 INFO = -2
167 ELSE IF( NRHS.LT.0 ) THEN
168 INFO = -3
169 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
170 INFO = -13
171 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
172 INFO = -15
173 END IF
174 IF( INFO.NE.0 ) THEN
175 CALL XERBLA( 'ZGTRFS', -INFO )
176 RETURN
177 END IF
178 *
179 * Quick return if possible
180 *
181 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
182 DO 10 J = 1, NRHS
183 FERR( J ) = ZERO
184 BERR( J ) = ZERO
185 10 CONTINUE
186 RETURN
187 END IF
188 *
189 IF( NOTRAN ) THEN
190 TRANSN = 'N'
191 TRANST = 'C'
192 ELSE
193 TRANSN = 'C'
194 TRANST = 'N'
195 END IF
196 *
197 * NZ = maximum number of nonzero elements in each row of A, plus 1
198 *
199 NZ = 4
200 EPS = DLAMCH( 'Epsilon' )
201 SAFMIN = DLAMCH( 'Safe minimum' )
202 SAFE1 = NZ*SAFMIN
203 SAFE2 = SAFE1 / EPS
204 *
205 * Do for each right hand side
206 *
207 DO 110 J = 1, NRHS
208 *
209 COUNT = 1
210 LSTRES = THREE
211 20 CONTINUE
212 *
213 * Loop until stopping criterion is satisfied.
214 *
215 * Compute residual R = B - op(A) * X,
216 * where op(A) = A, A**T, or A**H, depending on TRANS.
217 *
218 CALL ZCOPY( N, B( 1, J ), 1, WORK, 1 )
219 CALL ZLAGTM( TRANS, N, 1, -ONE, DL, D, DU, X( 1, J ), LDX, ONE,
220 $ WORK, N )
221 *
222 * Compute abs(op(A))*abs(x) + abs(b) for use in the backward
223 * error bound.
224 *
225 IF( NOTRAN ) THEN
226 IF( N.EQ.1 ) THEN
227 RWORK( 1 ) = CABS1( B( 1, J ) ) +
228 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
229 ELSE
230 RWORK( 1 ) = CABS1( B( 1, J ) ) +
231 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
232 $ CABS1( DU( 1 ) )*CABS1( X( 2, J ) )
233 DO 30 I = 2, N - 1
234 RWORK( I ) = CABS1( B( I, J ) ) +
235 $ CABS1( DL( I-1 ) )*CABS1( X( I-1, J ) ) +
236 $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
237 $ CABS1( DU( I ) )*CABS1( X( I+1, J ) )
238 30 CONTINUE
239 RWORK( N ) = CABS1( B( N, J ) ) +
240 $ CABS1( DL( N-1 ) )*CABS1( X( N-1, J ) ) +
241 $ CABS1( D( N ) )*CABS1( X( N, J ) )
242 END IF
243 ELSE
244 IF( N.EQ.1 ) THEN
245 RWORK( 1 ) = CABS1( B( 1, J ) ) +
246 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) )
247 ELSE
248 RWORK( 1 ) = CABS1( B( 1, J ) ) +
249 $ CABS1( D( 1 ) )*CABS1( X( 1, J ) ) +
250 $ CABS1( DL( 1 ) )*CABS1( X( 2, J ) )
251 DO 40 I = 2, N - 1
252 RWORK( I ) = CABS1( B( I, J ) ) +
253 $ CABS1( DU( I-1 ) )*CABS1( X( I-1, J ) ) +
254 $ CABS1( D( I ) )*CABS1( X( I, J ) ) +
255 $ CABS1( DL( I ) )*CABS1( X( I+1, J ) )
256 40 CONTINUE
257 RWORK( N ) = CABS1( B( N, J ) ) +
258 $ CABS1( DU( N-1 ) )*CABS1( X( N-1, J ) ) +
259 $ CABS1( D( N ) )*CABS1( X( N, J ) )
260 END IF
261 END IF
262 *
263 * Compute componentwise relative backward error from formula
264 *
265 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
266 *
267 * where abs(Z) is the componentwise absolute value of the matrix
268 * or vector Z. If the i-th component of the denominator is less
269 * than SAFE2, then SAFE1 is added to the i-th components of the
270 * numerator and denominator before dividing.
271 *
272 S = ZERO
273 DO 50 I = 1, N
274 IF( RWORK( I ).GT.SAFE2 ) THEN
275 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
276 ELSE
277 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
278 $ ( RWORK( I )+SAFE1 ) )
279 END IF
280 50 CONTINUE
281 BERR( J ) = S
282 *
283 * Test stopping criterion. Continue iterating if
284 * 1) The residual BERR(J) is larger than machine epsilon, and
285 * 2) BERR(J) decreased by at least a factor of 2 during the
286 * last iteration, and
287 * 3) At most ITMAX iterations tried.
288 *
289 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
290 $ COUNT.LE.ITMAX ) THEN
291 *
292 * Update solution and try again.
293 *
294 CALL ZGTTRS( TRANS, N, 1, DLF, DF, DUF, DU2, IPIV, WORK, N,
295 $ INFO )
296 CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
297 LSTRES = BERR( J )
298 COUNT = COUNT + 1
299 GO TO 20
300 END IF
301 *
302 * Bound error from formula
303 *
304 * norm(X - XTRUE) / norm(X) .le. FERR =
305 * norm( abs(inv(op(A)))*
306 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
307 *
308 * where
309 * norm(Z) is the magnitude of the largest component of Z
310 * inv(op(A)) is the inverse of op(A)
311 * abs(Z) is the componentwise absolute value of the matrix or
312 * vector Z
313 * NZ is the maximum number of nonzeros in any row of A, plus 1
314 * EPS is machine epsilon
315 *
316 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
317 * is incremented by SAFE1 if the i-th component of
318 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
319 *
320 * Use ZLACN2 to estimate the infinity-norm of the matrix
321 * inv(op(A)) * diag(W),
322 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
323 *
324 DO 60 I = 1, N
325 IF( RWORK( I ).GT.SAFE2 ) THEN
326 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
327 ELSE
328 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
329 $ SAFE1
330 END IF
331 60 CONTINUE
332 *
333 KASE = 0
334 70 CONTINUE
335 CALL ZLACN2( N, WORK( N+1 ), WORK, FERR( J ), KASE, ISAVE )
336 IF( KASE.NE.0 ) THEN
337 IF( KASE.EQ.1 ) THEN
338 *
339 * Multiply by diag(W)*inv(op(A)**H).
340 *
341 CALL ZGTTRS( TRANST, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
342 $ N, INFO )
343 DO 80 I = 1, N
344 WORK( I ) = RWORK( I )*WORK( I )
345 80 CONTINUE
346 ELSE
347 *
348 * Multiply by inv(op(A))*diag(W).
349 *
350 DO 90 I = 1, N
351 WORK( I ) = RWORK( I )*WORK( I )
352 90 CONTINUE
353 CALL ZGTTRS( TRANSN, N, 1, DLF, DF, DUF, DU2, IPIV, WORK,
354 $ N, INFO )
355 END IF
356 GO TO 70
357 END IF
358 *
359 * Normalize error.
360 *
361 LSTRES = ZERO
362 DO 100 I = 1, N
363 LSTRES = MAX( LSTRES, CABS1( X( I, J ) ) )
364 100 CONTINUE
365 IF( LSTRES.NE.ZERO )
366 $ FERR( J ) = FERR( J ) / LSTRES
367 *
368 110 CONTINUE
369 *
370 RETURN
371 *
372 * End of ZGTRFS
373 *
374 END