1 SUBROUTINE ZPTRFS( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
2 $ FERR, BERR, WORK, RWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER UPLO
11 INTEGER INFO, LDB, LDX, N, NRHS
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
15 $ RWORK( * )
16 COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
17 $ X( LDX, * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * ZPTRFS improves the computed solution to a system of linear
24 * equations when the coefficient matrix is Hermitian positive definite
25 * and tridiagonal, and provides error bounds and backward error
26 * estimates for the solution.
27 *
28 * Arguments
29 * =========
30 *
31 * UPLO (input) CHARACTER*1
32 * Specifies whether the superdiagonal or the subdiagonal of the
33 * tridiagonal matrix A is stored and the form of the
34 * factorization:
35 * = 'U': E is the superdiagonal of A, and A = U**H*D*U;
36 * = 'L': E is the subdiagonal of A, and A = L*D*L**H.
37 * (The two forms are equivalent if A is real.)
38 *
39 * N (input) INTEGER
40 * The order of the matrix A. N >= 0.
41 *
42 * NRHS (input) INTEGER
43 * The number of right hand sides, i.e., the number of columns
44 * of the matrix B. NRHS >= 0.
45 *
46 * D (input) DOUBLE PRECISION array, dimension (N)
47 * The n real diagonal elements of the tridiagonal matrix A.
48 *
49 * E (input) COMPLEX*16 array, dimension (N-1)
50 * The (n-1) off-diagonal elements of the tridiagonal matrix A
51 * (see UPLO).
52 *
53 * DF (input) DOUBLE PRECISION array, dimension (N)
54 * The n diagonal elements of the diagonal matrix D from
55 * the factorization computed by ZPTTRF.
56 *
57 * EF (input) COMPLEX*16 array, dimension (N-1)
58 * The (n-1) off-diagonal elements of the unit bidiagonal
59 * factor U or L from the factorization computed by ZPTTRF
60 * (see UPLO).
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 ZPTTRS.
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 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).
82 *
83 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
84 * The componentwise relative backward error of each solution
85 * vector X(j) (i.e., the smallest relative change in
86 * any element of A or B that makes X(j) an exact solution).
87 *
88 * WORK (workspace) COMPLEX*16 array, dimension (N)
89 *
90 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
91 *
92 * INFO (output) INTEGER
93 * = 0: successful exit
94 * < 0: if INFO = -i, the i-th argument had an illegal value
95 *
96 * Internal Parameters
97 * ===================
98 *
99 * ITMAX is the maximum number of steps of iterative refinement.
100 *
101 * =====================================================================
102 *
103 * .. Parameters ..
104 INTEGER ITMAX
105 PARAMETER ( ITMAX = 5 )
106 DOUBLE PRECISION ZERO
107 PARAMETER ( ZERO = 0.0D+0 )
108 DOUBLE PRECISION ONE
109 PARAMETER ( ONE = 1.0D+0 )
110 DOUBLE PRECISION TWO
111 PARAMETER ( TWO = 2.0D+0 )
112 DOUBLE PRECISION THREE
113 PARAMETER ( THREE = 3.0D+0 )
114 * ..
115 * .. Local Scalars ..
116 LOGICAL UPPER
117 INTEGER COUNT, I, IX, J, NZ
118 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
119 COMPLEX*16 BI, CX, DX, EX, ZDUM
120 * ..
121 * .. External Functions ..
122 LOGICAL LSAME
123 INTEGER IDAMAX
124 DOUBLE PRECISION DLAMCH
125 EXTERNAL LSAME, IDAMAX, DLAMCH
126 * ..
127 * .. External Subroutines ..
128 EXTERNAL XERBLA, ZAXPY, ZPTTRS
129 * ..
130 * .. Intrinsic Functions ..
131 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
132 * ..
133 * .. Statement Functions ..
134 DOUBLE PRECISION CABS1
135 * ..
136 * .. Statement Function definitions ..
137 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
138 * ..
139 * .. Executable Statements ..
140 *
141 * Test the input parameters.
142 *
143 INFO = 0
144 UPPER = LSAME( UPLO, 'U' )
145 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
146 INFO = -1
147 ELSE IF( N.LT.0 ) THEN
148 INFO = -2
149 ELSE IF( NRHS.LT.0 ) THEN
150 INFO = -3
151 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
152 INFO = -9
153 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
154 INFO = -11
155 END IF
156 IF( INFO.NE.0 ) THEN
157 CALL XERBLA( 'ZPTRFS', -INFO )
158 RETURN
159 END IF
160 *
161 * Quick return if possible
162 *
163 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
164 DO 10 J = 1, NRHS
165 FERR( J ) = ZERO
166 BERR( J ) = ZERO
167 10 CONTINUE
168 RETURN
169 END IF
170 *
171 * NZ = maximum number of nonzero elements in each row of A, plus 1
172 *
173 NZ = 4
174 EPS = DLAMCH( 'Epsilon' )
175 SAFMIN = DLAMCH( 'Safe minimum' )
176 SAFE1 = NZ*SAFMIN
177 SAFE2 = SAFE1 / EPS
178 *
179 * Do for each right hand side
180 *
181 DO 100 J = 1, NRHS
182 *
183 COUNT = 1
184 LSTRES = THREE
185 20 CONTINUE
186 *
187 * Loop until stopping criterion is satisfied.
188 *
189 * Compute residual R = B - A * X. Also compute
190 * abs(A)*abs(x) + abs(b) for use in the backward error bound.
191 *
192 IF( UPPER ) THEN
193 IF( N.EQ.1 ) THEN
194 BI = B( 1, J )
195 DX = D( 1 )*X( 1, J )
196 WORK( 1 ) = BI - DX
197 RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
198 ELSE
199 BI = B( 1, J )
200 DX = D( 1 )*X( 1, J )
201 EX = E( 1 )*X( 2, J )
202 WORK( 1 ) = BI - DX - EX
203 RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
204 $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
205 DO 30 I = 2, N - 1
206 BI = B( I, J )
207 CX = DCONJG( E( I-1 ) )*X( I-1, J )
208 DX = D( I )*X( I, J )
209 EX = E( I )*X( I+1, J )
210 WORK( I ) = BI - CX - DX - EX
211 RWORK( I ) = CABS1( BI ) +
212 $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
213 $ CABS1( DX ) + CABS1( E( I ) )*
214 $ CABS1( X( I+1, J ) )
215 30 CONTINUE
216 BI = B( N, J )
217 CX = DCONJG( E( N-1 ) )*X( N-1, J )
218 DX = D( N )*X( N, J )
219 WORK( N ) = BI - CX - DX
220 RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
221 $ CABS1( X( N-1, J ) ) + CABS1( DX )
222 END IF
223 ELSE
224 IF( N.EQ.1 ) THEN
225 BI = B( 1, J )
226 DX = D( 1 )*X( 1, J )
227 WORK( 1 ) = BI - DX
228 RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
229 ELSE
230 BI = B( 1, J )
231 DX = D( 1 )*X( 1, J )
232 EX = DCONJG( E( 1 ) )*X( 2, J )
233 WORK( 1 ) = BI - DX - EX
234 RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
235 $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
236 DO 40 I = 2, N - 1
237 BI = B( I, J )
238 CX = E( I-1 )*X( I-1, J )
239 DX = D( I )*X( I, J )
240 EX = DCONJG( E( I ) )*X( I+1, J )
241 WORK( I ) = BI - CX - DX - EX
242 RWORK( I ) = CABS1( BI ) +
243 $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
244 $ CABS1( DX ) + CABS1( E( I ) )*
245 $ CABS1( X( I+1, J ) )
246 40 CONTINUE
247 BI = B( N, J )
248 CX = E( N-1 )*X( N-1, J )
249 DX = D( N )*X( N, J )
250 WORK( N ) = BI - CX - DX
251 RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
252 $ CABS1( X( N-1, J ) ) + CABS1( DX )
253 END IF
254 END IF
255 *
256 * Compute componentwise relative backward error from formula
257 *
258 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
259 *
260 * where abs(Z) is the componentwise absolute value of the matrix
261 * or vector Z. If the i-th component of the denominator is less
262 * than SAFE2, then SAFE1 is added to the i-th components of the
263 * numerator and denominator before dividing.
264 *
265 S = ZERO
266 DO 50 I = 1, N
267 IF( RWORK( I ).GT.SAFE2 ) THEN
268 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
269 ELSE
270 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
271 $ ( RWORK( I )+SAFE1 ) )
272 END IF
273 50 CONTINUE
274 BERR( J ) = S
275 *
276 * Test stopping criterion. Continue iterating if
277 * 1) The residual BERR(J) is larger than machine epsilon, and
278 * 2) BERR(J) decreased by at least a factor of 2 during the
279 * last iteration, and
280 * 3) At most ITMAX iterations tried.
281 *
282 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
283 $ COUNT.LE.ITMAX ) THEN
284 *
285 * Update solution and try again.
286 *
287 CALL ZPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
288 CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
289 LSTRES = BERR( J )
290 COUNT = COUNT + 1
291 GO TO 20
292 END IF
293 *
294 * Bound error from formula
295 *
296 * norm(X - XTRUE) / norm(X) .le. FERR =
297 * norm( abs(inv(A))*
298 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
299 *
300 * where
301 * norm(Z) is the magnitude of the largest component of Z
302 * inv(A) is the inverse of A
303 * abs(Z) is the componentwise absolute value of the matrix or
304 * vector Z
305 * NZ is the maximum number of nonzeros in any row of A, plus 1
306 * EPS is machine epsilon
307 *
308 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
309 * is incremented by SAFE1 if the i-th component of
310 * abs(A)*abs(X) + abs(B) is less than SAFE2.
311 *
312 DO 60 I = 1, N
313 IF( RWORK( I ).GT.SAFE2 ) THEN
314 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
315 ELSE
316 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
317 $ SAFE1
318 END IF
319 60 CONTINUE
320 IX = IDAMAX( N, RWORK, 1 )
321 FERR( J ) = RWORK( IX )
322 *
323 * Estimate the norm of inv(A).
324 *
325 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
326 *
327 * m(i,j) = abs(A(i,j)), i = j,
328 * m(i,j) = -abs(A(i,j)), i .ne. j,
329 *
330 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
331 *
332 * Solve M(L) * x = e.
333 *
334 RWORK( 1 ) = ONE
335 DO 70 I = 2, N
336 RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
337 70 CONTINUE
338 *
339 * Solve D * M(L)**H * x = b.
340 *
341 RWORK( N ) = RWORK( N ) / DF( N )
342 DO 80 I = N - 1, 1, -1
343 RWORK( I ) = RWORK( I ) / DF( I ) +
344 $ RWORK( I+1 )*ABS( EF( I ) )
345 80 CONTINUE
346 *
347 * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
348 *
349 IX = IDAMAX( N, RWORK, 1 )
350 FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
351 *
352 * Normalize error.
353 *
354 LSTRES = ZERO
355 DO 90 I = 1, N
356 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
357 90 CONTINUE
358 IF( LSTRES.NE.ZERO )
359 $ FERR( J ) = FERR( J ) / LSTRES
360 *
361 100 CONTINUE
362 *
363 RETURN
364 *
365 * End of ZPTRFS
366 *
367 END
2 $ FERR, BERR, WORK, RWORK, INFO )
3 *
4 * -- LAPACK routine (version 3.3.1) --
5 * -- LAPACK is a software package provided by Univ. of Tennessee, --
6 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7 * -- April 2011 --
8 *
9 * .. Scalar Arguments ..
10 CHARACTER UPLO
11 INTEGER INFO, LDB, LDX, N, NRHS
12 * ..
13 * .. Array Arguments ..
14 DOUBLE PRECISION BERR( * ), D( * ), DF( * ), FERR( * ),
15 $ RWORK( * )
16 COMPLEX*16 B( LDB, * ), E( * ), EF( * ), WORK( * ),
17 $ X( LDX, * )
18 * ..
19 *
20 * Purpose
21 * =======
22 *
23 * ZPTRFS improves the computed solution to a system of linear
24 * equations when the coefficient matrix is Hermitian positive definite
25 * and tridiagonal, and provides error bounds and backward error
26 * estimates for the solution.
27 *
28 * Arguments
29 * =========
30 *
31 * UPLO (input) CHARACTER*1
32 * Specifies whether the superdiagonal or the subdiagonal of the
33 * tridiagonal matrix A is stored and the form of the
34 * factorization:
35 * = 'U': E is the superdiagonal of A, and A = U**H*D*U;
36 * = 'L': E is the subdiagonal of A, and A = L*D*L**H.
37 * (The two forms are equivalent if A is real.)
38 *
39 * N (input) INTEGER
40 * The order of the matrix A. N >= 0.
41 *
42 * NRHS (input) INTEGER
43 * The number of right hand sides, i.e., the number of columns
44 * of the matrix B. NRHS >= 0.
45 *
46 * D (input) DOUBLE PRECISION array, dimension (N)
47 * The n real diagonal elements of the tridiagonal matrix A.
48 *
49 * E (input) COMPLEX*16 array, dimension (N-1)
50 * The (n-1) off-diagonal elements of the tridiagonal matrix A
51 * (see UPLO).
52 *
53 * DF (input) DOUBLE PRECISION array, dimension (N)
54 * The n diagonal elements of the diagonal matrix D from
55 * the factorization computed by ZPTTRF.
56 *
57 * EF (input) COMPLEX*16 array, dimension (N-1)
58 * The (n-1) off-diagonal elements of the unit bidiagonal
59 * factor U or L from the factorization computed by ZPTTRF
60 * (see UPLO).
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 ZPTTRS.
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 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).
82 *
83 * BERR (output) DOUBLE PRECISION array, dimension (NRHS)
84 * The componentwise relative backward error of each solution
85 * vector X(j) (i.e., the smallest relative change in
86 * any element of A or B that makes X(j) an exact solution).
87 *
88 * WORK (workspace) COMPLEX*16 array, dimension (N)
89 *
90 * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
91 *
92 * INFO (output) INTEGER
93 * = 0: successful exit
94 * < 0: if INFO = -i, the i-th argument had an illegal value
95 *
96 * Internal Parameters
97 * ===================
98 *
99 * ITMAX is the maximum number of steps of iterative refinement.
100 *
101 * =====================================================================
102 *
103 * .. Parameters ..
104 INTEGER ITMAX
105 PARAMETER ( ITMAX = 5 )
106 DOUBLE PRECISION ZERO
107 PARAMETER ( ZERO = 0.0D+0 )
108 DOUBLE PRECISION ONE
109 PARAMETER ( ONE = 1.0D+0 )
110 DOUBLE PRECISION TWO
111 PARAMETER ( TWO = 2.0D+0 )
112 DOUBLE PRECISION THREE
113 PARAMETER ( THREE = 3.0D+0 )
114 * ..
115 * .. Local Scalars ..
116 LOGICAL UPPER
117 INTEGER COUNT, I, IX, J, NZ
118 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
119 COMPLEX*16 BI, CX, DX, EX, ZDUM
120 * ..
121 * .. External Functions ..
122 LOGICAL LSAME
123 INTEGER IDAMAX
124 DOUBLE PRECISION DLAMCH
125 EXTERNAL LSAME, IDAMAX, DLAMCH
126 * ..
127 * .. External Subroutines ..
128 EXTERNAL XERBLA, ZAXPY, ZPTTRS
129 * ..
130 * .. Intrinsic Functions ..
131 INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
132 * ..
133 * .. Statement Functions ..
134 DOUBLE PRECISION CABS1
135 * ..
136 * .. Statement Function definitions ..
137 CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
138 * ..
139 * .. Executable Statements ..
140 *
141 * Test the input parameters.
142 *
143 INFO = 0
144 UPPER = LSAME( UPLO, 'U' )
145 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
146 INFO = -1
147 ELSE IF( N.LT.0 ) THEN
148 INFO = -2
149 ELSE IF( NRHS.LT.0 ) THEN
150 INFO = -3
151 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
152 INFO = -9
153 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
154 INFO = -11
155 END IF
156 IF( INFO.NE.0 ) THEN
157 CALL XERBLA( 'ZPTRFS', -INFO )
158 RETURN
159 END IF
160 *
161 * Quick return if possible
162 *
163 IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
164 DO 10 J = 1, NRHS
165 FERR( J ) = ZERO
166 BERR( J ) = ZERO
167 10 CONTINUE
168 RETURN
169 END IF
170 *
171 * NZ = maximum number of nonzero elements in each row of A, plus 1
172 *
173 NZ = 4
174 EPS = DLAMCH( 'Epsilon' )
175 SAFMIN = DLAMCH( 'Safe minimum' )
176 SAFE1 = NZ*SAFMIN
177 SAFE2 = SAFE1 / EPS
178 *
179 * Do for each right hand side
180 *
181 DO 100 J = 1, NRHS
182 *
183 COUNT = 1
184 LSTRES = THREE
185 20 CONTINUE
186 *
187 * Loop until stopping criterion is satisfied.
188 *
189 * Compute residual R = B - A * X. Also compute
190 * abs(A)*abs(x) + abs(b) for use in the backward error bound.
191 *
192 IF( UPPER ) THEN
193 IF( N.EQ.1 ) THEN
194 BI = B( 1, J )
195 DX = D( 1 )*X( 1, J )
196 WORK( 1 ) = BI - DX
197 RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
198 ELSE
199 BI = B( 1, J )
200 DX = D( 1 )*X( 1, J )
201 EX = E( 1 )*X( 2, J )
202 WORK( 1 ) = BI - DX - EX
203 RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
204 $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
205 DO 30 I = 2, N - 1
206 BI = B( I, J )
207 CX = DCONJG( E( I-1 ) )*X( I-1, J )
208 DX = D( I )*X( I, J )
209 EX = E( I )*X( I+1, J )
210 WORK( I ) = BI - CX - DX - EX
211 RWORK( I ) = CABS1( BI ) +
212 $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
213 $ CABS1( DX ) + CABS1( E( I ) )*
214 $ CABS1( X( I+1, J ) )
215 30 CONTINUE
216 BI = B( N, J )
217 CX = DCONJG( E( N-1 ) )*X( N-1, J )
218 DX = D( N )*X( N, J )
219 WORK( N ) = BI - CX - DX
220 RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
221 $ CABS1( X( N-1, J ) ) + CABS1( DX )
222 END IF
223 ELSE
224 IF( N.EQ.1 ) THEN
225 BI = B( 1, J )
226 DX = D( 1 )*X( 1, J )
227 WORK( 1 ) = BI - DX
228 RWORK( 1 ) = CABS1( BI ) + CABS1( DX )
229 ELSE
230 BI = B( 1, J )
231 DX = D( 1 )*X( 1, J )
232 EX = DCONJG( E( 1 ) )*X( 2, J )
233 WORK( 1 ) = BI - DX - EX
234 RWORK( 1 ) = CABS1( BI ) + CABS1( DX ) +
235 $ CABS1( E( 1 ) )*CABS1( X( 2, J ) )
236 DO 40 I = 2, N - 1
237 BI = B( I, J )
238 CX = E( I-1 )*X( I-1, J )
239 DX = D( I )*X( I, J )
240 EX = DCONJG( E( I ) )*X( I+1, J )
241 WORK( I ) = BI - CX - DX - EX
242 RWORK( I ) = CABS1( BI ) +
243 $ CABS1( E( I-1 ) )*CABS1( X( I-1, J ) ) +
244 $ CABS1( DX ) + CABS1( E( I ) )*
245 $ CABS1( X( I+1, J ) )
246 40 CONTINUE
247 BI = B( N, J )
248 CX = E( N-1 )*X( N-1, J )
249 DX = D( N )*X( N, J )
250 WORK( N ) = BI - CX - DX
251 RWORK( N ) = CABS1( BI ) + CABS1( E( N-1 ) )*
252 $ CABS1( X( N-1, J ) ) + CABS1( DX )
253 END IF
254 END IF
255 *
256 * Compute componentwise relative backward error from formula
257 *
258 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
259 *
260 * where abs(Z) is the componentwise absolute value of the matrix
261 * or vector Z. If the i-th component of the denominator is less
262 * than SAFE2, then SAFE1 is added to the i-th components of the
263 * numerator and denominator before dividing.
264 *
265 S = ZERO
266 DO 50 I = 1, N
267 IF( RWORK( I ).GT.SAFE2 ) THEN
268 S = MAX( S, CABS1( WORK( I ) ) / RWORK( I ) )
269 ELSE
270 S = MAX( S, ( CABS1( WORK( I ) )+SAFE1 ) /
271 $ ( RWORK( I )+SAFE1 ) )
272 END IF
273 50 CONTINUE
274 BERR( J ) = S
275 *
276 * Test stopping criterion. Continue iterating if
277 * 1) The residual BERR(J) is larger than machine epsilon, and
278 * 2) BERR(J) decreased by at least a factor of 2 during the
279 * last iteration, and
280 * 3) At most ITMAX iterations tried.
281 *
282 IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
283 $ COUNT.LE.ITMAX ) THEN
284 *
285 * Update solution and try again.
286 *
287 CALL ZPTTRS( UPLO, N, 1, DF, EF, WORK, N, INFO )
288 CALL ZAXPY( N, DCMPLX( ONE ), WORK, 1, X( 1, J ), 1 )
289 LSTRES = BERR( J )
290 COUNT = COUNT + 1
291 GO TO 20
292 END IF
293 *
294 * Bound error from formula
295 *
296 * norm(X - XTRUE) / norm(X) .le. FERR =
297 * norm( abs(inv(A))*
298 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
299 *
300 * where
301 * norm(Z) is the magnitude of the largest component of Z
302 * inv(A) is the inverse of A
303 * abs(Z) is the componentwise absolute value of the matrix or
304 * vector Z
305 * NZ is the maximum number of nonzeros in any row of A, plus 1
306 * EPS is machine epsilon
307 *
308 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
309 * is incremented by SAFE1 if the i-th component of
310 * abs(A)*abs(X) + abs(B) is less than SAFE2.
311 *
312 DO 60 I = 1, N
313 IF( RWORK( I ).GT.SAFE2 ) THEN
314 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I )
315 ELSE
316 RWORK( I ) = CABS1( WORK( I ) ) + NZ*EPS*RWORK( I ) +
317 $ SAFE1
318 END IF
319 60 CONTINUE
320 IX = IDAMAX( N, RWORK, 1 )
321 FERR( J ) = RWORK( IX )
322 *
323 * Estimate the norm of inv(A).
324 *
325 * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
326 *
327 * m(i,j) = abs(A(i,j)), i = j,
328 * m(i,j) = -abs(A(i,j)), i .ne. j,
329 *
330 * and e = [ 1, 1, ..., 1 ]**T. Note M(A) = M(L)*D*M(L)**H.
331 *
332 * Solve M(L) * x = e.
333 *
334 RWORK( 1 ) = ONE
335 DO 70 I = 2, N
336 RWORK( I ) = ONE + RWORK( I-1 )*ABS( EF( I-1 ) )
337 70 CONTINUE
338 *
339 * Solve D * M(L)**H * x = b.
340 *
341 RWORK( N ) = RWORK( N ) / DF( N )
342 DO 80 I = N - 1, 1, -1
343 RWORK( I ) = RWORK( I ) / DF( I ) +
344 $ RWORK( I+1 )*ABS( EF( I ) )
345 80 CONTINUE
346 *
347 * Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
348 *
349 IX = IDAMAX( N, RWORK, 1 )
350 FERR( J ) = FERR( J )*ABS( RWORK( IX ) )
351 *
352 * Normalize error.
353 *
354 LSTRES = ZERO
355 DO 90 I = 1, N
356 LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
357 90 CONTINUE
358 IF( LSTRES.NE.ZERO )
359 $ FERR( J ) = FERR( J ) / LSTRES
360 *
361 100 CONTINUE
362 *
363 RETURN
364 *
365 * End of ZPTRFS
366 *
367 END