LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
zgbrfs.f
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1 *> \brief \b ZGBRFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
22 * IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK,
23 * INFO )
24 *
25 * .. Scalar Arguments ..
26 * CHARACTER TRANS
27 * INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
28 * ..
29 * .. Array Arguments ..
30 * INTEGER IPIV( * )
31 * DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
32 * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
33 * \$ WORK( * ), X( LDX, * )
34 * ..
35 *
36 *
37 *> \par Purpose:
38 * =============
39 *>
40 *> \verbatim
41 *>
42 *> ZGBRFS improves the computed solution to a system of linear
43 *> equations when the coefficient matrix is banded, and provides
44 *> error bounds and backward error estimates for the solution.
45 *> \endverbatim
46 *
47 * Arguments:
48 * ==========
49 *
50 *> \param[in] TRANS
51 *> \verbatim
52 *> TRANS is CHARACTER*1
53 *> Specifies the form of the system of equations:
54 *> = 'N': A * X = B (No transpose)
55 *> = 'T': A**T * X = B (Transpose)
56 *> = 'C': A**H * X = B (Conjugate transpose)
57 *> \endverbatim
58 *>
59 *> \param[in] N
60 *> \verbatim
61 *> N is INTEGER
62 *> The order of the matrix A. N >= 0.
63 *> \endverbatim
64 *>
65 *> \param[in] KL
66 *> \verbatim
67 *> KL is INTEGER
68 *> The number of subdiagonals within the band of A. KL >= 0.
69 *> \endverbatim
70 *>
71 *> \param[in] KU
72 *> \verbatim
73 *> KU is INTEGER
74 *> The number of superdiagonals within the band of A. KU >= 0.
75 *> \endverbatim
76 *>
77 *> \param[in] NRHS
78 *> \verbatim
79 *> NRHS is INTEGER
80 *> The number of right hand sides, i.e., the number of columns
81 *> of the matrices B and X. NRHS >= 0.
82 *> \endverbatim
83 *>
84 *> \param[in] AB
85 *> \verbatim
86 *> AB is COMPLEX*16 array, dimension (LDAB,N)
87 *> The original band matrix A, stored in rows 1 to KL+KU+1.
88 *> The j-th column of A is stored in the j-th column of the
89 *> array AB as follows:
90 *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
91 *> \endverbatim
92 *>
93 *> \param[in] LDAB
94 *> \verbatim
95 *> LDAB is INTEGER
96 *> The leading dimension of the array AB. LDAB >= KL+KU+1.
97 *> \endverbatim
98 *>
99 *> \param[in] AFB
100 *> \verbatim
101 *> AFB is COMPLEX*16 array, dimension (LDAFB,N)
102 *> Details of the LU factorization of the band matrix A, as
103 *> computed by ZGBTRF. U is stored as an upper triangular band
104 *> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and
105 *> the multipliers used during the factorization are stored in
106 *> rows KL+KU+2 to 2*KL+KU+1.
107 *> \endverbatim
108 *>
109 *> \param[in] LDAFB
110 *> \verbatim
111 *> LDAFB is INTEGER
112 *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1.
113 *> \endverbatim
114 *>
115 *> \param[in] IPIV
116 *> \verbatim
117 *> IPIV is INTEGER array, dimension (N)
118 *> The pivot indices from ZGBTRF; for 1<=i<=N, row i of the
119 *> matrix was interchanged with row IPIV(i).
120 *> \endverbatim
121 *>
122 *> \param[in] B
123 *> \verbatim
124 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
125 *> The right hand side matrix B.
126 *> \endverbatim
127 *>
128 *> \param[in] LDB
129 *> \verbatim
130 *> LDB is INTEGER
131 *> The leading dimension of the array B. LDB >= max(1,N).
132 *> \endverbatim
133 *>
134 *> \param[in,out] X
135 *> \verbatim
136 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
137 *> On entry, the solution matrix X, as computed by ZGBTRS.
138 *> On exit, the improved solution matrix X.
139 *> \endverbatim
140 *>
141 *> \param[in] LDX
142 *> \verbatim
143 *> LDX is INTEGER
144 *> The leading dimension of the array X. LDX >= max(1,N).
145 *> \endverbatim
146 *>
147 *> \param[out] FERR
148 *> \verbatim
149 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
150 *> The estimated forward error bound for each solution vector
151 *> X(j) (the j-th column of the solution matrix X).
152 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
153 *> is an estimated upper bound for the magnitude of the largest
154 *> element in (X(j) - XTRUE) divided by the magnitude of the
155 *> largest element in X(j). The estimate is as reliable as
156 *> the estimate for RCOND, and is almost always a slight
157 *> overestimate of the true error.
158 *> \endverbatim
159 *>
160 *> \param[out] BERR
161 *> \verbatim
162 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
163 *> The componentwise relative backward error of each solution
164 *> vector X(j) (i.e., the smallest relative change in
165 *> any element of A or B that makes X(j) an exact solution).
166 *> \endverbatim
167 *>
168 *> \param[out] WORK
169 *> \verbatim
170 *> WORK is COMPLEX*16 array, dimension (2*N)
171 *> \endverbatim
172 *>
173 *> \param[out] RWORK
174 *> \verbatim
175 *> RWORK is DOUBLE PRECISION array, dimension (N)
176 *> \endverbatim
177 *>
178 *> \param[out] INFO
179 *> \verbatim
180 *> INFO is INTEGER
181 *> = 0: successful exit
182 *> < 0: if INFO = -i, the i-th argument had an illegal value
183 *> \endverbatim
184 *
185 *> \par Internal Parameters:
186 * =========================
187 *>
188 *> \verbatim
189 *> ITMAX is the maximum number of steps of iterative refinement.
190 *> \endverbatim
191 *
192 * Authors:
193 * ========
194 *
195 *> \author Univ. of Tennessee
196 *> \author Univ. of California Berkeley
197 *> \author Univ. of Colorado Denver
198 *> \author NAG Ltd.
199 *
200 *> \date November 2011
201 *
202 *> \ingroup complex16GBcomputational
203 *
204 * =====================================================================
205  SUBROUTINE zgbrfs( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB,
206  \$ ipiv, b, ldb, x, ldx, ferr, berr, work, rwork,
207  \$ info )
208 *
209 * -- LAPACK computational routine (version 3.4.0) --
210 * -- LAPACK is a software package provided by Univ. of Tennessee, --
211 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
212 * November 2011
213 *
214 * .. Scalar Arguments ..
215  CHARACTER trans
216  INTEGER info, kl, ku, ldab, ldafb, ldb, ldx, n, nrhs
217 * ..
218 * .. Array Arguments ..
219  INTEGER ipiv( * )
220  DOUBLE PRECISION berr( * ), ferr( * ), rwork( * )
221  COMPLEX*16 ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
222  \$ work( * ), x( ldx, * )
223 * ..
224 *
225 * =====================================================================
226 *
227 * .. Parameters ..
228  INTEGER itmax
229  parameter( itmax = 5 )
230  DOUBLE PRECISION zero
231  parameter( zero = 0.0d+0 )
232  COMPLEX*16 cone
233  parameter( cone = ( 1.0d+0, 0.0d+0 ) )
234  DOUBLE PRECISION two
235  parameter( two = 2.0d+0 )
236  DOUBLE PRECISION three
237  parameter( three = 3.0d+0 )
238 * ..
239 * .. Local Scalars ..
240  LOGICAL notran
241  CHARACTER transn, transt
242  INTEGER count, i, j, k, kase, kk, nz
243  DOUBLE PRECISION eps, lstres, s, safe1, safe2, safmin, xk
244  COMPLEX*16 zdum
245 * ..
246 * .. Local Arrays ..
247  INTEGER isave( 3 )
248 * ..
249 * .. External Subroutines ..
250  EXTERNAL xerbla, zaxpy, zcopy, zgbmv, zgbtrs, zlacn2
251 * ..
252 * .. Intrinsic Functions ..
253  INTRINSIC abs, dble, dimag, max, min
254 * ..
255 * .. External Functions ..
256  LOGICAL lsame
257  DOUBLE PRECISION dlamch
258  EXTERNAL lsame, dlamch
259 * ..
260 * .. Statement Functions ..
261  DOUBLE PRECISION cabs1
262 * ..
263 * .. Statement Function definitions ..
264  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
265 * ..
266 * .. Executable Statements ..
267 *
268 * Test the input parameters.
269 *
270  info = 0
271  notran = lsame( trans, 'N' )
272  IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
273  \$ lsame( trans, 'C' ) ) THEN
274  info = -1
275  ELSE IF( n.LT.0 ) THEN
276  info = -2
277  ELSE IF( kl.LT.0 ) THEN
278  info = -3
279  ELSE IF( ku.LT.0 ) THEN
280  info = -4
281  ELSE IF( nrhs.LT.0 ) THEN
282  info = -5
283  ELSE IF( ldab.LT.kl+ku+1 ) THEN
284  info = -7
285  ELSE IF( ldafb.LT.2*kl+ku+1 ) THEN
286  info = -9
287  ELSE IF( ldb.LT.max( 1, n ) ) THEN
288  info = -12
289  ELSE IF( ldx.LT.max( 1, n ) ) THEN
290  info = -14
291  END IF
292  IF( info.NE.0 ) THEN
293  CALL xerbla( 'ZGBRFS', -info )
294  return
295  END IF
296 *
297 * Quick return if possible
298 *
299  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
300  DO 10 j = 1, nrhs
301  ferr( j ) = zero
302  berr( j ) = zero
303  10 continue
304  return
305  END IF
306 *
307  IF( notran ) THEN
308  transn = 'N'
309  transt = 'C'
310  ELSE
311  transn = 'C'
312  transt = 'N'
313  END IF
314 *
315 * NZ = maximum number of nonzero elements in each row of A, plus 1
316 *
317  nz = min( kl+ku+2, n+1 )
318  eps = dlamch( 'Epsilon' )
319  safmin = dlamch( 'Safe minimum' )
320  safe1 = nz*safmin
321  safe2 = safe1 / eps
322 *
323 * Do for each right hand side
324 *
325  DO 140 j = 1, nrhs
326 *
327  count = 1
328  lstres = three
329  20 continue
330 *
331 * Loop until stopping criterion is satisfied.
332 *
333 * Compute residual R = B - op(A) * X,
334 * where op(A) = A, A**T, or A**H, depending on TRANS.
335 *
336  CALL zcopy( n, b( 1, j ), 1, work, 1 )
337  CALL zgbmv( trans, n, n, kl, ku, -cone, ab, ldab, x( 1, j ), 1,
338  \$ cone, work, 1 )
339 *
340 * Compute componentwise relative backward error from formula
341 *
342 * max(i) ( abs(R(i)) / ( abs(op(A))*abs(X) + abs(B) )(i) )
343 *
344 * where abs(Z) is the componentwise absolute value of the matrix
345 * or vector Z. If the i-th component of the denominator is less
346 * than SAFE2, then SAFE1 is added to the i-th components of the
347 * numerator and denominator before dividing.
348 *
349  DO 30 i = 1, n
350  rwork( i ) = cabs1( b( i, j ) )
351  30 continue
352 *
353 * Compute abs(op(A))*abs(X) + abs(B).
354 *
355  IF( notran ) THEN
356  DO 50 k = 1, n
357  kk = ku + 1 - k
358  xk = cabs1( x( k, j ) )
359  DO 40 i = max( 1, k-ku ), min( n, k+kl )
360  rwork( i ) = rwork( i ) + cabs1( ab( kk+i, k ) )*xk
361  40 continue
362  50 continue
363  ELSE
364  DO 70 k = 1, n
365  s = zero
366  kk = ku + 1 - k
367  DO 60 i = max( 1, k-ku ), min( n, k+kl )
368  s = s + cabs1( ab( kk+i, k ) )*cabs1( x( i, j ) )
369  60 continue
370  rwork( k ) = rwork( k ) + s
371  70 continue
372  END IF
373  s = zero
374  DO 80 i = 1, n
375  IF( rwork( i ).GT.safe2 ) THEN
376  s = max( s, cabs1( work( i ) ) / rwork( i ) )
377  ELSE
378  s = max( s, ( cabs1( work( i ) )+safe1 ) /
379  \$ ( rwork( i )+safe1 ) )
380  END IF
381  80 continue
382  berr( j ) = s
383 *
384 * Test stopping criterion. Continue iterating if
385 * 1) The residual BERR(J) is larger than machine epsilon, and
386 * 2) BERR(J) decreased by at least a factor of 2 during the
387 * last iteration, and
388 * 3) At most ITMAX iterations tried.
389 *
390  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
391  \$ count.LE.itmax ) THEN
392 *
393 * Update solution and try again.
394 *
395  CALL zgbtrs( trans, n, kl, ku, 1, afb, ldafb, ipiv, work, n,
396  \$ info )
397  CALL zaxpy( n, cone, work, 1, x( 1, j ), 1 )
398  lstres = berr( j )
399  count = count + 1
400  go to 20
401  END IF
402 *
403 * Bound error from formula
404 *
405 * norm(X - XTRUE) / norm(X) .le. FERR =
406 * norm( abs(inv(op(A)))*
407 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
408 *
409 * where
410 * norm(Z) is the magnitude of the largest component of Z
411 * inv(op(A)) is the inverse of op(A)
412 * abs(Z) is the componentwise absolute value of the matrix or
413 * vector Z
414 * NZ is the maximum number of nonzeros in any row of A, plus 1
415 * EPS is machine epsilon
416 *
417 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
418 * is incremented by SAFE1 if the i-th component of
419 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
420 *
421 * Use ZLACN2 to estimate the infinity-norm of the matrix
422 * inv(op(A)) * diag(W),
423 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
424 *
425  DO 90 i = 1, n
426  IF( rwork( i ).GT.safe2 ) THEN
427  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
428  ELSE
429  rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
430  \$ safe1
431  END IF
432  90 continue
433 *
434  kase = 0
435  100 continue
436  CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
437  IF( kase.NE.0 ) THEN
438  IF( kase.EQ.1 ) THEN
439 *
440 * Multiply by diag(W)*inv(op(A)**H).
441 *
442  CALL zgbtrs( transt, n, kl, ku, 1, afb, ldafb, ipiv,
443  \$ work, n, info )
444  DO 110 i = 1, n
445  work( i ) = rwork( i )*work( i )
446  110 continue
447  ELSE
448 *
449 * Multiply by inv(op(A))*diag(W).
450 *
451  DO 120 i = 1, n
452  work( i ) = rwork( i )*work( i )
453  120 continue
454  CALL zgbtrs( transn, n, kl, ku, 1, afb, ldafb, ipiv,
455  \$ work, n, info )
456  END IF
457  go to 100
458  END IF
459 *
460 * Normalize error.
461 *
462  lstres = zero
463  DO 130 i = 1, n
464  lstres = max( lstres, cabs1( x( i, j ) ) )
465  130 continue
466  IF( lstres.NE.zero )
467  \$ ferr( j ) = ferr( j ) / lstres
468 *
469  140 continue
470 *
471  return
472 *
473 * End of ZGBRFS
474 *
475  END