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spbrfs.f
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1 *> \brief \b SPBRFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download SPBRFS + dependencies
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11 *> [TGZ]</a>
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13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/spbrfs.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
22 * LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IWORK( * )
30 * REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
31 * $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> SPBRFS improves the computed solution to a system of linear
41 *> equations when the coefficient matrix is symmetric positive definite
42 *> and banded, and provides error bounds and backward error estimates
43 *> for the solution.
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] UPLO
50 *> \verbatim
51 *> UPLO is CHARACTER*1
52 *> = 'U': Upper triangle of A is stored;
53 *> = 'L': Lower triangle of A is stored.
54 *> \endverbatim
55 *>
56 *> \param[in] N
57 *> \verbatim
58 *> N is INTEGER
59 *> The order of the matrix A. N >= 0.
60 *> \endverbatim
61 *>
62 *> \param[in] KD
63 *> \verbatim
64 *> KD is INTEGER
65 *> The number of superdiagonals of the matrix A if UPLO = 'U',
66 *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in] NRHS
70 *> \verbatim
71 *> NRHS is INTEGER
72 *> The number of right hand sides, i.e., the number of columns
73 *> of the matrices B and X. NRHS >= 0.
74 *> \endverbatim
75 *>
76 *> \param[in] AB
77 *> \verbatim
78 *> AB is REAL array, dimension (LDAB,N)
79 *> The upper or lower triangle of the symmetric band matrix A,
80 *> stored in the first KD+1 rows of the array. The j-th column
81 *> of A is stored in the j-th column of the array AB as follows:
82 *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
83 *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
84 *> \endverbatim
85 *>
86 *> \param[in] LDAB
87 *> \verbatim
88 *> LDAB is INTEGER
89 *> The leading dimension of the array AB. LDAB >= KD+1.
90 *> \endverbatim
91 *>
92 *> \param[in] AFB
93 *> \verbatim
94 *> AFB is REAL array, dimension (LDAFB,N)
95 *> The triangular factor U or L from the Cholesky factorization
96 *> A = U**T*U or A = L*L**T of the band matrix A as computed by
97 *> SPBTRF, in the same storage format as A (see AB).
98 *> \endverbatim
99 *>
100 *> \param[in] LDAFB
101 *> \verbatim
102 *> LDAFB is INTEGER
103 *> The leading dimension of the array AFB. LDAFB >= KD+1.
104 *> \endverbatim
105 *>
106 *> \param[in] B
107 *> \verbatim
108 *> B is REAL array, dimension (LDB,NRHS)
109 *> The right hand side matrix B.
110 *> \endverbatim
111 *>
112 *> \param[in] LDB
113 *> \verbatim
114 *> LDB is INTEGER
115 *> The leading dimension of the array B. LDB >= max(1,N).
116 *> \endverbatim
117 *>
118 *> \param[in,out] X
119 *> \verbatim
120 *> X is REAL array, dimension (LDX,NRHS)
121 *> On entry, the solution matrix X, as computed by SPBTRS.
122 *> On exit, the improved solution matrix X.
123 *> \endverbatim
124 *>
125 *> \param[in] LDX
126 *> \verbatim
127 *> LDX is INTEGER
128 *> The leading dimension of the array X. LDX >= max(1,N).
129 *> \endverbatim
130 *>
131 *> \param[out] FERR
132 *> \verbatim
133 *> FERR is REAL array, dimension (NRHS)
134 *> The estimated forward error bound for each solution vector
135 *> X(j) (the j-th column of the solution matrix X).
136 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
137 *> is an estimated upper bound for the magnitude of the largest
138 *> element in (X(j) - XTRUE) divided by the magnitude of the
139 *> largest element in X(j). The estimate is as reliable as
140 *> the estimate for RCOND, and is almost always a slight
141 *> overestimate of the true error.
142 *> \endverbatim
143 *>
144 *> \param[out] BERR
145 *> \verbatim
146 *> BERR is REAL array, dimension (NRHS)
147 *> The componentwise relative backward error of each solution
148 *> vector X(j) (i.e., the smallest relative change in
149 *> any element of A or B that makes X(j) an exact solution).
150 *> \endverbatim
151 *>
152 *> \param[out] WORK
153 *> \verbatim
154 *> WORK is REAL array, dimension (3*N)
155 *> \endverbatim
156 *>
157 *> \param[out] IWORK
158 *> \verbatim
159 *> IWORK is INTEGER array, dimension (N)
160 *> \endverbatim
161 *>
162 *> \param[out] INFO
163 *> \verbatim
164 *> INFO is INTEGER
165 *> = 0: successful exit
166 *> < 0: if INFO = -i, the i-th argument had an illegal value
167 *> \endverbatim
168 *
169 *> \par Internal Parameters:
170 * =========================
171 *>
172 *> \verbatim
173 *> ITMAX is the maximum number of steps of iterative refinement.
174 *> \endverbatim
175 *
176 * Authors:
177 * ========
178 *
179 *> \author Univ. of Tennessee
180 *> \author Univ. of California Berkeley
181 *> \author Univ. of Colorado Denver
182 *> \author NAG Ltd.
183 *
184 *> \date November 2011
185 *
186 *> \ingroup realOTHERcomputational
187 *
188 * =====================================================================
189  SUBROUTINE spbrfs( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
190  $ ldb, x, ldx, ferr, berr, work, iwork, info )
191 *
192 * -- LAPACK computational routine (version 3.4.0) --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 * November 2011
196 *
197 * .. Scalar Arguments ..
198  CHARACTER uplo
199  INTEGER info, kd, ldab, ldafb, ldb, ldx, n, nrhs
200 * ..
201 * .. Array Arguments ..
202  INTEGER iwork( * )
203  REAL ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
204  $ berr( * ), ferr( * ), work( * ), x( ldx, * )
205 * ..
206 *
207 * =====================================================================
208 *
209 * .. Parameters ..
210  INTEGER itmax
211  parameter( itmax = 5 )
212  REAL zero
213  parameter( zero = 0.0e+0 )
214  REAL one
215  parameter( one = 1.0e+0 )
216  REAL two
217  parameter( two = 2.0e+0 )
218  REAL three
219  parameter( three = 3.0e+0 )
220 * ..
221 * .. Local Scalars ..
222  LOGICAL upper
223  INTEGER count, i, j, k, kase, l, nz
224  REAL eps, lstres, s, safe1, safe2, safmin, xk
225 * ..
226 * .. Local Arrays ..
227  INTEGER isave( 3 )
228 * ..
229 * .. External Subroutines ..
230  EXTERNAL saxpy, scopy, slacn2, spbtrs, ssbmv, xerbla
231 * ..
232 * .. Intrinsic Functions ..
233  INTRINSIC abs, max, min
234 * ..
235 * .. External Functions ..
236  LOGICAL lsame
237  REAL slamch
238  EXTERNAL lsame, slamch
239 * ..
240 * .. Executable Statements ..
241 *
242 * Test the input parameters.
243 *
244  info = 0
245  upper = lsame( uplo, 'U' )
246  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
247  info = -1
248  ELSE IF( n.LT.0 ) THEN
249  info = -2
250  ELSE IF( kd.LT.0 ) THEN
251  info = -3
252  ELSE IF( nrhs.LT.0 ) THEN
253  info = -4
254  ELSE IF( ldab.LT.kd+1 ) THEN
255  info = -6
256  ELSE IF( ldafb.LT.kd+1 ) THEN
257  info = -8
258  ELSE IF( ldb.LT.max( 1, n ) ) THEN
259  info = -10
260  ELSE IF( ldx.LT.max( 1, n ) ) THEN
261  info = -12
262  END IF
263  IF( info.NE.0 ) THEN
264  CALL xerbla( 'SPBRFS', -info )
265  return
266  END IF
267 *
268 * Quick return if possible
269 *
270  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
271  DO 10 j = 1, nrhs
272  ferr( j ) = zero
273  berr( j ) = zero
274  10 continue
275  return
276  END IF
277 *
278 * NZ = maximum number of nonzero elements in each row of A, plus 1
279 *
280  nz = min( n+1, 2*kd+2 )
281  eps = slamch( 'Epsilon' )
282  safmin = slamch( 'Safe minimum' )
283  safe1 = nz*safmin
284  safe2 = safe1 / eps
285 *
286 * Do for each right hand side
287 *
288  DO 140 j = 1, nrhs
289 *
290  count = 1
291  lstres = three
292  20 continue
293 *
294 * Loop until stopping criterion is satisfied.
295 *
296 * Compute residual R = B - A * X
297 *
298  CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
299  CALL ssbmv( uplo, n, kd, -one, ab, ldab, x( 1, j ), 1, one,
300  $ work( n+1 ), 1 )
301 *
302 * Compute componentwise relative backward error from formula
303 *
304 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
305 *
306 * where abs(Z) is the componentwise absolute value of the matrix
307 * or vector Z. If the i-th component of the denominator is less
308 * than SAFE2, then SAFE1 is added to the i-th components of the
309 * numerator and denominator before dividing.
310 *
311  DO 30 i = 1, n
312  work( i ) = abs( b( i, j ) )
313  30 continue
314 *
315 * Compute abs(A)*abs(X) + abs(B).
316 *
317  IF( upper ) THEN
318  DO 50 k = 1, n
319  s = zero
320  xk = abs( x( k, j ) )
321  l = kd + 1 - k
322  DO 40 i = max( 1, k-kd ), k - 1
323  work( i ) = work( i ) + abs( ab( l+i, k ) )*xk
324  s = s + abs( ab( l+i, k ) )*abs( x( i, j ) )
325  40 continue
326  work( k ) = work( k ) + abs( ab( kd+1, k ) )*xk + s
327  50 continue
328  ELSE
329  DO 70 k = 1, n
330  s = zero
331  xk = abs( x( k, j ) )
332  work( k ) = work( k ) + abs( ab( 1, k ) )*xk
333  l = 1 - k
334  DO 60 i = k + 1, min( n, k+kd )
335  work( i ) = work( i ) + abs( ab( l+i, k ) )*xk
336  s = s + abs( ab( l+i, k ) )*abs( x( i, j ) )
337  60 continue
338  work( k ) = work( k ) + s
339  70 continue
340  END IF
341  s = zero
342  DO 80 i = 1, n
343  IF( work( i ).GT.safe2 ) THEN
344  s = max( s, abs( work( n+i ) ) / work( i ) )
345  ELSE
346  s = max( s, ( abs( work( n+i ) )+safe1 ) /
347  $ ( work( i )+safe1 ) )
348  END IF
349  80 continue
350  berr( j ) = s
351 *
352 * Test stopping criterion. Continue iterating if
353 * 1) The residual BERR(J) is larger than machine epsilon, and
354 * 2) BERR(J) decreased by at least a factor of 2 during the
355 * last iteration, and
356 * 3) At most ITMAX iterations tried.
357 *
358  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
359  $ count.LE.itmax ) THEN
360 *
361 * Update solution and try again.
362 *
363  CALL spbtrs( uplo, n, kd, 1, afb, ldafb, work( n+1 ), n,
364  $ info )
365  CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
366  lstres = berr( j )
367  count = count + 1
368  go to 20
369  END IF
370 *
371 * Bound error from formula
372 *
373 * norm(X - XTRUE) / norm(X) .le. FERR =
374 * norm( abs(inv(A))*
375 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
376 *
377 * where
378 * norm(Z) is the magnitude of the largest component of Z
379 * inv(A) is the inverse of A
380 * abs(Z) is the componentwise absolute value of the matrix or
381 * vector Z
382 * NZ is the maximum number of nonzeros in any row of A, plus 1
383 * EPS is machine epsilon
384 *
385 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
386 * is incremented by SAFE1 if the i-th component of
387 * abs(A)*abs(X) + abs(B) is less than SAFE2.
388 *
389 * Use SLACN2 to estimate the infinity-norm of the matrix
390 * inv(A) * diag(W),
391 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
392 *
393  DO 90 i = 1, n
394  IF( work( i ).GT.safe2 ) THEN
395  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
396  ELSE
397  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
398  END IF
399  90 continue
400 *
401  kase = 0
402  100 continue
403  CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
404  $ kase, isave )
405  IF( kase.NE.0 ) THEN
406  IF( kase.EQ.1 ) THEN
407 *
408 * Multiply by diag(W)*inv(A**T).
409 *
410  CALL spbtrs( uplo, n, kd, 1, afb, ldafb, work( n+1 ), n,
411  $ info )
412  DO 110 i = 1, n
413  work( n+i ) = work( n+i )*work( i )
414  110 continue
415  ELSE IF( kase.EQ.2 ) THEN
416 *
417 * Multiply by inv(A)*diag(W).
418 *
419  DO 120 i = 1, n
420  work( n+i ) = work( n+i )*work( i )
421  120 continue
422  CALL spbtrs( uplo, n, kd, 1, afb, ldafb, work( n+1 ), n,
423  $ info )
424  END IF
425  go to 100
426  END IF
427 *
428 * Normalize error.
429 *
430  lstres = zero
431  DO 130 i = 1, n
432  lstres = max( lstres, abs( x( i, j ) ) )
433  130 continue
434  IF( lstres.NE.zero )
435  $ ferr( j ) = ferr( j ) / lstres
436 *
437  140 continue
438 *
439  return
440 *
441 * End of SPBRFS
442 *
443  END