LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
sporfs.f
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1 *> \brief \b SPORFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
22 * LDX, FERR, BERR, WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IWORK( * )
30 * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31 * \$ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
32 * ..
33 *
34 *
35 *> \par Purpose:
36 * =============
37 *>
38 *> \verbatim
39 *>
40 *> SPORFS improves the computed solution to a system of linear
41 *> equations when the coefficient matrix is symmetric positive definite,
42 *> and provides error bounds and backward error estimates for the
43 *> 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] NRHS
63 *> \verbatim
64 *> NRHS is INTEGER
65 *> The number of right hand sides, i.e., the number of columns
66 *> of the matrices B and X. NRHS >= 0.
67 *> \endverbatim
68 *>
69 *> \param[in] A
70 *> \verbatim
71 *> A is REAL array, dimension (LDA,N)
72 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
73 *> upper triangular part of A contains the upper triangular part
74 *> of the matrix A, and the strictly lower triangular part of A
75 *> is not referenced. If UPLO = 'L', the leading N-by-N lower
76 *> triangular part of A contains the lower triangular part of
77 *> the matrix A, and the strictly upper triangular part of A is
78 *> not referenced.
79 *> \endverbatim
80 *>
81 *> \param[in] LDA
82 *> \verbatim
83 *> LDA is INTEGER
84 *> The leading dimension of the array A. LDA >= max(1,N).
85 *> \endverbatim
86 *>
87 *> \param[in] AF
88 *> \verbatim
89 *> AF is REAL array, dimension (LDAF,N)
90 *> The triangular factor U or L from the Cholesky factorization
91 *> A = U**T*U or A = L*L**T, as computed by SPOTRF.
92 *> \endverbatim
93 *>
94 *> \param[in] LDAF
95 *> \verbatim
96 *> LDAF is INTEGER
97 *> The leading dimension of the array AF. LDAF >= max(1,N).
98 *> \endverbatim
99 *>
100 *> \param[in] B
101 *> \verbatim
102 *> B is REAL array, dimension (LDB,NRHS)
103 *> The right hand side matrix B.
104 *> \endverbatim
105 *>
106 *> \param[in] LDB
107 *> \verbatim
108 *> LDB is INTEGER
109 *> The leading dimension of the array B. LDB >= max(1,N).
110 *> \endverbatim
111 *>
112 *> \param[in,out] X
113 *> \verbatim
114 *> X is REAL array, dimension (LDX,NRHS)
115 *> On entry, the solution matrix X, as computed by SPOTRS.
116 *> On exit, the improved solution matrix X.
117 *> \endverbatim
118 *>
119 *> \param[in] LDX
120 *> \verbatim
121 *> LDX is INTEGER
122 *> The leading dimension of the array X. LDX >= max(1,N).
123 *> \endverbatim
124 *>
125 *> \param[out] FERR
126 *> \verbatim
127 *> FERR is REAL array, dimension (NRHS)
128 *> The estimated forward error bound for each solution vector
129 *> X(j) (the j-th column of the solution matrix X).
130 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
131 *> is an estimated upper bound for the magnitude of the largest
132 *> element in (X(j) - XTRUE) divided by the magnitude of the
133 *> largest element in X(j). The estimate is as reliable as
134 *> the estimate for RCOND, and is almost always a slight
135 *> overestimate of the true error.
136 *> \endverbatim
137 *>
138 *> \param[out] BERR
139 *> \verbatim
140 *> BERR is REAL array, dimension (NRHS)
141 *> The componentwise relative backward error of each solution
142 *> vector X(j) (i.e., the smallest relative change in
143 *> any element of A or B that makes X(j) an exact solution).
144 *> \endverbatim
145 *>
146 *> \param[out] WORK
147 *> \verbatim
148 *> WORK is REAL array, dimension (3*N)
149 *> \endverbatim
150 *>
151 *> \param[out] IWORK
152 *> \verbatim
153 *> IWORK is INTEGER array, dimension (N)
154 *> \endverbatim
155 *>
156 *> \param[out] INFO
157 *> \verbatim
158 *> INFO is INTEGER
159 *> = 0: successful exit
160 *> < 0: if INFO = -i, the i-th argument had an illegal value
161 *> \endverbatim
162 *
163 *> \par Internal Parameters:
164 * =========================
165 *>
166 *> \verbatim
167 *> ITMAX is the maximum number of steps of iterative refinement.
168 *> \endverbatim
169 *
170 * Authors:
171 * ========
172 *
173 *> \author Univ. of Tennessee
174 *> \author Univ. of California Berkeley
175 *> \author Univ. of Colorado Denver
176 *> \author NAG Ltd.
177 *
178 *> \date November 2011
179 *
180 *> \ingroup realPOcomputational
181 *
182 * =====================================================================
183  SUBROUTINE sporfs( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
184  \$ ldx, ferr, berr, work, iwork, info )
185 *
186 * -- LAPACK computational routine (version 3.4.0) --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 * November 2011
190 *
191 * .. Scalar Arguments ..
192  CHARACTER uplo
193  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs
194 * ..
195 * .. Array Arguments ..
196  INTEGER iwork( * )
197  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
198  \$ berr( * ), ferr( * ), work( * ), x( ldx, * )
199 * ..
200 *
201 * =====================================================================
202 *
203 * .. Parameters ..
204  INTEGER itmax
205  parameter( itmax = 5 )
206  REAL zero
207  parameter( zero = 0.0e+0 )
208  REAL one
209  parameter( one = 1.0e+0 )
210  REAL two
211  parameter( two = 2.0e+0 )
212  REAL three
213  parameter( three = 3.0e+0 )
214 * ..
215 * .. Local Scalars ..
216  LOGICAL upper
217  INTEGER count, i, j, k, kase, nz
218  REAL eps, lstres, s, safe1, safe2, safmin, xk
219 * ..
220 * .. Local Arrays ..
221  INTEGER isave( 3 )
222 * ..
223 * .. External Subroutines ..
224  EXTERNAL saxpy, scopy, slacn2, spotrs, ssymv, xerbla
225 * ..
226 * .. Intrinsic Functions ..
227  INTRINSIC abs, max
228 * ..
229 * .. External Functions ..
230  LOGICAL lsame
231  REAL slamch
232  EXTERNAL lsame, slamch
233 * ..
234 * .. Executable Statements ..
235 *
236 * Test the input parameters.
237 *
238  info = 0
239  upper = lsame( uplo, 'U' )
240  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
241  info = -1
242  ELSE IF( n.LT.0 ) THEN
243  info = -2
244  ELSE IF( nrhs.LT.0 ) THEN
245  info = -3
246  ELSE IF( lda.LT.max( 1, n ) ) THEN
247  info = -5
248  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
249  info = -7
250  ELSE IF( ldb.LT.max( 1, n ) ) THEN
251  info = -9
252  ELSE IF( ldx.LT.max( 1, n ) ) THEN
253  info = -11
254  END IF
255  IF( info.NE.0 ) THEN
256  CALL xerbla( 'SPORFS', -info )
257  return
258  END IF
259 *
260 * Quick return if possible
261 *
262  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
263  DO 10 j = 1, nrhs
264  ferr( j ) = zero
265  berr( j ) = zero
266  10 continue
267  return
268  END IF
269 *
270 * NZ = maximum number of nonzero elements in each row of A, plus 1
271 *
272  nz = n + 1
273  eps = slamch( 'Epsilon' )
274  safmin = slamch( 'Safe minimum' )
275  safe1 = nz*safmin
276  safe2 = safe1 / eps
277 *
278 * Do for each right hand side
279 *
280  DO 140 j = 1, nrhs
281 *
282  count = 1
283  lstres = three
284  20 continue
285 *
286 * Loop until stopping criterion is satisfied.
287 *
288 * Compute residual R = B - A * X
289 *
290  CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
291  CALL ssymv( uplo, n, -one, a, lda, x( 1, j ), 1, one,
292  \$ work( n+1 ), 1 )
293 *
294 * Compute componentwise relative backward error from formula
295 *
296 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
297 *
298 * where abs(Z) is the componentwise absolute value of the matrix
299 * or vector Z. If the i-th component of the denominator is less
300 * than SAFE2, then SAFE1 is added to the i-th components of the
301 * numerator and denominator before dividing.
302 *
303  DO 30 i = 1, n
304  work( i ) = abs( b( i, j ) )
305  30 continue
306 *
307 * Compute abs(A)*abs(X) + abs(B).
308 *
309  IF( upper ) THEN
310  DO 50 k = 1, n
311  s = zero
312  xk = abs( x( k, j ) )
313  DO 40 i = 1, k - 1
314  work( i ) = work( i ) + abs( a( i, k ) )*xk
315  s = s + abs( a( i, k ) )*abs( x( i, j ) )
316  40 continue
317  work( k ) = work( k ) + abs( a( k, k ) )*xk + s
318  50 continue
319  ELSE
320  DO 70 k = 1, n
321  s = zero
322  xk = abs( x( k, j ) )
323  work( k ) = work( k ) + abs( a( k, k ) )*xk
324  DO 60 i = k + 1, n
325  work( i ) = work( i ) + abs( a( i, k ) )*xk
326  s = s + abs( a( i, k ) )*abs( x( i, j ) )
327  60 continue
328  work( k ) = work( k ) + s
329  70 continue
330  END IF
331  s = zero
332  DO 80 i = 1, n
333  IF( work( i ).GT.safe2 ) THEN
334  s = max( s, abs( work( n+i ) ) / work( i ) )
335  ELSE
336  s = max( s, ( abs( work( n+i ) )+safe1 ) /
337  \$ ( work( i )+safe1 ) )
338  END IF
339  80 continue
340  berr( j ) = s
341 *
342 * Test stopping criterion. Continue iterating if
343 * 1) The residual BERR(J) is larger than machine epsilon, and
344 * 2) BERR(J) decreased by at least a factor of 2 during the
345 * last iteration, and
346 * 3) At most ITMAX iterations tried.
347 *
348  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
349  \$ count.LE.itmax ) THEN
350 *
351 * Update solution and try again.
352 *
353  CALL spotrs( uplo, n, 1, af, ldaf, work( n+1 ), n, info )
354  CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
355  lstres = berr( j )
356  count = count + 1
357  go to 20
358  END IF
359 *
360 * Bound error from formula
361 *
362 * norm(X - XTRUE) / norm(X) .le. FERR =
363 * norm( abs(inv(A))*
364 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
365 *
366 * where
367 * norm(Z) is the magnitude of the largest component of Z
368 * inv(A) is the inverse of A
369 * abs(Z) is the componentwise absolute value of the matrix or
370 * vector Z
371 * NZ is the maximum number of nonzeros in any row of A, plus 1
372 * EPS is machine epsilon
373 *
374 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
375 * is incremented by SAFE1 if the i-th component of
376 * abs(A)*abs(X) + abs(B) is less than SAFE2.
377 *
378 * Use SLACN2 to estimate the infinity-norm of the matrix
379 * inv(A) * diag(W),
380 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
381 *
382  DO 90 i = 1, n
383  IF( work( i ).GT.safe2 ) THEN
384  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
385  ELSE
386  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
387  END IF
388  90 continue
389 *
390  kase = 0
391  100 continue
392  CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
393  \$ kase, isave )
394  IF( kase.NE.0 ) THEN
395  IF( kase.EQ.1 ) THEN
396 *
397 * Multiply by diag(W)*inv(A**T).
398 *
399  CALL spotrs( uplo, n, 1, af, ldaf, work( n+1 ), n, info )
400  DO 110 i = 1, n
401  work( n+i ) = work( i )*work( n+i )
402  110 continue
403  ELSE IF( kase.EQ.2 ) THEN
404 *
405 * Multiply by inv(A)*diag(W).
406 *
407  DO 120 i = 1, n
408  work( n+i ) = work( i )*work( n+i )
409  120 continue
410  CALL spotrs( uplo, n, 1, af, ldaf, work( n+1 ), n, info )
411  END IF
412  go to 100
413  END IF
414 *
415 * Normalize error.
416 *
417  lstres = zero
418  DO 130 i = 1, n
419  lstres = max( lstres, abs( x( i, j ) ) )
420  130 continue
421  IF( lstres.NE.zero )
422  \$ ferr( j ) = ferr( j ) / lstres
423 *
424  140 continue
425 *
426  return
427 *
428 * End of SPORFS
429 *
430  END