LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sgerfs.f
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1 *> \brief \b SGERFS
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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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/sgerfs.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGERFS( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22 * X, LDX, FERR, BERR, WORK, IWORK, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER TRANS
26 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * INTEGER IPIV( * ), 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 *> SGERFS improves the computed solution to a system of linear
41 *> equations and provides error bounds and backward error estimates for
42 *> the solution.
43 *> \endverbatim
44 *
45 * Arguments:
46 * ==========
47 *
48 *> \param[in] TRANS
49 *> \verbatim
50 *> TRANS is CHARACTER*1
51 *> Specifies the form of the system of equations:
52 *> = 'N': A * X = B (No transpose)
53 *> = 'T': A**T * X = B (Transpose)
54 *> = 'C': A**H * X = B (Conjugate transpose = Transpose)
55 *> \endverbatim
56 *>
57 *> \param[in] N
58 *> \verbatim
59 *> N is INTEGER
60 *> The order of the matrix A. N >= 0.
61 *> \endverbatim
62 *>
63 *> \param[in] NRHS
64 *> \verbatim
65 *> NRHS is INTEGER
66 *> The number of right hand sides, i.e., the number of columns
67 *> of the matrices B and X. NRHS >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in] A
71 *> \verbatim
72 *> A is REAL array, dimension (LDA,N)
73 *> The original N-by-N matrix A.
74 *> \endverbatim
75 *>
76 *> \param[in] LDA
77 *> \verbatim
78 *> LDA is INTEGER
79 *> The leading dimension of the array A. LDA >= max(1,N).
80 *> \endverbatim
81 *>
82 *> \param[in] AF
83 *> \verbatim
84 *> AF is REAL array, dimension (LDAF,N)
85 *> The factors L and U from the factorization A = P*L*U
86 *> as computed by SGETRF.
87 *> \endverbatim
88 *>
89 *> \param[in] LDAF
90 *> \verbatim
91 *> LDAF is INTEGER
92 *> The leading dimension of the array AF. LDAF >= max(1,N).
93 *> \endverbatim
94 *>
95 *> \param[in] IPIV
96 *> \verbatim
97 *> IPIV is INTEGER array, dimension (N)
98 *> The pivot indices from SGETRF; for 1<=i<=N, row i of the
99 *> matrix was interchanged with row IPIV(i).
100 *> \endverbatim
101 *>
102 *> \param[in] B
103 *> \verbatim
104 *> B is REAL array, dimension (LDB,NRHS)
105 *> The right hand side matrix B.
106 *> \endverbatim
107 *>
108 *> \param[in] LDB
109 *> \verbatim
110 *> LDB is INTEGER
111 *> The leading dimension of the array B. LDB >= max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[in,out] X
115 *> \verbatim
116 *> X is REAL array, dimension (LDX,NRHS)
117 *> On entry, the solution matrix X, as computed by SGETRS.
118 *> On exit, the improved solution matrix X.
119 *> \endverbatim
120 *>
121 *> \param[in] LDX
122 *> \verbatim
123 *> LDX is INTEGER
124 *> The leading dimension of the array X. LDX >= max(1,N).
125 *> \endverbatim
126 *>
127 *> \param[out] FERR
128 *> \verbatim
129 *> FERR is REAL array, dimension (NRHS)
130 *> The estimated forward error bound for each solution vector
131 *> X(j) (the j-th column of the solution matrix X).
132 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
133 *> is an estimated upper bound for the magnitude of the largest
134 *> element in (X(j) - XTRUE) divided by the magnitude of the
135 *> largest element in X(j). The estimate is as reliable as
136 *> the estimate for RCOND, and is almost always a slight
137 *> overestimate of the true error.
138 *> \endverbatim
139 *>
140 *> \param[out] BERR
141 *> \verbatim
142 *> BERR is REAL array, dimension (NRHS)
143 *> The componentwise relative backward error of each solution
144 *> vector X(j) (i.e., the smallest relative change in
145 *> any element of A or B that makes X(j) an exact solution).
146 *> \endverbatim
147 *>
148 *> \param[out] WORK
149 *> \verbatim
150 *> WORK is REAL array, dimension (3*N)
151 *> \endverbatim
152 *>
153 *> \param[out] IWORK
154 *> \verbatim
155 *> IWORK is INTEGER array, dimension (N)
156 *> \endverbatim
157 *>
158 *> \param[out] INFO
159 *> \verbatim
160 *> INFO is INTEGER
161 *> = 0: successful exit
162 *> < 0: if INFO = -i, the i-th argument had an illegal value
163 *> \endverbatim
164 *
165 *> \par Internal Parameters:
166 * =========================
167 *>
168 *> \verbatim
169 *> ITMAX is the maximum number of steps of iterative refinement.
170 *> \endverbatim
171 *
172 * Authors:
173 * ========
174 *
175 *> \author Univ. of Tennessee
176 *> \author Univ. of California Berkeley
177 *> \author Univ. of Colorado Denver
178 *> \author NAG Ltd.
179 *
180 *> \ingroup realGEcomputational
181 *
182 * =====================================================================
183  SUBROUTINE sgerfs( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
184  $ X, LDX, FERR, BERR, WORK, IWORK, INFO )
185 *
186 * -- LAPACK computational routine --
187 * -- LAPACK is a software package provided by Univ. of Tennessee, --
188 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
189 *
190 * .. Scalar Arguments ..
191  CHARACTER TRANS
192  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
193 * ..
194 * .. Array Arguments ..
195  INTEGER IPIV( * ), IWORK( * )
196  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
197  $ berr( * ), ferr( * ), work( * ), x( ldx, * )
198 * ..
199 *
200 * =====================================================================
201 *
202 * .. Parameters ..
203  INTEGER ITMAX
204  parameter( itmax = 5 )
205  REAL ZERO
206  parameter( zero = 0.0e+0 )
207  REAL ONE
208  parameter( one = 1.0e+0 )
209  REAL TWO
210  parameter( two = 2.0e+0 )
211  REAL THREE
212  parameter( three = 3.0e+0 )
213 * ..
214 * .. Local Scalars ..
215  LOGICAL NOTRAN
216  CHARACTER TRANST
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, sgemv, sgetrs, slacn2, 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  notran = lsame( trans, 'N' )
240  IF( .NOT.notran .AND. .NOT.lsame( trans, 'T' ) .AND. .NOT.
241  $ lsame( trans, 'C' ) ) THEN
242  info = -1
243  ELSE IF( n.LT.0 ) THEN
244  info = -2
245  ELSE IF( nrhs.LT.0 ) THEN
246  info = -3
247  ELSE IF( lda.LT.max( 1, n ) ) THEN
248  info = -5
249  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
250  info = -7
251  ELSE IF( ldb.LT.max( 1, n ) ) THEN
252  info = -10
253  ELSE IF( ldx.LT.max( 1, n ) ) THEN
254  info = -12
255  END IF
256  IF( info.NE.0 ) THEN
257  CALL xerbla( 'SGERFS', -info )
258  RETURN
259  END IF
260 *
261 * Quick return if possible
262 *
263  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
264  DO 10 j = 1, nrhs
265  ferr( j ) = zero
266  berr( j ) = zero
267  10 CONTINUE
268  RETURN
269  END IF
270 *
271  IF( notran ) THEN
272  transt = 'T'
273  ELSE
274  transt = 'N'
275  END IF
276 *
277 * NZ = maximum number of nonzero elements in each row of A, plus 1
278 *
279  nz = n + 1
280  eps = slamch( 'Epsilon' )
281  safmin = slamch( 'Safe minimum' )
282  safe1 = nz*safmin
283  safe2 = safe1 / eps
284 *
285 * Do for each right hand side
286 *
287  DO 140 j = 1, nrhs
288 *
289  count = 1
290  lstres = three
291  20 CONTINUE
292 *
293 * Loop until stopping criterion is satisfied.
294 *
295 * Compute residual R = B - op(A) * X,
296 * where op(A) = A, A**T, or A**H, depending on TRANS.
297 *
298  CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
299  CALL sgemv( trans, n, n, -one, a, lda, 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(op(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(op(A))*abs(X) + abs(B).
316 *
317  IF( notran ) THEN
318  DO 50 k = 1, n
319  xk = abs( x( k, j ) )
320  DO 40 i = 1, n
321  work( i ) = work( i ) + abs( a( i, k ) )*xk
322  40 CONTINUE
323  50 CONTINUE
324  ELSE
325  DO 70 k = 1, n
326  s = zero
327  DO 60 i = 1, n
328  s = s + abs( a( i, k ) )*abs( x( i, j ) )
329  60 CONTINUE
330  work( k ) = work( k ) + s
331  70 CONTINUE
332  END IF
333  s = zero
334  DO 80 i = 1, n
335  IF( work( i ).GT.safe2 ) THEN
336  s = max( s, abs( work( n+i ) ) / work( i ) )
337  ELSE
338  s = max( s, ( abs( work( n+i ) )+safe1 ) /
339  $ ( work( i )+safe1 ) )
340  END IF
341  80 CONTINUE
342  berr( j ) = s
343 *
344 * Test stopping criterion. Continue iterating if
345 * 1) The residual BERR(J) is larger than machine epsilon, and
346 * 2) BERR(J) decreased by at least a factor of 2 during the
347 * last iteration, and
348 * 3) At most ITMAX iterations tried.
349 *
350  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
351  $ count.LE.itmax ) THEN
352 *
353 * Update solution and try again.
354 *
355  CALL sgetrs( trans, n, 1, af, ldaf, ipiv, work( n+1 ), n,
356  $ info )
357  CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
358  lstres = berr( j )
359  count = count + 1
360  GO TO 20
361  END IF
362 *
363 * Bound error from formula
364 *
365 * norm(X - XTRUE) / norm(X) .le. FERR =
366 * norm( abs(inv(op(A)))*
367 * ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X)
368 *
369 * where
370 * norm(Z) is the magnitude of the largest component of Z
371 * inv(op(A)) is the inverse of op(A)
372 * abs(Z) is the componentwise absolute value of the matrix or
373 * vector Z
374 * NZ is the maximum number of nonzeros in any row of A, plus 1
375 * EPS is machine epsilon
376 *
377 * The i-th component of abs(R)+NZ*EPS*(abs(op(A))*abs(X)+abs(B))
378 * is incremented by SAFE1 if the i-th component of
379 * abs(op(A))*abs(X) + abs(B) is less than SAFE2.
380 *
381 * Use SLACN2 to estimate the infinity-norm of the matrix
382 * inv(op(A)) * diag(W),
383 * where W = abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) )))
384 *
385  DO 90 i = 1, n
386  IF( work( i ).GT.safe2 ) THEN
387  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
388  ELSE
389  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
390  END IF
391  90 CONTINUE
392 *
393  kase = 0
394  100 CONTINUE
395  CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
396  $ kase, isave )
397  IF( kase.NE.0 ) THEN
398  IF( kase.EQ.1 ) THEN
399 *
400 * Multiply by diag(W)*inv(op(A)**T).
401 *
402  CALL sgetrs( transt, n, 1, af, ldaf, ipiv, work( n+1 ),
403  $ n, info )
404  DO 110 i = 1, n
405  work( n+i ) = work( i )*work( n+i )
406  110 CONTINUE
407  ELSE
408 *
409 * Multiply by inv(op(A))*diag(W).
410 *
411  DO 120 i = 1, n
412  work( n+i ) = work( i )*work( n+i )
413  120 CONTINUE
414  CALL sgetrs( trans, n, 1, af, ldaf, ipiv, work( n+1 ), n,
415  $ info )
416  END IF
417  GO TO 100
418  END IF
419 *
420 * Normalize error.
421 *
422  lstres = zero
423  DO 130 i = 1, n
424  lstres = max( lstres, abs( x( i, j ) ) )
425  130 CONTINUE
426  IF( lstres.NE.zero )
427  $ ferr( j ) = ferr( j ) / lstres
428 *
429  140 CONTINUE
430 *
431  RETURN
432 *
433 * End of SGERFS
434 *
435  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgerfs(TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SGERFS
Definition: sgerfs.f:185
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156