LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
ssyrfs.f
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1 *> \brief \b SSYRFS
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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13 *> [ZIP]</a>
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22 * X, 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 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 *> SSYRFS improves the computed solution to a system of linear
41 *> equations when the coefficient matrix is symmetric indefinite, and
42 *> provides error bounds and backward error estimates for the solution.
43 *> \endverbatim
44 *
45 * Arguments:
46 * ==========
47 *
48 *> \param[in] UPLO
49 *> \verbatim
50 *> UPLO is CHARACTER*1
51 *> = 'U': Upper triangle of A is stored;
52 *> = 'L': Lower triangle of A is stored.
53 *> \endverbatim
54 *>
55 *> \param[in] N
56 *> \verbatim
57 *> N is INTEGER
58 *> The order of the matrix A. N >= 0.
59 *> \endverbatim
60 *>
61 *> \param[in] NRHS
62 *> \verbatim
63 *> NRHS is INTEGER
64 *> The number of right hand sides, i.e., the number of columns
65 *> of the matrices B and X. NRHS >= 0.
66 *> \endverbatim
67 *>
68 *> \param[in] A
69 *> \verbatim
70 *> A is REAL array, dimension (LDA,N)
71 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
72 *> upper triangular part of A contains the upper triangular part
73 *> of the matrix A, and the strictly lower triangular part of A
74 *> is not referenced. If UPLO = 'L', the leading N-by-N lower
75 *> triangular part of A contains the lower triangular part of
76 *> the matrix A, and the strictly upper triangular part of A is
77 *> not referenced.
78 *> \endverbatim
79 *>
80 *> \param[in] LDA
81 *> \verbatim
82 *> LDA is INTEGER
83 *> The leading dimension of the array A. LDA >= max(1,N).
84 *> \endverbatim
85 *>
86 *> \param[in] AF
87 *> \verbatim
88 *> AF is REAL array, dimension (LDAF,N)
89 *> The factored form of the matrix A. AF contains the block
90 *> diagonal matrix D and the multipliers used to obtain the
91 *> factor U or L from the factorization A = U*D*U**T or
92 *> A = L*D*L**T as computed by SSYTRF.
93 *> \endverbatim
94 *>
95 *> \param[in] LDAF
96 *> \verbatim
97 *> LDAF is INTEGER
98 *> The leading dimension of the array AF. LDAF >= max(1,N).
99 *> \endverbatim
100 *>
101 *> \param[in] IPIV
102 *> \verbatim
103 *> IPIV is INTEGER array, dimension (N)
104 *> Details of the interchanges and the block structure of D
105 *> as determined by SSYTRF.
106 *> \endverbatim
107 *>
108 *> \param[in] B
109 *> \verbatim
110 *> B is REAL array, dimension (LDB,NRHS)
111 *> The right hand side matrix B.
112 *> \endverbatim
113 *>
114 *> \param[in] LDB
115 *> \verbatim
116 *> LDB is INTEGER
117 *> The leading dimension of the array B. LDB >= max(1,N).
118 *> \endverbatim
119 *>
120 *> \param[in,out] X
121 *> \verbatim
122 *> X is REAL array, dimension (LDX,NRHS)
123 *> On entry, the solution matrix X, as computed by SSYTRS.
124 *> On exit, the improved solution matrix X.
125 *> \endverbatim
126 *>
127 *> \param[in] LDX
128 *> \verbatim
129 *> LDX is INTEGER
130 *> The leading dimension of the array X. LDX >= max(1,N).
131 *> \endverbatim
132 *>
133 *> \param[out] FERR
134 *> \verbatim
135 *> FERR is REAL array, dimension (NRHS)
136 *> The estimated forward error bound for each solution vector
137 *> X(j) (the j-th column of the solution matrix X).
138 *> If XTRUE is the true solution corresponding to X(j), FERR(j)
139 *> is an estimated upper bound for the magnitude of the largest
140 *> element in (X(j) - XTRUE) divided by the magnitude of the
141 *> largest element in X(j). The estimate is as reliable as
142 *> the estimate for RCOND, and is almost always a slight
143 *> overestimate of the true error.
144 *> \endverbatim
145 *>
146 *> \param[out] BERR
147 *> \verbatim
148 *> BERR is REAL array, dimension (NRHS)
149 *> The componentwise relative backward error of each solution
150 *> vector X(j) (i.e., the smallest relative change in
151 *> any element of A or B that makes X(j) an exact solution).
152 *> \endverbatim
153 *>
154 *> \param[out] WORK
155 *> \verbatim
156 *> WORK is REAL array, dimension (3*N)
157 *> \endverbatim
158 *>
159 *> \param[out] IWORK
160 *> \verbatim
161 *> IWORK is INTEGER array, dimension (N)
162 *> \endverbatim
163 *>
164 *> \param[out] INFO
165 *> \verbatim
166 *> INFO is INTEGER
167 *> = 0: successful exit
168 *> < 0: if INFO = -i, the i-th argument had an illegal value
169 *> \endverbatim
170 *
171 *> \par Internal Parameters:
172 * =========================
173 *>
174 *> \verbatim
175 *> ITMAX is the maximum number of steps of iterative refinement.
176 *> \endverbatim
177 *
178 * Authors:
179 * ========
180 *
181 *> \author Univ. of Tennessee
182 *> \author Univ. of California Berkeley
183 *> \author Univ. of Colorado Denver
184 *> \author NAG Ltd.
185 *
186 *> \ingroup realSYcomputational
187 *
188 * =====================================================================
189  SUBROUTINE ssyrfs( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
190  $ X, LDX, FERR, BERR, WORK, IWORK, INFO )
191 *
192 * -- LAPACK computational routine --
193 * -- LAPACK is a software package provided by Univ. of Tennessee, --
194 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
195 *
196 * .. Scalar Arguments ..
197  CHARACTER UPLO
198  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
199 * ..
200 * .. Array Arguments ..
201  INTEGER IPIV( * ), IWORK( * )
202  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
203  $ berr( * ), ferr( * ), work( * ), x( ldx, * )
204 * ..
205 *
206 * =====================================================================
207 *
208 * .. Parameters ..
209  INTEGER ITMAX
210  parameter( itmax = 5 )
211  REAL ZERO
212  parameter( zero = 0.0e+0 )
213  REAL ONE
214  parameter( one = 1.0e+0 )
215  REAL TWO
216  parameter( two = 2.0e+0 )
217  REAL THREE
218  parameter( three = 3.0e+0 )
219 * ..
220 * .. Local Scalars ..
221  LOGICAL UPPER
222  INTEGER COUNT, I, J, K, KASE, NZ
223  REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
224 * ..
225 * .. Local Arrays ..
226  INTEGER ISAVE( 3 )
227 * ..
228 * .. External Subroutines ..
229  EXTERNAL saxpy, scopy, slacn2, ssymv, ssytrs, xerbla
230 * ..
231 * .. Intrinsic Functions ..
232  INTRINSIC abs, max
233 * ..
234 * .. External Functions ..
235  LOGICAL LSAME
236  REAL SLAMCH
237  EXTERNAL lsame, slamch
238 * ..
239 * .. Executable Statements ..
240 *
241 * Test the input parameters.
242 *
243  info = 0
244  upper = lsame( uplo, 'U' )
245  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
246  info = -1
247  ELSE IF( n.LT.0 ) THEN
248  info = -2
249  ELSE IF( nrhs.LT.0 ) THEN
250  info = -3
251  ELSE IF( lda.LT.max( 1, n ) ) THEN
252  info = -5
253  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
254  info = -7
255  ELSE IF( ldb.LT.max( 1, n ) ) THEN
256  info = -10
257  ELSE IF( ldx.LT.max( 1, n ) ) THEN
258  info = -12
259  END IF
260  IF( info.NE.0 ) THEN
261  CALL xerbla( 'SSYRFS', -info )
262  RETURN
263  END IF
264 *
265 * Quick return if possible
266 *
267  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
268  DO 10 j = 1, nrhs
269  ferr( j ) = zero
270  berr( j ) = zero
271  10 CONTINUE
272  RETURN
273  END IF
274 *
275 * NZ = maximum number of nonzero elements in each row of A, plus 1
276 *
277  nz = n + 1
278  eps = slamch( 'Epsilon' )
279  safmin = slamch( 'Safe minimum' )
280  safe1 = nz*safmin
281  safe2 = safe1 / eps
282 *
283 * Do for each right hand side
284 *
285  DO 140 j = 1, nrhs
286 *
287  count = 1
288  lstres = three
289  20 CONTINUE
290 *
291 * Loop until stopping criterion is satisfied.
292 *
293 * Compute residual R = B - A * X
294 *
295  CALL scopy( n, b( 1, j ), 1, work( n+1 ), 1 )
296  CALL ssymv( uplo, n, -one, a, lda, x( 1, j ), 1, one,
297  $ work( n+1 ), 1 )
298 *
299 * Compute componentwise relative backward error from formula
300 *
301 * max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
302 *
303 * where abs(Z) is the componentwise absolute value of the matrix
304 * or vector Z. If the i-th component of the denominator is less
305 * than SAFE2, then SAFE1 is added to the i-th components of the
306 * numerator and denominator before dividing.
307 *
308  DO 30 i = 1, n
309  work( i ) = abs( b( i, j ) )
310  30 CONTINUE
311 *
312 * Compute abs(A)*abs(X) + abs(B).
313 *
314  IF( upper ) THEN
315  DO 50 k = 1, n
316  s = zero
317  xk = abs( x( k, j ) )
318  DO 40 i = 1, k - 1
319  work( i ) = work( i ) + abs( a( i, k ) )*xk
320  s = s + abs( a( i, k ) )*abs( x( i, j ) )
321  40 CONTINUE
322  work( k ) = work( k ) + abs( a( k, k ) )*xk + s
323  50 CONTINUE
324  ELSE
325  DO 70 k = 1, n
326  s = zero
327  xk = abs( x( k, j ) )
328  work( k ) = work( k ) + abs( a( k, k ) )*xk
329  DO 60 i = k + 1, n
330  work( i ) = work( i ) + abs( a( i, k ) )*xk
331  s = s + abs( a( i, k ) )*abs( x( i, j ) )
332  60 CONTINUE
333  work( k ) = work( k ) + s
334  70 CONTINUE
335  END IF
336  s = zero
337  DO 80 i = 1, n
338  IF( work( i ).GT.safe2 ) THEN
339  s = max( s, abs( work( n+i ) ) / work( i ) )
340  ELSE
341  s = max( s, ( abs( work( n+i ) )+safe1 ) /
342  $ ( work( i )+safe1 ) )
343  END IF
344  80 CONTINUE
345  berr( j ) = s
346 *
347 * Test stopping criterion. Continue iterating if
348 * 1) The residual BERR(J) is larger than machine epsilon, and
349 * 2) BERR(J) decreased by at least a factor of 2 during the
350 * last iteration, and
351 * 3) At most ITMAX iterations tried.
352 *
353  IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
354  $ count.LE.itmax ) THEN
355 *
356 * Update solution and try again.
357 *
358  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
359  $ info )
360  CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
361  lstres = berr( j )
362  count = count + 1
363  GO TO 20
364  END IF
365 *
366 * Bound error from formula
367 *
368 * norm(X - XTRUE) / norm(X) .le. FERR =
369 * norm( abs(inv(A))*
370 * ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
371 *
372 * where
373 * norm(Z) is the magnitude of the largest component of Z
374 * inv(A) is the inverse of A
375 * abs(Z) is the componentwise absolute value of the matrix or
376 * vector Z
377 * NZ is the maximum number of nonzeros in any row of A, plus 1
378 * EPS is machine epsilon
379 *
380 * The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
381 * is incremented by SAFE1 if the i-th component of
382 * abs(A)*abs(X) + abs(B) is less than SAFE2.
383 *
384 * Use SLACN2 to estimate the infinity-norm of the matrix
385 * inv(A) * diag(W),
386 * where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
387 *
388  DO 90 i = 1, n
389  IF( work( i ).GT.safe2 ) THEN
390  work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
391  ELSE
392  work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
393  END IF
394  90 CONTINUE
395 *
396  kase = 0
397  100 CONTINUE
398  CALL slacn2( n, work( 2*n+1 ), work( n+1 ), iwork, ferr( j ),
399  $ kase, isave )
400  IF( kase.NE.0 ) THEN
401  IF( kase.EQ.1 ) THEN
402 *
403 * Multiply by diag(W)*inv(A**T).
404 *
405  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
406  $ info )
407  DO 110 i = 1, n
408  work( n+i ) = work( i )*work( n+i )
409  110 CONTINUE
410  ELSE IF( kase.EQ.2 ) THEN
411 *
412 * Multiply by inv(A)*diag(W).
413 *
414  DO 120 i = 1, n
415  work( n+i ) = work( i )*work( n+i )
416  120 CONTINUE
417  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
418  $ info )
419  END IF
420  GO TO 100
421  END IF
422 *
423 * Normalize error.
424 *
425  lstres = zero
426  DO 130 i = 1, n
427  lstres = max( lstres, abs( x( i, j ) ) )
428  130 CONTINUE
429  IF( lstres.NE.zero )
430  $ ferr( j ) = ferr( j ) / lstres
431 *
432  140 CONTINUE
433 *
434  RETURN
435 *
436 * End of SSYRFS
437 *
438  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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 ssyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SSYRFS
Definition: ssyrfs.f:191
subroutine ssytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SSYTRS
Definition: ssytrs.f:120
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 ssymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SSYMV
Definition: ssymv.f:152