LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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dsyrfs.f
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1*> \brief \b DSYRFS
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download DSYRFS + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyrfs.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyrfs.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyrfs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE DSYRFS( 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* DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31* $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> DSYRFS 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSYTRF.
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 DSYTRF.
106*> \endverbatim
107*>
108*> \param[in] B
109*> \verbatim
110*> B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
123*> On entry, the solution matrix X, as computed by DSYTRS.
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 doubleSYcomputational
187*
188* =====================================================================
189 SUBROUTINE dsyrfs( 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 DOUBLE PRECISION 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 DOUBLE PRECISION ZERO
212 parameter( zero = 0.0d+0 )
213 DOUBLE PRECISION ONE
214 parameter( one = 1.0d+0 )
215 DOUBLE PRECISION TWO
216 parameter( two = 2.0d+0 )
217 DOUBLE PRECISION THREE
218 parameter( three = 3.0d+0 )
219* ..
220* .. Local Scalars ..
221 LOGICAL UPPER
222 INTEGER COUNT, I, J, K, KASE, NZ
223 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
224* ..
225* .. Local Arrays ..
226 INTEGER ISAVE( 3 )
227* ..
228* .. External Subroutines ..
229 EXTERNAL daxpy, dcopy, dlacn2, dsymv, dsytrs, xerbla
230* ..
231* .. Intrinsic Functions ..
232 INTRINSIC abs, max
233* ..
234* .. External Functions ..
235 LOGICAL LSAME
236 DOUBLE PRECISION DLAMCH
237 EXTERNAL lsame, dlamch
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( 'DSYRFS', -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 = dlamch( 'Epsilon' )
279 safmin = dlamch( '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 dcopy( n, b( 1, j ), 1, work( n+1 ), 1 )
296 CALL dsymv( 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 dsytrs( uplo, n, 1, af, ldaf, ipiv, work( n+1 ), n,
359 $ info )
360 CALL daxpy( 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 DLACN2 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 dlacn2( 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 dsytrs( 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 dsytrs( 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 DSYRFS
437*
438 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DSYMV
Definition: dsymv.f:152
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
subroutine dsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DSYTRS
Definition: dsytrs.f:120
subroutine dsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
DSYRFS
Definition: dsyrfs.f:191