LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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zsyrfs.f
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1*> \brief \b ZSYRFS
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download ZSYRFS + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zsyrfs.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zsyrfs.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zsyrfs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
22* X, LDX, FERR, BERR, WORK, RWORK, INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER UPLO
26* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
27* ..
28* .. Array Arguments ..
29* INTEGER IPIV( * )
30* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
31* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
32* $ WORK( * ), X( LDX, * )
33* ..
34*
35*
36*> \par Purpose:
37* =============
38*>
39*> \verbatim
40*>
41*> ZSYRFS improves the computed solution to a system of linear
42*> equations when the coefficient matrix is symmetric indefinite, and
43*> provides error bounds and backward error estimates 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] 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDAF,N)
90*> The factored form of the matrix A. AF contains the block
91*> diagonal matrix D and the multipliers used to obtain the
92*> factor U or L from the factorization A = U*D*U**T or
93*> A = L*D*L**T as computed by ZSYTRF.
94*> \endverbatim
95*>
96*> \param[in] LDAF
97*> \verbatim
98*> LDAF is INTEGER
99*> The leading dimension of the array AF. LDAF >= max(1,N).
100*> \endverbatim
101*>
102*> \param[in] IPIV
103*> \verbatim
104*> IPIV is INTEGER array, dimension (N)
105*> Details of the interchanges and the block structure of D
106*> as determined by ZSYTRF.
107*> \endverbatim
108*>
109*> \param[in] B
110*> \verbatim
111*> B is COMPLEX*16 array, dimension (LDB,NRHS)
112*> The right hand side matrix B.
113*> \endverbatim
114*>
115*> \param[in] LDB
116*> \verbatim
117*> LDB is INTEGER
118*> The leading dimension of the array B. LDB >= max(1,N).
119*> \endverbatim
120*>
121*> \param[in,out] X
122*> \verbatim
123*> X is COMPLEX*16 array, dimension (LDX,NRHS)
124*> On entry, the solution matrix X, as computed by ZSYTRS.
125*> On exit, the improved solution matrix X.
126*> \endverbatim
127*>
128*> \param[in] LDX
129*> \verbatim
130*> LDX is INTEGER
131*> The leading dimension of the array X. LDX >= max(1,N).
132*> \endverbatim
133*>
134*> \param[out] FERR
135*> \verbatim
136*> FERR is DOUBLE PRECISION array, dimension (NRHS)
137*> The estimated forward error bound for each solution vector
138*> X(j) (the j-th column of the solution matrix X).
139*> If XTRUE is the true solution corresponding to X(j), FERR(j)
140*> is an estimated upper bound for the magnitude of the largest
141*> element in (X(j) - XTRUE) divided by the magnitude of the
142*> largest element in X(j). The estimate is as reliable as
143*> the estimate for RCOND, and is almost always a slight
144*> overestimate of the true error.
145*> \endverbatim
146*>
147*> \param[out] BERR
148*> \verbatim
149*> BERR is DOUBLE PRECISION array, dimension (NRHS)
150*> The componentwise relative backward error of each solution
151*> vector X(j) (i.e., the smallest relative change in
152*> any element of A or B that makes X(j) an exact solution).
153*> \endverbatim
154*>
155*> \param[out] WORK
156*> \verbatim
157*> WORK is COMPLEX*16 array, dimension (2*N)
158*> \endverbatim
159*>
160*> \param[out] RWORK
161*> \verbatim
162*> RWORK is DOUBLE PRECISION array, dimension (N)
163*> \endverbatim
164*>
165*> \param[out] INFO
166*> \verbatim
167*> INFO is INTEGER
168*> = 0: successful exit
169*> < 0: if INFO = -i, the i-th argument had an illegal value
170*> \endverbatim
171*
172*> \par Internal Parameters:
173* =========================
174*>
175*> \verbatim
176*> ITMAX is the maximum number of steps of iterative refinement.
177*> \endverbatim
178*
179* Authors:
180* ========
181*
182*> \author Univ. of Tennessee
183*> \author Univ. of California Berkeley
184*> \author Univ. of Colorado Denver
185*> \author NAG Ltd.
186*
187*> \ingroup complex16SYcomputational
188*
189* =====================================================================
190 SUBROUTINE zsyrfs( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB,
191 $ X, LDX, FERR, BERR, WORK, RWORK, INFO )
192*
193* -- LAPACK computational routine --
194* -- LAPACK is a software package provided by Univ. of Tennessee, --
195* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
196*
197* .. Scalar Arguments ..
198 CHARACTER UPLO
199 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
200* ..
201* .. Array Arguments ..
202 INTEGER IPIV( * )
203 DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
204 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
205 $ work( * ), x( ldx, * )
206* ..
207*
208* =====================================================================
209*
210* .. Parameters ..
211 INTEGER ITMAX
212 parameter( itmax = 5 )
213 DOUBLE PRECISION ZERO
214 parameter( zero = 0.0d+0 )
215 COMPLEX*16 ONE
216 parameter( one = ( 1.0d+0, 0.0d+0 ) )
217 DOUBLE PRECISION TWO
218 parameter( two = 2.0d+0 )
219 DOUBLE PRECISION THREE
220 parameter( three = 3.0d+0 )
221* ..
222* .. Local Scalars ..
223 LOGICAL UPPER
224 INTEGER COUNT, I, J, K, KASE, NZ
225 DOUBLE PRECISION EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
226 COMPLEX*16 ZDUM
227* ..
228* .. Local Arrays ..
229 INTEGER ISAVE( 3 )
230* ..
231* .. External Subroutines ..
232 EXTERNAL xerbla, zaxpy, zcopy, zlacn2, zsymv, zsytrs
233* ..
234* .. Intrinsic Functions ..
235 INTRINSIC abs, dble, dimag, max
236* ..
237* .. External Functions ..
238 LOGICAL LSAME
239 DOUBLE PRECISION DLAMCH
240 EXTERNAL lsame, dlamch
241* ..
242* .. Statement Functions ..
243 DOUBLE PRECISION CABS1
244* ..
245* .. Statement Function definitions ..
246 cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
247* ..
248* .. Executable Statements ..
249*
250* Test the input parameters.
251*
252 info = 0
253 upper = lsame( uplo, 'U' )
254 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
255 info = -1
256 ELSE IF( n.LT.0 ) THEN
257 info = -2
258 ELSE IF( nrhs.LT.0 ) THEN
259 info = -3
260 ELSE IF( lda.LT.max( 1, n ) ) THEN
261 info = -5
262 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
263 info = -7
264 ELSE IF( ldb.LT.max( 1, n ) ) THEN
265 info = -10
266 ELSE IF( ldx.LT.max( 1, n ) ) THEN
267 info = -12
268 END IF
269 IF( info.NE.0 ) THEN
270 CALL xerbla( 'ZSYRFS', -info )
271 RETURN
272 END IF
273*
274* Quick return if possible
275*
276 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
277 DO 10 j = 1, nrhs
278 ferr( j ) = zero
279 berr( j ) = zero
280 10 CONTINUE
281 RETURN
282 END IF
283*
284* NZ = maximum number of nonzero elements in each row of A, plus 1
285*
286 nz = n + 1
287 eps = dlamch( 'Epsilon' )
288 safmin = dlamch( 'Safe minimum' )
289 safe1 = nz*safmin
290 safe2 = safe1 / eps
291*
292* Do for each right hand side
293*
294 DO 140 j = 1, nrhs
295*
296 count = 1
297 lstres = three
298 20 CONTINUE
299*
300* Loop until stopping criterion is satisfied.
301*
302* Compute residual R = B - A * X
303*
304 CALL zcopy( n, b( 1, j ), 1, work, 1 )
305 CALL zsymv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
306*
307* Compute componentwise relative backward error from formula
308*
309* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
310*
311* where abs(Z) is the componentwise absolute value of the matrix
312* or vector Z. If the i-th component of the denominator is less
313* than SAFE2, then SAFE1 is added to the i-th components of the
314* numerator and denominator before dividing.
315*
316 DO 30 i = 1, n
317 rwork( i ) = cabs1( b( i, j ) )
318 30 CONTINUE
319*
320* Compute abs(A)*abs(X) + abs(B).
321*
322 IF( upper ) THEN
323 DO 50 k = 1, n
324 s = zero
325 xk = cabs1( x( k, j ) )
326 DO 40 i = 1, k - 1
327 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
328 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
329 40 CONTINUE
330 rwork( k ) = rwork( k ) + cabs1( a( k, k ) )*xk + s
331 50 CONTINUE
332 ELSE
333 DO 70 k = 1, n
334 s = zero
335 xk = cabs1( x( k, j ) )
336 rwork( k ) = rwork( k ) + cabs1( a( k, k ) )*xk
337 DO 60 i = k + 1, n
338 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
339 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
340 60 CONTINUE
341 rwork( k ) = rwork( k ) + s
342 70 CONTINUE
343 END IF
344 s = zero
345 DO 80 i = 1, n
346 IF( rwork( i ).GT.safe2 ) THEN
347 s = max( s, cabs1( work( i ) ) / rwork( i ) )
348 ELSE
349 s = max( s, ( cabs1( work( i ) )+safe1 ) /
350 $ ( rwork( i )+safe1 ) )
351 END IF
352 80 CONTINUE
353 berr( j ) = s
354*
355* Test stopping criterion. Continue iterating if
356* 1) The residual BERR(J) is larger than machine epsilon, and
357* 2) BERR(J) decreased by at least a factor of 2 during the
358* last iteration, and
359* 3) At most ITMAX iterations tried.
360*
361 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
362 $ count.LE.itmax ) THEN
363*
364* Update solution and try again.
365*
366 CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
367 CALL zaxpy( n, one, work, 1, x( 1, j ), 1 )
368 lstres = berr( j )
369 count = count + 1
370 GO TO 20
371 END IF
372*
373* Bound error from formula
374*
375* norm(X - XTRUE) / norm(X) .le. FERR =
376* norm( abs(inv(A))*
377* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
378*
379* where
380* norm(Z) is the magnitude of the largest component of Z
381* inv(A) is the inverse of A
382* abs(Z) is the componentwise absolute value of the matrix or
383* vector Z
384* NZ is the maximum number of nonzeros in any row of A, plus 1
385* EPS is machine epsilon
386*
387* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
388* is incremented by SAFE1 if the i-th component of
389* abs(A)*abs(X) + abs(B) is less than SAFE2.
390*
391* Use ZLACN2 to estimate the infinity-norm of the matrix
392* inv(A) * diag(W),
393* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
394*
395 DO 90 i = 1, n
396 IF( rwork( i ).GT.safe2 ) THEN
397 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
398 ELSE
399 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
400 $ safe1
401 END IF
402 90 CONTINUE
403*
404 kase = 0
405 100 CONTINUE
406 CALL zlacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
407 IF( kase.NE.0 ) THEN
408 IF( kase.EQ.1 ) THEN
409*
410* Multiply by diag(W)*inv(A**T).
411*
412 CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, info )
413 DO 110 i = 1, n
414 work( i ) = rwork( i )*work( i )
415 110 CONTINUE
416 ELSE IF( kase.EQ.2 ) THEN
417*
418* Multiply by inv(A)*diag(W).
419*
420 DO 120 i = 1, n
421 work( i ) = rwork( i )*work( i )
422 120 CONTINUE
423 CALL zsytrs( uplo, n, 1, af, ldaf, ipiv, work, n, 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, cabs1( 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 ZSYRFS
442*
443 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zlacn2(N, V, X, EST, KASE, ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: zlacn2.f:133
subroutine zsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZSYMV computes a matrix-vector product for a complex symmetric matrix.
Definition: zsymv.f:157
subroutine zsyrfs(UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
ZSYRFS
Definition: zsyrfs.f:192
subroutine zsytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZSYTRS
Definition: zsytrs.f:120