LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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cporfs.f
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1*> \brief \b CPORFS
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>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cporfs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
22* 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* REAL BERR( * ), FERR( * ), RWORK( * )
30* COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
31* \$ WORK( * ), X( LDX, * )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> CPORFS improves the computed solution to a system of linear
41*> equations when the coefficient matrix is Hermitian 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 COMPLEX array, dimension (LDA,N)
72*> The Hermitian 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 array, dimension (LDAF,N)
90*> The triangular factor U or L from the Cholesky factorization
91*> A = U**H*U or A = L*L**H, as computed by CPOTRF.
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 COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)
115*> On entry, the solution matrix X, as computed by CPOTRS.
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 COMPLEX array, dimension (2*N)
149*> \endverbatim
150*>
151*> \param[out] RWORK
152*> \verbatim
153*> RWORK is REAL 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*> \ingroup complexPOcomputational
179*
180* =====================================================================
181 SUBROUTINE cporfs( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X,
182 \$ LDX, FERR, BERR, WORK, RWORK, INFO )
183*
184* -- LAPACK computational routine --
185* -- LAPACK is a software package provided by Univ. of Tennessee, --
186* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187*
188* .. Scalar Arguments ..
189 CHARACTER UPLO
190 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS
191* ..
192* .. Array Arguments ..
193 REAL BERR( * ), FERR( * ), RWORK( * )
194 COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
195 \$ work( * ), x( ldx, * )
196* ..
197*
198* ====================================================================
199*
200* .. Parameters ..
201 INTEGER ITMAX
202 parameter( itmax = 5 )
203 REAL ZERO
204 parameter( zero = 0.0e+0 )
205 COMPLEX ONE
206 parameter( one = ( 1.0e+0, 0.0e+0 ) )
207 REAL TWO
208 parameter( two = 2.0e+0 )
209 REAL THREE
210 parameter( three = 3.0e+0 )
211* ..
212* .. Local Scalars ..
213 LOGICAL UPPER
214 INTEGER COUNT, I, J, K, KASE, NZ
215 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
216 COMPLEX ZDUM
217* ..
218* .. Local Arrays ..
219 INTEGER ISAVE( 3 )
220* ..
221* .. External Subroutines ..
222 EXTERNAL caxpy, ccopy, chemv, clacn2, cpotrs, xerbla
223* ..
224* .. Intrinsic Functions ..
225 INTRINSIC abs, aimag, max, real
226* ..
227* .. External Functions ..
228 LOGICAL LSAME
229 REAL SLAMCH
230 EXTERNAL lsame, slamch
231* ..
232* .. Statement Functions ..
233 REAL CABS1
234* ..
235* .. Statement Function definitions ..
236 cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
237* ..
238* .. Executable Statements ..
239*
240* Test the input parameters.
241*
242 info = 0
243 upper = lsame( uplo, 'U' )
244 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
245 info = -1
246 ELSE IF( n.LT.0 ) THEN
247 info = -2
248 ELSE IF( nrhs.LT.0 ) THEN
249 info = -3
250 ELSE IF( lda.LT.max( 1, n ) ) THEN
251 info = -5
252 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
253 info = -7
254 ELSE IF( ldb.LT.max( 1, n ) ) THEN
255 info = -9
256 ELSE IF( ldx.LT.max( 1, n ) ) THEN
257 info = -11
258 END IF
259 IF( info.NE.0 ) THEN
260 CALL xerbla( 'CPORFS', -info )
261 RETURN
262 END IF
263*
264* Quick return if possible
265*
266 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
267 DO 10 j = 1, nrhs
268 ferr( j ) = zero
269 berr( j ) = zero
270 10 CONTINUE
271 RETURN
272 END IF
273*
274* NZ = maximum number of nonzero elements in each row of A, plus 1
275*
276 nz = n + 1
277 eps = slamch( 'Epsilon' )
278 safmin = slamch( 'Safe minimum' )
279 safe1 = nz*safmin
280 safe2 = safe1 / eps
281*
282* Do for each right hand side
283*
284 DO 140 j = 1, nrhs
285*
286 count = 1
287 lstres = three
288 20 CONTINUE
289*
290* Loop until stopping criterion is satisfied.
291*
292* Compute residual R = B - A * X
293*
294 CALL ccopy( n, b( 1, j ), 1, work, 1 )
295 CALL chemv( uplo, n, -one, a, lda, x( 1, j ), 1, one, work, 1 )
296*
297* Compute componentwise relative backward error from formula
298*
299* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
300*
301* where abs(Z) is the componentwise absolute value of the matrix
302* or vector Z. If the i-th component of the denominator is less
303* than SAFE2, then SAFE1 is added to the i-th components of the
304* numerator and denominator before dividing.
305*
306 DO 30 i = 1, n
307 rwork( i ) = cabs1( b( i, j ) )
308 30 CONTINUE
309*
310* Compute abs(A)*abs(X) + abs(B).
311*
312 IF( upper ) THEN
313 DO 50 k = 1, n
314 s = zero
315 xk = cabs1( x( k, j ) )
316 DO 40 i = 1, k - 1
317 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
318 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
319 40 CONTINUE
320 rwork( k ) = rwork( k ) + abs( real( a( k, k ) ) )*xk + s
321 50 CONTINUE
322 ELSE
323 DO 70 k = 1, n
324 s = zero
325 xk = cabs1( x( k, j ) )
326 rwork( k ) = rwork( k ) + abs( real( a( k, k ) ) )*xk
327 DO 60 i = k + 1, n
328 rwork( i ) = rwork( i ) + cabs1( a( i, k ) )*xk
329 s = s + cabs1( a( i, k ) )*cabs1( x( i, j ) )
330 60 CONTINUE
331 rwork( k ) = rwork( k ) + s
332 70 CONTINUE
333 END IF
334 s = zero
335 DO 80 i = 1, n
336 IF( rwork( i ).GT.safe2 ) THEN
337 s = max( s, cabs1( work( i ) ) / rwork( i ) )
338 ELSE
339 s = max( s, ( cabs1( work( i ) )+safe1 ) /
340 \$ ( rwork( i )+safe1 ) )
341 END IF
342 80 CONTINUE
343 berr( j ) = s
344*
345* Test stopping criterion. Continue iterating if
346* 1) The residual BERR(J) is larger than machine epsilon, and
347* 2) BERR(J) decreased by at least a factor of 2 during the
348* last iteration, and
349* 3) At most ITMAX iterations tried.
350*
351 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
352 \$ count.LE.itmax ) THEN
353*
354* Update solution and try again.
355*
356 CALL cpotrs( uplo, n, 1, af, ldaf, work, n, info )
357 CALL caxpy( n, one, work, 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(A))*
367* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
368*
369* where
370* norm(Z) is the magnitude of the largest component of Z
371* inv(A) is the inverse of 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(A)*abs(X)+abs(B))
378* is incremented by SAFE1 if the i-th component of
379* abs(A)*abs(X) + abs(B) is less than SAFE2.
380*
381* Use CLACN2 to estimate the infinity-norm of the matrix
382* inv(A) * diag(W),
383* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
384*
385 DO 90 i = 1, n
386 IF( rwork( i ).GT.safe2 ) THEN
387 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
388 ELSE
389 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
390 \$ safe1
391 END IF
392 90 CONTINUE
393*
394 kase = 0
395 100 CONTINUE
396 CALL clacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
397 IF( kase.NE.0 ) THEN
398 IF( kase.EQ.1 ) THEN
399*
400* Multiply by diag(W)*inv(A**H).
401*
402 CALL cpotrs( uplo, n, 1, af, ldaf, work, n, info )
403 DO 110 i = 1, n
404 work( i ) = rwork( i )*work( i )
405 110 CONTINUE
406 ELSE IF( kase.EQ.2 ) THEN
407*
408* Multiply by inv(A)*diag(W).
409*
410 DO 120 i = 1, n
411 work( i ) = rwork( i )*work( i )
412 120 CONTINUE
413 CALL cpotrs( uplo, n, 1, af, ldaf, work, n, info )
414 END IF
415 GO TO 100
416 END IF
417*
418* Normalize error.
419*
420 lstres = zero
421 DO 130 i = 1, n
422 lstres = max( lstres, cabs1( x( i, j ) ) )
423 130 CONTINUE
424 IF( lstres.NE.zero )
425 \$ ferr( j ) = ferr( j ) / lstres
426*
427 140 CONTINUE
428*
429 RETURN
430*
431* End of CPORFS
432*
433 END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine chemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CHEMV
Definition: chemv.f:154
subroutine clacn2(N, V, X, EST, KASE, ISAVE)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: clacn2.f:133
subroutine cporfs(UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPORFS
Definition: cporfs.f:183
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:110