LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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cpprfs.f
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1*> \brief \b CPPRFS
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
9*> Download CPPRFS + dependencies
10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpprfs.f">
11*> [TGZ]</a>
12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cpprfs.f">
13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cpprfs.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE CPPRFS( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
22* BERR, WORK, RWORK, INFO )
23*
24* .. Scalar Arguments ..
25* CHARACTER UPLO
26* INTEGER INFO, LDB, LDX, N, NRHS
27* ..
28* .. Array Arguments ..
29* REAL BERR( * ), FERR( * ), RWORK( * )
30* COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
31* $ X( LDX, * )
32* ..
33*
34*
35*> \par Purpose:
36* =============
37*>
38*> \verbatim
39*>
40*> CPPRFS improves the computed solution to a system of linear
41*> equations when the coefficient matrix is Hermitian positive definite
42*> and packed, and provides error bounds and backward error estimates
43*> 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] AP
70*> \verbatim
71*> AP is COMPLEX array, dimension (N*(N+1)/2)
72*> The upper or lower triangle of the Hermitian matrix A, packed
73*> columnwise in a linear array. The j-th column of A is stored
74*> in the array AP as follows:
75*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
76*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
77*> \endverbatim
78*>
79*> \param[in] AFP
80*> \verbatim
81*> AFP is COMPLEX array, dimension (N*(N+1)/2)
82*> The triangular factor U or L from the Cholesky factorization
83*> A = U**H*U or A = L*L**H, as computed by SPPTRF/CPPTRF,
84*> packed columnwise in a linear array in the same format as A
85*> (see AP).
86*> \endverbatim
87*>
88*> \param[in] B
89*> \verbatim
90*> B is COMPLEX array, dimension (LDB,NRHS)
91*> The right hand side matrix B.
92*> \endverbatim
93*>
94*> \param[in] LDB
95*> \verbatim
96*> LDB is INTEGER
97*> The leading dimension of the array B. LDB >= max(1,N).
98*> \endverbatim
99*>
100*> \param[in,out] X
101*> \verbatim
102*> X is COMPLEX array, dimension (LDX,NRHS)
103*> On entry, the solution matrix X, as computed by CPPTRS.
104*> On exit, the improved solution matrix X.
105*> \endverbatim
106*>
107*> \param[in] LDX
108*> \verbatim
109*> LDX is INTEGER
110*> The leading dimension of the array X. LDX >= max(1,N).
111*> \endverbatim
112*>
113*> \param[out] FERR
114*> \verbatim
115*> FERR is REAL array, dimension (NRHS)
116*> The estimated forward error bound for each solution vector
117*> X(j) (the j-th column of the solution matrix X).
118*> If XTRUE is the true solution corresponding to X(j), FERR(j)
119*> is an estimated upper bound for the magnitude of the largest
120*> element in (X(j) - XTRUE) divided by the magnitude of the
121*> largest element in X(j). The estimate is as reliable as
122*> the estimate for RCOND, and is almost always a slight
123*> overestimate of the true error.
124*> \endverbatim
125*>
126*> \param[out] BERR
127*> \verbatim
128*> BERR is REAL array, dimension (NRHS)
129*> The componentwise relative backward error of each solution
130*> vector X(j) (i.e., the smallest relative change in
131*> any element of A or B that makes X(j) an exact solution).
132*> \endverbatim
133*>
134*> \param[out] WORK
135*> \verbatim
136*> WORK is COMPLEX array, dimension (2*N)
137*> \endverbatim
138*>
139*> \param[out] RWORK
140*> \verbatim
141*> RWORK is REAL array, dimension (N)
142*> \endverbatim
143*>
144*> \param[out] INFO
145*> \verbatim
146*> INFO is INTEGER
147*> = 0: successful exit
148*> < 0: if INFO = -i, the i-th argument had an illegal value
149*> \endverbatim
150*
151*> \par Internal Parameters:
152* =========================
153*>
154*> \verbatim
155*> ITMAX is the maximum number of steps of iterative refinement.
156*> \endverbatim
157*
158* Authors:
159* ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \ingroup complexOTHERcomputational
167*
168* =====================================================================
169 SUBROUTINE cpprfs( UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR,
170 $ BERR, WORK, RWORK, INFO )
171*
172* -- LAPACK computational routine --
173* -- LAPACK is a software package provided by Univ. of Tennessee, --
174* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176* .. Scalar Arguments ..
177 CHARACTER UPLO
178 INTEGER INFO, LDB, LDX, N, NRHS
179* ..
180* .. Array Arguments ..
181 REAL BERR( * ), FERR( * ), RWORK( * )
182 COMPLEX AFP( * ), AP( * ), B( LDB, * ), WORK( * ),
183 $ x( ldx, * )
184* ..
185*
186* ====================================================================
187*
188* .. Parameters ..
189 INTEGER ITMAX
190 parameter( itmax = 5 )
191 REAL ZERO
192 parameter( zero = 0.0e+0 )
193 COMPLEX CONE
194 parameter( cone = ( 1.0e+0, 0.0e+0 ) )
195 REAL TWO
196 parameter( two = 2.0e+0 )
197 REAL THREE
198 parameter( three = 3.0e+0 )
199* ..
200* .. Local Scalars ..
201 LOGICAL UPPER
202 INTEGER COUNT, I, IK, J, K, KASE, KK, NZ
203 REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
204 COMPLEX ZDUM
205* ..
206* .. Local Arrays ..
207 INTEGER ISAVE( 3 )
208* ..
209* .. External Subroutines ..
210 EXTERNAL caxpy, ccopy, chpmv, clacn2, cpptrs, xerbla
211* ..
212* .. Intrinsic Functions ..
213 INTRINSIC abs, aimag, max, real
214* ..
215* .. External Functions ..
216 LOGICAL LSAME
217 REAL SLAMCH
218 EXTERNAL lsame, slamch
219* ..
220* .. Statement Functions ..
221 REAL CABS1
222* ..
223* .. Statement Function definitions ..
224 cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
225* ..
226* .. Executable Statements ..
227*
228* Test the input parameters.
229*
230 info = 0
231 upper = lsame( uplo, 'U' )
232 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
233 info = -1
234 ELSE IF( n.LT.0 ) THEN
235 info = -2
236 ELSE IF( nrhs.LT.0 ) THEN
237 info = -3
238 ELSE IF( ldb.LT.max( 1, n ) ) THEN
239 info = -7
240 ELSE IF( ldx.LT.max( 1, n ) ) THEN
241 info = -9
242 END IF
243 IF( info.NE.0 ) THEN
244 CALL xerbla( 'CPPRFS', -info )
245 RETURN
246 END IF
247*
248* Quick return if possible
249*
250 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
251 DO 10 j = 1, nrhs
252 ferr( j ) = zero
253 berr( j ) = zero
254 10 CONTINUE
255 RETURN
256 END IF
257*
258* NZ = maximum number of nonzero elements in each row of A, plus 1
259*
260 nz = n + 1
261 eps = slamch( 'Epsilon' )
262 safmin = slamch( 'Safe minimum' )
263 safe1 = nz*safmin
264 safe2 = safe1 / eps
265*
266* Do for each right hand side
267*
268 DO 140 j = 1, nrhs
269*
270 count = 1
271 lstres = three
272 20 CONTINUE
273*
274* Loop until stopping criterion is satisfied.
275*
276* Compute residual R = B - A * X
277*
278 CALL ccopy( n, b( 1, j ), 1, work, 1 )
279 CALL chpmv( uplo, n, -cone, ap, x( 1, j ), 1, cone, work, 1 )
280*
281* Compute componentwise relative backward error from formula
282*
283* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
284*
285* where abs(Z) is the componentwise absolute value of the matrix
286* or vector Z. If the i-th component of the denominator is less
287* than SAFE2, then SAFE1 is added to the i-th components of the
288* numerator and denominator before dividing.
289*
290 DO 30 i = 1, n
291 rwork( i ) = cabs1( b( i, j ) )
292 30 CONTINUE
293*
294* Compute abs(A)*abs(X) + abs(B).
295*
296 kk = 1
297 IF( upper ) THEN
298 DO 50 k = 1, n
299 s = zero
300 xk = cabs1( x( k, j ) )
301 ik = kk
302 DO 40 i = 1, k - 1
303 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
304 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
305 ik = ik + 1
306 40 CONTINUE
307 rwork( k ) = rwork( k ) + abs( real( ap( kk+k-1 ) ) )*
308 $ xk + s
309 kk = kk + k
310 50 CONTINUE
311 ELSE
312 DO 70 k = 1, n
313 s = zero
314 xk = cabs1( x( k, j ) )
315 rwork( k ) = rwork( k ) + abs( real( ap( kk ) ) )*xk
316 ik = kk + 1
317 DO 60 i = k + 1, n
318 rwork( i ) = rwork( i ) + cabs1( ap( ik ) )*xk
319 s = s + cabs1( ap( ik ) )*cabs1( x( i, j ) )
320 ik = ik + 1
321 60 CONTINUE
322 rwork( k ) = rwork( k ) + s
323 kk = kk + ( n-k+1 )
324 70 CONTINUE
325 END IF
326 s = zero
327 DO 80 i = 1, n
328 IF( rwork( i ).GT.safe2 ) THEN
329 s = max( s, cabs1( work( i ) ) / rwork( i ) )
330 ELSE
331 s = max( s, ( cabs1( work( i ) )+safe1 ) /
332 $ ( rwork( i )+safe1 ) )
333 END IF
334 80 CONTINUE
335 berr( j ) = s
336*
337* Test stopping criterion. Continue iterating if
338* 1) The residual BERR(J) is larger than machine epsilon, and
339* 2) BERR(J) decreased by at least a factor of 2 during the
340* last iteration, and
341* 3) At most ITMAX iterations tried.
342*
343 IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
344 $ count.LE.itmax ) THEN
345*
346* Update solution and try again.
347*
348 CALL cpptrs( uplo, n, 1, afp, work, n, info )
349 CALL caxpy( n, cone, work, 1, x( 1, j ), 1 )
350 lstres = berr( j )
351 count = count + 1
352 GO TO 20
353 END IF
354*
355* Bound error from formula
356*
357* norm(X - XTRUE) / norm(X) .le. FERR =
358* norm( abs(inv(A))*
359* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
360*
361* where
362* norm(Z) is the magnitude of the largest component of Z
363* inv(A) is the inverse of A
364* abs(Z) is the componentwise absolute value of the matrix or
365* vector Z
366* NZ is the maximum number of nonzeros in any row of A, plus 1
367* EPS is machine epsilon
368*
369* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
370* is incremented by SAFE1 if the i-th component of
371* abs(A)*abs(X) + abs(B) is less than SAFE2.
372*
373* Use CLACN2 to estimate the infinity-norm of the matrix
374* inv(A) * diag(W),
375* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
376*
377 DO 90 i = 1, n
378 IF( rwork( i ).GT.safe2 ) THEN
379 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i )
380 ELSE
381 rwork( i ) = cabs1( work( i ) ) + nz*eps*rwork( i ) +
382 $ safe1
383 END IF
384 90 CONTINUE
385*
386 kase = 0
387 100 CONTINUE
388 CALL clacn2( n, work( n+1 ), work, ferr( j ), kase, isave )
389 IF( kase.NE.0 ) THEN
390 IF( kase.EQ.1 ) THEN
391*
392* Multiply by diag(W)*inv(A**H).
393*
394 CALL cpptrs( uplo, n, 1, afp, work, n, info )
395 DO 110 i = 1, n
396 work( i ) = rwork( i )*work( i )
397 110 CONTINUE
398 ELSE IF( kase.EQ.2 ) THEN
399*
400* Multiply by inv(A)*diag(W).
401*
402 DO 120 i = 1, n
403 work( i ) = rwork( i )*work( i )
404 120 CONTINUE
405 CALL cpptrs( uplo, n, 1, afp, work, n, info )
406 END IF
407 GO TO 100
408 END IF
409*
410* Normalize error.
411*
412 lstres = zero
413 DO 130 i = 1, n
414 lstres = max( lstres, cabs1( x( i, j ) ) )
415 130 CONTINUE
416 IF( lstres.NE.zero )
417 $ ferr( j ) = ferr( j ) / lstres
418*
419 140 CONTINUE
420*
421 RETURN
422*
423* End of CPPRFS
424*
425 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 chpmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CHPMV
Definition: chpmv.f:149
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 cpptrs(UPLO, N, NRHS, AP, B, LDB, INFO)
CPPTRS
Definition: cpptrs.f:108
subroutine cpprfs(UPLO, N, NRHS, AP, AFP, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPPRFS
Definition: cpprfs.f:171