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zcposv.f
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1 *> \brief <b> ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE ZCPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
22 * SWORK, RWORK, ITER, INFO )
23 *
24 * .. Scalar Arguments ..
25 * CHARACTER UPLO
26 * INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
27 * ..
28 * .. Array Arguments ..
29 * DOUBLE PRECISION RWORK( * )
30 * COMPLEX SWORK( * )
31 * COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ),
32 * $ X( LDX, * )
33 * ..
34 *
35 *
36 *> \par Purpose:
37 * =============
38 *>
39 *> \verbatim
40 *>
41 *> ZCPOSV computes the solution to a complex system of linear equations
42 *> A * X = B,
43 *> where A is an N-by-N Hermitian positive definite matrix and X and B
44 *> are N-by-NRHS matrices.
45 *>
46 *> ZCPOSV first attempts to factorize the matrix in COMPLEX and use this
47 *> factorization within an iterative refinement procedure to produce a
48 *> solution with COMPLEX*16 normwise backward error quality (see below).
49 *> If the approach fails the method switches to a COMPLEX*16
50 *> factorization and solve.
51 *>
52 *> The iterative refinement is not going to be a winning strategy if
53 *> the ratio COMPLEX performance over COMPLEX*16 performance is too
54 *> small. A reasonable strategy should take the number of right-hand
55 *> sides and the size of the matrix into account. This might be done
56 *> with a call to ILAENV in the future. Up to now, we always try
57 *> iterative refinement.
58 *>
59 *> The iterative refinement process is stopped if
60 *> ITER > ITERMAX
61 *> or for all the RHS we have:
62 *> RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
63 *> where
64 *> o ITER is the number of the current iteration in the iterative
65 *> refinement process
66 *> o RNRM is the infinity-norm of the residual
67 *> o XNRM is the infinity-norm of the solution
68 *> o ANRM is the infinity-operator-norm of the matrix A
69 *> o EPS is the machine epsilon returned by DLAMCH('Epsilon')
70 *> The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
71 *> respectively.
72 *> \endverbatim
73 *
74 * Arguments:
75 * ==========
76 *
77 *> \param[in] UPLO
78 *> \verbatim
79 *> UPLO is CHARACTER*1
80 *> = 'U': Upper triangle of A is stored;
81 *> = 'L': Lower triangle of A is stored.
82 *> \endverbatim
83 *>
84 *> \param[in] N
85 *> \verbatim
86 *> N is INTEGER
87 *> The number of linear equations, i.e., the order of the
88 *> matrix A. N >= 0.
89 *> \endverbatim
90 *>
91 *> \param[in] NRHS
92 *> \verbatim
93 *> NRHS is INTEGER
94 *> The number of right hand sides, i.e., the number of columns
95 *> of the matrix B. NRHS >= 0.
96 *> \endverbatim
97 *>
98 *> \param[in,out] A
99 *> \verbatim
100 *> A is COMPLEX*16 array,
101 *> dimension (LDA,N)
102 *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
103 *> N-by-N upper triangular part of A contains the upper
104 *> triangular part of the matrix A, and the strictly lower
105 *> triangular part of A is not referenced. If UPLO = 'L', the
106 *> leading N-by-N lower triangular part of A contains the lower
107 *> triangular part of the matrix A, and the strictly upper
108 *> triangular part of A is not referenced.
109 *>
110 *> Note that the imaginary parts of the diagonal
111 *> elements need not be set and are assumed to be zero.
112 *>
113 *> On exit, if iterative refinement has been successfully used
114 *> (INFO.EQ.0 and ITER.GE.0, see description below), then A is
115 *> unchanged, if double precision factorization has been used
116 *> (INFO.EQ.0 and ITER.LT.0, see description below), then the
117 *> array A contains the factor U or L from the Cholesky
118 *> factorization A = U**H*U or A = L*L**H.
119 *> \endverbatim
120 *>
121 *> \param[in] LDA
122 *> \verbatim
123 *> LDA is INTEGER
124 *> The leading dimension of the array A. LDA >= max(1,N).
125 *> \endverbatim
126 *>
127 *> \param[in] B
128 *> \verbatim
129 *> B is COMPLEX*16 array, dimension (LDB,NRHS)
130 *> The N-by-NRHS right hand side matrix B.
131 *> \endverbatim
132 *>
133 *> \param[in] LDB
134 *> \verbatim
135 *> LDB is INTEGER
136 *> The leading dimension of the array B. LDB >= max(1,N).
137 *> \endverbatim
138 *>
139 *> \param[out] X
140 *> \verbatim
141 *> X is COMPLEX*16 array, dimension (LDX,NRHS)
142 *> If INFO = 0, the N-by-NRHS solution matrix X.
143 *> \endverbatim
144 *>
145 *> \param[in] LDX
146 *> \verbatim
147 *> LDX is INTEGER
148 *> The leading dimension of the array X. LDX >= max(1,N).
149 *> \endverbatim
150 *>
151 *> \param[out] WORK
152 *> \verbatim
153 *> WORK is COMPLEX*16 array, dimension (N*NRHS)
154 *> This array is used to hold the residual vectors.
155 *> \endverbatim
156 *>
157 *> \param[out] SWORK
158 *> \verbatim
159 *> SWORK is COMPLEX array, dimension (N*(N+NRHS))
160 *> This array is used to use the single precision matrix and the
161 *> right-hand sides or solutions in single precision.
162 *> \endverbatim
163 *>
164 *> \param[out] RWORK
165 *> \verbatim
166 *> RWORK is DOUBLE PRECISION array, dimension (N)
167 *> \endverbatim
168 *>
169 *> \param[out] ITER
170 *> \verbatim
171 *> ITER is INTEGER
172 *> < 0: iterative refinement has failed, COMPLEX*16
173 *> factorization has been performed
174 *> -1 : the routine fell back to full precision for
175 *> implementation- or machine-specific reasons
176 *> -2 : narrowing the precision induced an overflow,
177 *> the routine fell back to full precision
178 *> -3 : failure of CPOTRF
179 *> -31: stop the iterative refinement after the 30th
180 *> iterations
181 *> > 0: iterative refinement has been sucessfully used.
182 *> Returns the number of iterations
183 *> \endverbatim
184 *>
185 *> \param[out] INFO
186 *> \verbatim
187 *> INFO is INTEGER
188 *> = 0: successful exit
189 *> < 0: if INFO = -i, the i-th argument had an illegal value
190 *> > 0: if INFO = i, the leading minor of order i of
191 *> (COMPLEX*16) A is not positive definite, so the
192 *> factorization could not be completed, and the solution
193 *> has not been computed.
194 *> \endverbatim
195 *
196 * Authors:
197 * ========
198 *
199 *> \author Univ. of Tennessee
200 *> \author Univ. of California Berkeley
201 *> \author Univ. of Colorado Denver
202 *> \author NAG Ltd.
203 *
204 *> \date November 2011
205 *
206 *> \ingroup complex16POsolve
207 *
208 * =====================================================================
209  SUBROUTINE zcposv( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
210  $ swork, rwork, iter, info )
211 *
212 * -- LAPACK driver routine (version 3.4.0) --
213 * -- LAPACK is a software package provided by Univ. of Tennessee, --
214 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
215 * November 2011
216 *
217 * .. Scalar Arguments ..
218  CHARACTER uplo
219  INTEGER info, iter, lda, ldb, ldx, n, nrhs
220 * ..
221 * .. Array Arguments ..
222  DOUBLE PRECISION rwork( * )
223  COMPLEX swork( * )
224  COMPLEX*16 a( lda, * ), b( ldb, * ), work( n, * ),
225  $ x( ldx, * )
226 * ..
227 *
228 * =====================================================================
229 *
230 * .. Parameters ..
231  LOGICAL doitref
232  parameter( doitref = .true. )
233 *
234  INTEGER itermax
235  parameter( itermax = 30 )
236 *
237  DOUBLE PRECISION bwdmax
238  parameter( bwdmax = 1.0e+00 )
239 *
240  COMPLEX*16 negone, one
241  parameter( negone = ( -1.0d+00, 0.0d+00 ),
242  $ one = ( 1.0d+00, 0.0d+00 ) )
243 *
244 * .. Local Scalars ..
245  INTEGER i, iiter, ptsa, ptsx
246  DOUBLE PRECISION anrm, cte, eps, rnrm, xnrm
247  COMPLEX*16 zdum
248 *
249 * .. External Subroutines ..
250  EXTERNAL zaxpy, zhemm, zlacpy, zlat2c, zlag2c, clag2z,
251  $ cpotrf, cpotrs, xerbla
252 * ..
253 * .. External Functions ..
254  INTEGER izamax
255  DOUBLE PRECISION dlamch, zlanhe
256  LOGICAL lsame
257  EXTERNAL izamax, dlamch, zlanhe, lsame
258 * ..
259 * .. Intrinsic Functions ..
260  INTRINSIC abs, dble, max, sqrt
261 * .. Statement Functions ..
262  DOUBLE PRECISION cabs1
263 * ..
264 * .. Statement Function definitions ..
265  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
266 * ..
267 * .. Executable Statements ..
268 *
269  info = 0
270  iter = 0
271 *
272 * Test the input parameters.
273 *
274  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
275  info = -1
276  ELSE IF( n.LT.0 ) THEN
277  info = -2
278  ELSE IF( nrhs.LT.0 ) THEN
279  info = -3
280  ELSE IF( lda.LT.max( 1, n ) ) THEN
281  info = -5
282  ELSE IF( ldb.LT.max( 1, n ) ) THEN
283  info = -7
284  ELSE IF( ldx.LT.max( 1, n ) ) THEN
285  info = -9
286  END IF
287  IF( info.NE.0 ) THEN
288  CALL xerbla( 'ZCPOSV', -info )
289  RETURN
290  END IF
291 *
292 * Quick return if (N.EQ.0).
293 *
294  IF( n.EQ.0 )
295  $ RETURN
296 *
297 * Skip single precision iterative refinement if a priori slower
298 * than double precision factorization.
299 *
300  IF( .NOT.doitref ) THEN
301  iter = -1
302  go to 40
303  END IF
304 *
305 * Compute some constants.
306 *
307  anrm = zlanhe( 'I', uplo, n, a, lda, rwork )
308  eps = dlamch( 'Epsilon' )
309  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
310 *
311 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
312 *
313  ptsa = 1
314  ptsx = ptsa + n*n
315 *
316 * Convert B from double precision to single precision and store the
317 * result in SX.
318 *
319  CALL zlag2c( n, nrhs, b, ldb, swork( ptsx ), n, info )
320 *
321  IF( info.NE.0 ) THEN
322  iter = -2
323  go to 40
324  END IF
325 *
326 * Convert A from double precision to single precision and store the
327 * result in SA.
328 *
329  CALL zlat2c( uplo, n, a, lda, swork( ptsa ), n, info )
330 *
331  IF( info.NE.0 ) THEN
332  iter = -2
333  go to 40
334  END IF
335 *
336 * Compute the Cholesky factorization of SA.
337 *
338  CALL cpotrf( uplo, n, swork( ptsa ), n, info )
339 *
340  IF( info.NE.0 ) THEN
341  iter = -3
342  go to 40
343  END IF
344 *
345 * Solve the system SA*SX = SB.
346 *
347  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
348  $ info )
349 *
350 * Convert SX back to COMPLEX*16
351 *
352  CALL clag2z( n, nrhs, swork( ptsx ), n, x, ldx, info )
353 *
354 * Compute R = B - AX (R is WORK).
355 *
356  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
357 *
358  CALL zhemm( 'Left', uplo, n, nrhs, negone, a, lda, x, ldx, one,
359  $ work, n )
360 *
361 * Check whether the NRHS normwise backward errors satisfy the
362 * stopping criterion. If yes, set ITER=0 and return.
363 *
364  DO i = 1, nrhs
365  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
366  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
367  IF( rnrm.GT.xnrm*cte )
368  $ go to 10
369  END DO
370 *
371 * If we are here, the NRHS normwise backward errors satisfy the
372 * stopping criterion. We are good to exit.
373 *
374  iter = 0
375  RETURN
376 *
377  10 CONTINUE
378 *
379  DO 30 iiter = 1, itermax
380 *
381 * Convert R (in WORK) from double precision to single precision
382 * and store the result in SX.
383 *
384  CALL zlag2c( n, nrhs, work, n, swork( ptsx ), n, info )
385 *
386  IF( info.NE.0 ) THEN
387  iter = -2
388  go to 40
389  END IF
390 *
391 * Solve the system SA*SX = SR.
392 *
393  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
394  $ info )
395 *
396 * Convert SX back to double precision and update the current
397 * iterate.
398 *
399  CALL clag2z( n, nrhs, swork( ptsx ), n, work, n, info )
400 *
401  DO i = 1, nrhs
402  CALL zaxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
403  END DO
404 *
405 * Compute R = B - AX (R is WORK).
406 *
407  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
408 *
409  CALL zhemm( 'L', uplo, n, nrhs, negone, a, lda, x, ldx, one,
410  $ work, n )
411 *
412 * Check whether the NRHS normwise backward errors satisfy the
413 * stopping criterion. If yes, set ITER=IITER>0 and return.
414 *
415  DO i = 1, nrhs
416  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
417  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
418  IF( rnrm.GT.xnrm*cte )
419  $ go to 20
420  END DO
421 *
422 * If we are here, the NRHS normwise backward errors satisfy the
423 * stopping criterion, we are good to exit.
424 *
425  iter = iiter
426 *
427  RETURN
428 *
429  20 CONTINUE
430 *
431  30 CONTINUE
432 *
433 * If we are at this place of the code, this is because we have
434 * performed ITER=ITERMAX iterations and never satisified the
435 * stopping criterion, set up the ITER flag accordingly and follow
436 * up on double precision routine.
437 *
438  iter = -itermax - 1
439 *
440  40 CONTINUE
441 *
442 * Single-precision iterative refinement failed to converge to a
443 * satisfactory solution, so we resort to double precision.
444 *
445  CALL zpotrf( uplo, n, a, lda, info )
446 *
447  IF( info.NE.0 )
448  $ RETURN
449 *
450  CALL zlacpy( 'All', n, nrhs, b, ldb, x, ldx )
451  CALL zpotrs( uplo, n, nrhs, a, lda, x, ldx, info )
452 *
453  RETURN
454 *
455 * End of ZCPOSV.
456 *
457  END