LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
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
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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 = 0 and ITER >= 0, see description below), then A is
115 *> unchanged, if double precision factorization has been used
116 *> (INFO = 0 and ITER < 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 successfully 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 *> \ingroup complex16POsolve
205 *
206 * =====================================================================
207  SUBROUTINE zcposv( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK,
208  $ SWORK, RWORK, ITER, INFO )
209 *
210 * -- LAPACK driver routine --
211 * -- LAPACK is a software package provided by Univ. of Tennessee, --
212 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
213 *
214 * .. Scalar Arguments ..
215  CHARACTER UPLO
216  INTEGER INFO, ITER, LDA, LDB, LDX, N, NRHS
217 * ..
218 * .. Array Arguments ..
219  DOUBLE PRECISION RWORK( * )
220  COMPLEX SWORK( * )
221  COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( N, * ),
222  $ x( ldx, * )
223 * ..
224 *
225 * =====================================================================
226 *
227 * .. Parameters ..
228  LOGICAL DOITREF
229  parameter( doitref = .true. )
230 *
231  INTEGER ITERMAX
232  parameter( itermax = 30 )
233 *
234  DOUBLE PRECISION BWDMAX
235  parameter( bwdmax = 1.0e+00 )
236 *
237  COMPLEX*16 NEGONE, ONE
238  parameter( negone = ( -1.0d+00, 0.0d+00 ),
239  $ one = ( 1.0d+00, 0.0d+00 ) )
240 *
241 * .. Local Scalars ..
242  INTEGER I, IITER, PTSA, PTSX
243  DOUBLE PRECISION ANRM, CTE, EPS, RNRM, XNRM
244  COMPLEX*16 ZDUM
245 *
246 * .. External Subroutines ..
247  EXTERNAL zaxpy, zhemm, zlacpy, zlat2c, zlag2c, clag2z,
249 * ..
250 * .. External Functions ..
251  INTEGER IZAMAX
252  DOUBLE PRECISION DLAMCH, ZLANHE
253  LOGICAL LSAME
254  EXTERNAL izamax, dlamch, zlanhe, lsame
255 * ..
256 * .. Intrinsic Functions ..
257  INTRINSIC abs, dble, max, sqrt
258 * .. Statement Functions ..
259  DOUBLE PRECISION CABS1
260 * ..
261 * .. Statement Function definitions ..
262  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
263 * ..
264 * .. Executable Statements ..
265 *
266  info = 0
267  iter = 0
268 *
269 * Test the input parameters.
270 *
271  IF( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
272  info = -1
273  ELSE IF( n.LT.0 ) THEN
274  info = -2
275  ELSE IF( nrhs.LT.0 ) THEN
276  info = -3
277  ELSE IF( lda.LT.max( 1, n ) ) THEN
278  info = -5
279  ELSE IF( ldb.LT.max( 1, n ) ) THEN
280  info = -7
281  ELSE IF( ldx.LT.max( 1, n ) ) THEN
282  info = -9
283  END IF
284  IF( info.NE.0 ) THEN
285  CALL xerbla( 'ZCPOSV', -info )
286  RETURN
287  END IF
288 *
289 * Quick return if (N.EQ.0).
290 *
291  IF( n.EQ.0 )
292  $ RETURN
293 *
294 * Skip single precision iterative refinement if a priori slower
295 * than double precision factorization.
296 *
297  IF( .NOT.doitref ) THEN
298  iter = -1
299  GO TO 40
300  END IF
301 *
302 * Compute some constants.
303 *
304  anrm = zlanhe( 'I', uplo, n, a, lda, rwork )
305  eps = dlamch( 'Epsilon' )
306  cte = anrm*eps*sqrt( dble( n ) )*bwdmax
307 *
308 * Set the indices PTSA, PTSX for referencing SA and SX in SWORK.
309 *
310  ptsa = 1
311  ptsx = ptsa + n*n
312 *
313 * Convert B from double precision to single precision and store the
314 * result in SX.
315 *
316  CALL zlag2c( n, nrhs, b, ldb, swork( ptsx ), n, info )
317 *
318  IF( info.NE.0 ) THEN
319  iter = -2
320  GO TO 40
321  END IF
322 *
323 * Convert A from double precision to single precision and store the
324 * result in SA.
325 *
326  CALL zlat2c( uplo, n, a, lda, swork( ptsa ), n, info )
327 *
328  IF( info.NE.0 ) THEN
329  iter = -2
330  GO TO 40
331  END IF
332 *
333 * Compute the Cholesky factorization of SA.
334 *
335  CALL cpotrf( uplo, n, swork( ptsa ), n, info )
336 *
337  IF( info.NE.0 ) THEN
338  iter = -3
339  GO TO 40
340  END IF
341 *
342 * Solve the system SA*SX = SB.
343 *
344  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
345  $ info )
346 *
347 * Convert SX back to COMPLEX*16
348 *
349  CALL clag2z( n, nrhs, swork( ptsx ), n, x, ldx, info )
350 *
351 * Compute R = B - AX (R is WORK).
352 *
353  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
354 *
355  CALL zhemm( 'Left', uplo, n, nrhs, negone, a, lda, x, ldx, one,
356  $ work, n )
357 *
358 * Check whether the NRHS normwise backward errors satisfy the
359 * stopping criterion. If yes, set ITER=0 and return.
360 *
361  DO i = 1, nrhs
362  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
363  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
364  IF( rnrm.GT.xnrm*cte )
365  $ GO TO 10
366  END DO
367 *
368 * If we are here, the NRHS normwise backward errors satisfy the
369 * stopping criterion. We are good to exit.
370 *
371  iter = 0
372  RETURN
373 *
374  10 CONTINUE
375 *
376  DO 30 iiter = 1, itermax
377 *
378 * Convert R (in WORK) from double precision to single precision
379 * and store the result in SX.
380 *
381  CALL zlag2c( n, nrhs, work, n, swork( ptsx ), n, info )
382 *
383  IF( info.NE.0 ) THEN
384  iter = -2
385  GO TO 40
386  END IF
387 *
388 * Solve the system SA*SX = SR.
389 *
390  CALL cpotrs( uplo, n, nrhs, swork( ptsa ), n, swork( ptsx ), n,
391  $ info )
392 *
393 * Convert SX back to double precision and update the current
394 * iterate.
395 *
396  CALL clag2z( n, nrhs, swork( ptsx ), n, work, n, info )
397 *
398  DO i = 1, nrhs
399  CALL zaxpy( n, one, work( 1, i ), 1, x( 1, i ), 1 )
400  END DO
401 *
402 * Compute R = B - AX (R is WORK).
403 *
404  CALL zlacpy( 'All', n, nrhs, b, ldb, work, n )
405 *
406  CALL zhemm( 'L', uplo, n, nrhs, negone, a, lda, x, ldx, one,
407  $ work, n )
408 *
409 * Check whether the NRHS normwise backward errors satisfy the
410 * stopping criterion. If yes, set ITER=IITER>0 and return.
411 *
412  DO i = 1, nrhs
413  xnrm = cabs1( x( izamax( n, x( 1, i ), 1 ), i ) )
414  rnrm = cabs1( work( izamax( n, work( 1, i ), 1 ), i ) )
415  IF( rnrm.GT.xnrm*cte )
416  $ GO TO 20
417  END DO
418 *
419 * If we are here, the NRHS normwise backward errors satisfy the
420 * stopping criterion, we are good to exit.
421 *
422  iter = iiter
423 *
424  RETURN
425 *
426  20 CONTINUE
427 *
428  30 CONTINUE
429 *
430 * If we are at this place of the code, this is because we have
431 * performed ITER=ITERMAX iterations and never satisfied the
432 * stopping criterion, set up the ITER flag accordingly and follow
433 * up on double precision routine.
434 *
435  iter = -itermax - 1
436 *
437  40 CONTINUE
438 *
439 * Single-precision iterative refinement failed to converge to a
440 * satisfactory solution, so we resort to double precision.
441 *
442  CALL zpotrf( uplo, n, a, lda, info )
443 *
444  IF( info.NE.0 )
445  $ RETURN
446 *
447  CALL zlacpy( 'All', n, nrhs, b, ldb, x, ldx )
448  CALL zpotrs( uplo, n, nrhs, a, lda, x, ldx, info )
449 *
450  RETURN
451 *
452 * End of ZCPOSV
453 *
454  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zhemm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
ZHEMM
Definition: zhemm.f:191
subroutine zlacpy(UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Definition: zlacpy.f:103
subroutine zlat2c(UPLO, N, A, LDA, SA, LDSA, INFO)
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
Definition: zlat2c.f:111
subroutine zlag2c(M, N, A, LDA, SA, LDSA, INFO)
ZLAG2C converts a complex double precision matrix to a complex single precision matrix.
Definition: zlag2c.f:107
subroutine clag2z(M, N, SA, LDSA, A, LDA, INFO)
CLAG2Z converts a complex single precision matrix to a complex double precision matrix.
Definition: clag2z.f:103
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:110
subroutine zcposv(UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK, RWORK, ITER, INFO)
ZCPOSV computes the solution to system of linear equations A * X = B for PO matrices
Definition: zcposv.f:209
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:107
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:110
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102