LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
cposvxx.f
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1 *> \brief <b> CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices</b>
2 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
22 * S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
23 * N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
24 * NPARAMS, PARAMS, WORK, RWORK, INFO )
25 *
26 * .. Scalar Arguments ..
27 * CHARACTER EQUED, FACT, UPLO
28 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
29 * \$ N_ERR_BNDS
30 * REAL RCOND, RPVGRW
31 * ..
32 * .. Array Arguments ..
33 * COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
34 * \$ WORK( * ), X( LDX, * )
35 * REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
36 * \$ ERR_BNDS_NORM( NRHS, * ),
37 * \$ ERR_BNDS_COMP( NRHS, * )
38 * ..
39 *
40 *
41 *> \par Purpose:
42 * =============
43 *>
44 *> \verbatim
45 *>
46 *> CPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
47 *> to compute the solution to a complex system of linear equations
48 *> A * X = B, where A is an N-by-N symmetric positive definite matrix
49 *> and X and B are N-by-NRHS matrices.
50 *>
51 *> If requested, both normwise and maximum componentwise error bounds
52 *> are returned. CPOSVXX will return a solution with a tiny
53 *> guaranteed error (O(eps) where eps is the working machine
54 *> precision) unless the matrix is very ill-conditioned, in which
55 *> case a warning is returned. Relevant condition numbers also are
56 *> calculated and returned.
57 *>
58 *> CPOSVXX accepts user-provided factorizations and equilibration
59 *> factors; see the definitions of the FACT and EQUED options.
60 *> Solving with refinement and using a factorization from a previous
61 *> CPOSVXX call will also produce a solution with either O(eps)
62 *> errors or warnings, but we cannot make that claim for general
63 *> user-provided factorizations and equilibration factors if they
64 *> differ from what CPOSVXX would itself produce.
65 *> \endverbatim
66 *
67 *> \par Description:
68 * =================
69 *>
70 *> \verbatim
71 *>
72 *> The following steps are performed:
73 *>
74 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate
75 *> the system:
76 *>
77 *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
78 *>
79 *> Whether or not the system will be equilibrated depends on the
80 *> scaling of the matrix A, but if equilibration is used, A is
81 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
82 *>
83 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
84 *> factor the matrix A (after equilibration if FACT = 'E') as
85 *> A = U**T* U, if UPLO = 'U', or
86 *> A = L * L**T, if UPLO = 'L',
87 *> where U is an upper triangular matrix and L is a lower triangular
88 *> matrix.
89 *>
90 *> 3. If the leading i-by-i principal minor is not positive definite,
91 *> then the routine returns with INFO = i. Otherwise, the factored
92 *> form of A is used to estimate the condition number of the matrix
93 *> A (see argument RCOND). If the reciprocal of the condition number
94 *> is less than machine precision, the routine still goes on to solve
95 *> for X and compute error bounds as described below.
96 *>
97 *> 4. The system of equations is solved for X using the factored form
98 *> of A.
99 *>
100 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
101 *> the routine will use iterative refinement to try to get a small
102 *> error and error bounds. Refinement calculates the residual to at
103 *> least twice the working precision.
104 *>
105 *> 6. If equilibration was used, the matrix X is premultiplied by
106 *> diag(S) so that it solves the original system before
107 *> equilibration.
108 *> \endverbatim
109 *
110 * Arguments:
111 * ==========
112 *
113 *> \verbatim
114 *> Some optional parameters are bundled in the PARAMS array. These
115 *> settings determine how refinement is performed, but often the
116 *> defaults are acceptable. If the defaults are acceptable, users
117 *> can pass NPARAMS = 0 which prevents the source code from accessing
118 *> the PARAMS argument.
119 *> \endverbatim
120 *>
121 *> \param[in] FACT
122 *> \verbatim
123 *> FACT is CHARACTER*1
124 *> Specifies whether or not the factored form of the matrix A is
125 *> supplied on entry, and if not, whether the matrix A should be
126 *> equilibrated before it is factored.
127 *> = 'F': On entry, AF contains the factored form of A.
128 *> If EQUED is not 'N', the matrix A has been
129 *> equilibrated with scaling factors given by S.
130 *> A and AF are not modified.
131 *> = 'N': The matrix A will be copied to AF and factored.
132 *> = 'E': The matrix A will be equilibrated if necessary, then
133 *> copied to AF and factored.
134 *> \endverbatim
135 *>
136 *> \param[in] UPLO
137 *> \verbatim
138 *> UPLO is CHARACTER*1
139 *> = 'U': Upper triangle of A is stored;
140 *> = 'L': Lower triangle of A is stored.
141 *> \endverbatim
142 *>
143 *> \param[in] N
144 *> \verbatim
145 *> N is INTEGER
146 *> The number of linear equations, i.e., the order of the
147 *> matrix A. N >= 0.
148 *> \endverbatim
149 *>
150 *> \param[in] NRHS
151 *> \verbatim
152 *> NRHS is INTEGER
153 *> The number of right hand sides, i.e., the number of columns
154 *> of the matrices B and X. NRHS >= 0.
155 *> \endverbatim
156 *>
157 *> \param[in,out] A
158 *> \verbatim
159 *> A is COMPLEX array, dimension (LDA,N)
160 *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
161 *> 'Y', then A must contain the equilibrated matrix
162 *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
163 *> triangular part of A contains the upper triangular part of the
164 *> matrix A, and the strictly lower triangular part of A is not
165 *> referenced. If UPLO = 'L', the leading N-by-N lower triangular
166 *> part of A contains the lower triangular part of the matrix A, and
167 *> the strictly upper triangular part of A is not referenced. A is
168 *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
169 *> 'N' on exit.
170 *>
171 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
172 *> diag(S)*A*diag(S).
173 *> \endverbatim
174 *>
175 *> \param[in] LDA
176 *> \verbatim
177 *> LDA is INTEGER
178 *> The leading dimension of the array A. LDA >= max(1,N).
179 *> \endverbatim
180 *>
181 *> \param[in,out] AF
182 *> \verbatim
183 *> AF is COMPLEX array, dimension (LDAF,N)
184 *> If FACT = 'F', then AF is an input argument and on entry
185 *> contains the triangular factor U or L from the Cholesky
186 *> factorization A = U**T*U or A = L*L**T, in the same storage
187 *> format as A. If EQUED .ne. 'N', then AF is the factored
188 *> form of the equilibrated matrix diag(S)*A*diag(S).
189 *>
190 *> If FACT = 'N', then AF is an output argument and on exit
191 *> returns the triangular factor U or L from the Cholesky
192 *> factorization A = U**T*U or A = L*L**T of the original
193 *> matrix A.
194 *>
195 *> If FACT = 'E', then AF is an output argument and on exit
196 *> returns the triangular factor U or L from the Cholesky
197 *> factorization A = U**T*U or A = L*L**T of the equilibrated
198 *> matrix A (see the description of A for the form of the
199 *> equilibrated matrix).
200 *> \endverbatim
201 *>
202 *> \param[in] LDAF
203 *> \verbatim
204 *> LDAF is INTEGER
205 *> The leading dimension of the array AF. LDAF >= max(1,N).
206 *> \endverbatim
207 *>
208 *> \param[in,out] EQUED
209 *> \verbatim
210 *> EQUED is CHARACTER*1
211 *> Specifies the form of equilibration that was done.
212 *> = 'N': No equilibration (always true if FACT = 'N').
213 *> = 'Y': Both row and column equilibration, i.e., A has been
214 *> replaced by diag(S) * A * diag(S).
215 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
216 *> output argument.
217 *> \endverbatim
218 *>
219 *> \param[in,out] S
220 *> \verbatim
221 *> S is REAL array, dimension (N)
222 *> The row scale factors for A. If EQUED = 'Y', A is multiplied on
223 *> the left and right by diag(S). S is an input argument if FACT =
224 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
225 *> = 'Y', each element of S must be positive. If S is output, each
226 *> element of S is a power of the radix. If S is input, each element
227 *> of S should be a power of the radix to ensure a reliable solution
228 *> and error estimates. Scaling by powers of the radix does not cause
229 *> rounding errors unless the result underflows or overflows.
230 *> Rounding errors during scaling lead to refining with a matrix that
231 *> is not equivalent to the input matrix, producing error estimates
232 *> that may not be reliable.
233 *> \endverbatim
234 *>
235 *> \param[in,out] B
236 *> \verbatim
237 *> B is COMPLEX array, dimension (LDB,NRHS)
238 *> On entry, the N-by-NRHS right hand side matrix B.
239 *> On exit,
240 *> if EQUED = 'N', B is not modified;
241 *> if EQUED = 'Y', B is overwritten by diag(S)*B;
242 *> \endverbatim
243 *>
244 *> \param[in] LDB
245 *> \verbatim
246 *> LDB is INTEGER
247 *> The leading dimension of the array B. LDB >= max(1,N).
248 *> \endverbatim
249 *>
250 *> \param[out] X
251 *> \verbatim
252 *> X is COMPLEX array, dimension (LDX,NRHS)
253 *> If INFO = 0, the N-by-NRHS solution matrix X to the original
254 *> system of equations. Note that A and B are modified on exit if
255 *> EQUED .ne. 'N', and the solution to the equilibrated system is
256 *> inv(diag(S))*X.
257 *> \endverbatim
258 *>
259 *> \param[in] LDX
260 *> \verbatim
261 *> LDX is INTEGER
262 *> The leading dimension of the array X. LDX >= max(1,N).
263 *> \endverbatim
264 *>
265 *> \param[out] RCOND
266 *> \verbatim
267 *> RCOND is REAL
268 *> Reciprocal scaled condition number. This is an estimate of the
269 *> reciprocal Skeel condition number of the matrix A after
270 *> equilibration (if done). If this is less than the machine
271 *> precision (in particular, if it is zero), the matrix is singular
272 *> to working precision. Note that the error may still be small even
273 *> if this number is very small and the matrix appears ill-
274 *> conditioned.
275 *> \endverbatim
276 *>
277 *> \param[out] RPVGRW
278 *> \verbatim
279 *> RPVGRW is REAL
280 *> Reciprocal pivot growth. On exit, this contains the reciprocal
281 *> pivot growth factor norm(A)/norm(U). The "max absolute element"
282 *> norm is used. If this is much less than 1, then the stability of
283 *> the LU factorization of the (equilibrated) matrix A could be poor.
284 *> This also means that the solution X, estimated condition numbers,
285 *> and error bounds could be unreliable. If factorization fails with
286 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
287 *> for the leading INFO columns of A.
288 *> \endverbatim
289 *>
290 *> \param[out] BERR
291 *> \verbatim
292 *> BERR is REAL array, dimension (NRHS)
293 *> Componentwise relative backward error. This is the
294 *> componentwise relative backward error of each solution vector X(j)
295 *> (i.e., the smallest relative change in any element of A or B that
296 *> makes X(j) an exact solution).
297 *> \endverbatim
298 *>
299 *> \param[in] N_ERR_BNDS
300 *> \verbatim
301 *> N_ERR_BNDS is INTEGER
302 *> Number of error bounds to return for each right hand side
303 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
304 *> ERR_BNDS_COMP below.
305 *> \endverbatim
306 *>
307 *> \param[out] ERR_BNDS_NORM
308 *> \verbatim
309 *> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
310 *> For each right-hand side, this array contains information about
311 *> various error bounds and condition numbers corresponding to the
312 *> normwise relative error, which is defined as follows:
313 *>
314 *> Normwise relative error in the ith solution vector:
315 *> max_j (abs(XTRUE(j,i) - X(j,i)))
316 *> ------------------------------
317 *> max_j abs(X(j,i))
318 *>
319 *> The array is indexed by the type of error information as described
320 *> below. There currently are up to three pieces of information
321 *> returned.
322 *>
323 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
324 *> right-hand side.
325 *>
326 *> The second index in ERR_BNDS_NORM(:,err) contains the following
327 *> three fields:
328 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
329 *> reciprocal condition number is less than the threshold
330 *> sqrt(n) * slamch('Epsilon').
331 *>
332 *> err = 2 "Guaranteed" error bound: The estimated forward error,
333 *> almost certainly within a factor of 10 of the true error
334 *> so long as the next entry is greater than the threshold
335 *> sqrt(n) * slamch('Epsilon'). This error bound should only
336 *> be trusted if the previous boolean is true.
337 *>
338 *> err = 3 Reciprocal condition number: Estimated normwise
339 *> reciprocal condition number. Compared with the threshold
340 *> sqrt(n) * slamch('Epsilon') to determine if the error
341 *> estimate is "guaranteed". These reciprocal condition
342 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
343 *> appropriately scaled matrix Z.
344 *> Let Z = S*A, where S scales each row by a power of the
345 *> radix so all absolute row sums of Z are approximately 1.
346 *>
347 *> See Lapack Working Note 165 for further details and extra
348 *> cautions.
349 *> \endverbatim
350 *>
351 *> \param[out] ERR_BNDS_COMP
352 *> \verbatim
353 *> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
354 *> For each right-hand side, this array contains information about
355 *> various error bounds and condition numbers corresponding to the
356 *> componentwise relative error, which is defined as follows:
357 *>
358 *> Componentwise relative error in the ith solution vector:
359 *> abs(XTRUE(j,i) - X(j,i))
360 *> max_j ----------------------
361 *> abs(X(j,i))
362 *>
363 *> The array is indexed by the right-hand side i (on which the
364 *> componentwise relative error depends), and the type of error
365 *> information as described below. There currently are up to three
366 *> pieces of information returned for each right-hand side. If
367 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
368 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
369 *> the first (:,N_ERR_BNDS) entries are returned.
370 *>
371 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
372 *> right-hand side.
373 *>
374 *> The second index in ERR_BNDS_COMP(:,err) contains the following
375 *> three fields:
376 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
377 *> reciprocal condition number is less than the threshold
378 *> sqrt(n) * slamch('Epsilon').
379 *>
380 *> err = 2 "Guaranteed" error bound: The estimated forward error,
381 *> almost certainly within a factor of 10 of the true error
382 *> so long as the next entry is greater than the threshold
383 *> sqrt(n) * slamch('Epsilon'). This error bound should only
384 *> be trusted if the previous boolean is true.
385 *>
386 *> err = 3 Reciprocal condition number: Estimated componentwise
387 *> reciprocal condition number. Compared with the threshold
388 *> sqrt(n) * slamch('Epsilon') to determine if the error
389 *> estimate is "guaranteed". These reciprocal condition
390 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
391 *> appropriately scaled matrix Z.
392 *> Let Z = S*(A*diag(x)), where x is the solution for the
393 *> current right-hand side and S scales each row of
394 *> A*diag(x) by a power of the radix so all absolute row
395 *> sums of Z are approximately 1.
396 *>
397 *> See Lapack Working Note 165 for further details and extra
398 *> cautions.
399 *> \endverbatim
400 *>
401 *> \param[in] NPARAMS
402 *> \verbatim
403 *> NPARAMS is INTEGER
404 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
405 *> PARAMS array is never referenced and default values are used.
406 *> \endverbatim
407 *>
408 *> \param[in,out] PARAMS
409 *> \verbatim
410 *> PARAMS is REAL array, dimension NPARAMS
411 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
412 *> that entry will be filled with default value used for that
413 *> parameter. Only positions up to NPARAMS are accessed; defaults
414 *> are used for higher-numbered parameters.
415 *>
416 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
417 *> refinement or not.
418 *> Default: 1.0
419 *> = 0.0 : No refinement is performed, and no error bounds are
420 *> computed.
421 *> = 1.0 : Use the double-precision refinement algorithm,
422 *> possibly with doubled-single computations if the
423 *> compilation environment does not support DOUBLE
424 *> PRECISION.
425 *> (other values are reserved for future use)
426 *>
427 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
428 *> computations allowed for refinement.
429 *> Default: 10
430 *> Aggressive: Set to 100 to permit convergence using approximate
431 *> factorizations or factorizations other than LU. If
432 *> the factorization uses a technique other than
433 *> Gaussian elimination, the guarantees in
434 *> err_bnds_norm and err_bnds_comp may no longer be
435 *> trustworthy.
436 *>
437 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
438 *> will attempt to find a solution with small componentwise
439 *> relative error in the double-precision algorithm. Positive
440 *> is true, 0.0 is false.
441 *> Default: 1.0 (attempt componentwise convergence)
442 *> \endverbatim
443 *>
444 *> \param[out] WORK
445 *> \verbatim
446 *> WORK is COMPLEX array, dimension (2*N)
447 *> \endverbatim
448 *>
449 *> \param[out] RWORK
450 *> \verbatim
451 *> RWORK is REAL array, dimension (2*N)
452 *> \endverbatim
453 *>
454 *> \param[out] INFO
455 *> \verbatim
456 *> INFO is INTEGER
457 *> = 0: Successful exit. The solution to every right-hand side is
458 *> guaranteed.
459 *> < 0: If INFO = -i, the i-th argument had an illegal value
460 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
461 *> has been completed, but the factor U is exactly singular, so
462 *> the solution and error bounds could not be computed. RCOND = 0
463 *> is returned.
464 *> = N+J: The solution corresponding to the Jth right-hand side is
465 *> not guaranteed. The solutions corresponding to other right-
466 *> hand sides K with K > J may not be guaranteed as well, but
467 *> only the first such right-hand side is reported. If a small
468 *> componentwise error is not requested (PARAMS(3) = 0.0) then
469 *> the Jth right-hand side is the first with a normwise error
470 *> bound that is not guaranteed (the smallest J such
471 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
472 *> the Jth right-hand side is the first with either a normwise or
473 *> componentwise error bound that is not guaranteed (the smallest
474 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
475 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
476 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
477 *> about all of the right-hand sides check ERR_BNDS_NORM or
478 *> ERR_BNDS_COMP.
479 *> \endverbatim
480 *
481 * Authors:
482 * ========
483 *
484 *> \author Univ. of Tennessee
485 *> \author Univ. of California Berkeley
486 *> \author Univ. of Colorado Denver
487 *> \author NAG Ltd.
488 *
489 *> \date April 2012
490 *
491 *> \ingroup complexPOsolve
492 *
493 * =====================================================================
494  SUBROUTINE cposvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
495  \$ s, b, ldb, x, ldx, rcond, rpvgrw, berr,
496  \$ n_err_bnds, err_bnds_norm, err_bnds_comp,
497  \$ nparams, params, work, rwork, info )
498 *
499 * -- LAPACK driver routine (version 3.4.1) --
500 * -- LAPACK is a software package provided by Univ. of Tennessee, --
501 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
502 * April 2012
503 *
504 * .. Scalar Arguments ..
505  CHARACTER EQUED, FACT, UPLO
506  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
507  \$ n_err_bnds
508  REAL RCOND, RPVGRW
509 * ..
510 * .. Array Arguments ..
511  COMPLEX A( lda, * ), AF( ldaf, * ), B( ldb, * ),
512  \$ work( * ), x( ldx, * )
513  REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
514  \$ err_bnds_norm( nrhs, * ),
515  \$ err_bnds_comp( nrhs, * )
516 * ..
517 *
518 * ==================================================================
519 *
520 * .. Parameters ..
521  REAL ZERO, ONE
522  parameter ( zero = 0.0e+0, one = 1.0e+0 )
523  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
524  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
525  INTEGER CMP_ERR_I, PIV_GROWTH_I
526  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
527  \$ berr_i = 3 )
528  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
529  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
530  \$ piv_growth_i = 9 )
531 * ..
532 * .. Local Scalars ..
533  LOGICAL EQUIL, NOFACT, RCEQU
534  INTEGER INFEQU, J
535  REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
536 * ..
537 * .. External Functions ..
538  EXTERNAL lsame, slamch, cla_porpvgrw
539  LOGICAL LSAME
540  REAL SLAMCH, CLA_PORPVGRW
541 * ..
542 * .. External Subroutines ..
543  EXTERNAL cpocon, cpoequb, cpotrf, cpotrs, clacpy,
545 * ..
546 * .. Intrinsic Functions ..
547  INTRINSIC max, min
548 * ..
549 * .. Executable Statements ..
550 *
551  info = 0
552  nofact = lsame( fact, 'N' )
553  equil = lsame( fact, 'E' )
554  smlnum = slamch( 'Safe minimum' )
555  bignum = one / smlnum
556  IF( nofact .OR. equil ) THEN
557  equed = 'N'
558  rcequ = .false.
559  ELSE
560  rcequ = lsame( equed, 'Y' )
561  ENDIF
562 *
563 * Default is failure. If an input parameter is wrong or
564 * factorization fails, make everything look horrible. Only the
565 * pivot growth is set here, the rest is initialized in CPORFSX.
566 *
567  rpvgrw = zero
568 *
569 * Test the input parameters. PARAMS is not tested until CPORFSX.
570 *
571  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
572  \$ lsame( fact, 'F' ) ) THEN
573  info = -1
574  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
575  \$ .NOT.lsame( uplo, 'L' ) ) THEN
576  info = -2
577  ELSE IF( n.LT.0 ) THEN
578  info = -3
579  ELSE IF( nrhs.LT.0 ) THEN
580  info = -4
581  ELSE IF( lda.LT.max( 1, n ) ) THEN
582  info = -6
583  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
584  info = -8
585  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
586  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
587  info = -9
588  ELSE
589  IF ( rcequ ) THEN
590  smin = bignum
591  smax = zero
592  DO 10 j = 1, n
593  smin = min( smin, s( j ) )
594  smax = max( smax, s( j ) )
595  10 CONTINUE
596  IF( smin.LE.zero ) THEN
597  info = -10
598  ELSE IF( n.GT.0 ) THEN
599  scond = max( smin, smlnum ) / min( smax, bignum )
600  ELSE
601  scond = one
602  END IF
603  END IF
604  IF( info.EQ.0 ) THEN
605  IF( ldb.LT.max( 1, n ) ) THEN
606  info = -12
607  ELSE IF( ldx.LT.max( 1, n ) ) THEN
608  info = -14
609  END IF
610  END IF
611  END IF
612 *
613  IF( info.NE.0 ) THEN
614  CALL xerbla( 'CPOSVXX', -info )
615  RETURN
616  END IF
617 *
618  IF( equil ) THEN
619 *
620 * Compute row and column scalings to equilibrate the matrix A.
621 *
622  CALL cpoequb( n, a, lda, s, scond, amax, infequ )
623  IF( infequ.EQ.0 ) THEN
624 *
625 * Equilibrate the matrix.
626 *
627  CALL claqhe( uplo, n, a, lda, s, scond, amax, equed )
628  rcequ = lsame( equed, 'Y' )
629  END IF
630  END IF
631 *
632 * Scale the right-hand side.
633 *
634  IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
635 *
636  IF( nofact .OR. equil ) THEN
637 *
638 * Compute the Cholesky factorization of A.
639 *
640  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
641  CALL cpotrf( uplo, n, af, ldaf, info )
642 *
643 * Return if INFO is non-zero.
644 *
645  IF( info.GT.0 ) THEN
646 *
647 * Pivot in column INFO is exactly 0
648 * Compute the reciprocal pivot growth factor of the
649 * leading rank-deficient INFO columns of A.
650 *
651  rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
652  RETURN
653  END IF
654  END IF
655 *
656 * Compute the reciprocal pivot growth factor RPVGRW.
657 *
658  rpvgrw = cla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
659 *
660 * Compute the solution matrix X.
661 *
662  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
663  CALL cpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
664 *
665 * Use iterative refinement to improve the computed solution and
666 * compute error bounds and backward error estimates for it.
667 *
668  CALL cporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
669  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
670  \$ err_bnds_comp, nparams, params, work, rwork, info )
671
672 *
673 * Scale solutions.
674 *
675  IF ( rcequ ) THEN
676  CALL clascl2( n, nrhs, s, x, ldx )
677  END IF
678 *
679  RETURN
680 *
681 * End of CPOSVXX
682 *
683  END
subroutine cposvxx(FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CPOSVXX computes the solution to system of linear equations A * X = B for PO matrices ...
Definition: cposvxx.f:498
subroutine claqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
CLAQHE scales a Hermitian matrix.
Definition: claqhe.f:136
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine cpocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, RWORK, INFO)
CPOCON
Definition: cpocon.f:123
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:112
subroutine clascl2(M, N, D, X, LDX)
CLASCL2 performs diagonal scaling on a vector.
Definition: clascl2.f:93
subroutine cpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
CPOEQUB
Definition: cpoequb.f:115
subroutine cpotrf(UPLO, N, A, LDA, INFO)
CPOTRF
Definition: cpotrf.f:109
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:105
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
real function cla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
CLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: cla_porpvgrw.f:107
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine cporfsx(UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO)
CPORFSX
Definition: cporfsx.f:395