LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
zposvxx.f
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1 *> \brief <b> ZPOSVXX 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 ZPOSVXX( 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 * DOUBLE PRECISION RCOND, RPVGRW
31 * ..
32 * .. Array Arguments ..
33 * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
34 * \$ WORK( * ), X( LDX, * )
35 * DOUBLE PRECISION 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 *> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
47 *> to compute the solution to a complex*16 system of linear equations
48 *> A * X = B, where A is an N-by-N Hermitian 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. ZPOSVXX 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 *> ZPOSVXX 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 *> ZPOSVXX 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 ZPOSVXX 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', double precision 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*16 array, dimension (LDA,N)
160 *> On entry, the Hermitian 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*16 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 DOUBLE PRECISION 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*16 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*16 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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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 < 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) * dlamch('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) * dlamch('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) * dlamch('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 <= 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 DOUBLE PRECISION array, dimension NPARAMS
411 *> Specifies algorithm parameters. If an entry is < 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: 1.0D+0
419 *> = 0.0: No refinement is performed, and no error bounds are
420 *> computed.
421 *> = 1.0: Use the extra-precise refinement algorithm.
422 *> (other values are reserved for future use)
423 *>
424 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
425 *> computations allowed for refinement.
426 *> Default: 10
427 *> Aggressive: Set to 100 to permit convergence using approximate
428 *> factorizations or factorizations other than LU. If
429 *> the factorization uses a technique other than
430 *> Gaussian elimination, the guarantees in
431 *> err_bnds_norm and err_bnds_comp may no longer be
432 *> trustworthy.
433 *>
434 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
435 *> will attempt to find a solution with small componentwise
436 *> relative error in the double-precision algorithm. Positive
437 *> is true, 0.0 is false.
438 *> Default: 1.0 (attempt componentwise convergence)
439 *> \endverbatim
440 *>
441 *> \param[out] WORK
442 *> \verbatim
443 *> WORK is COMPLEX*16 array, dimension (2*N)
444 *> \endverbatim
445 *>
446 *> \param[out] RWORK
447 *> \verbatim
448 *> RWORK is DOUBLE PRECISION array, dimension (2*N)
449 *> \endverbatim
450 *>
451 *> \param[out] INFO
452 *> \verbatim
453 *> INFO is INTEGER
454 *> = 0: Successful exit. The solution to every right-hand side is
455 *> guaranteed.
456 *> < 0: If INFO = -i, the i-th argument had an illegal value
457 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
458 *> has been completed, but the factor U is exactly singular, so
459 *> the solution and error bounds could not be computed. RCOND = 0
460 *> is returned.
461 *> = N+J: The solution corresponding to the Jth right-hand side is
462 *> not guaranteed. The solutions corresponding to other right-
463 *> hand sides K with K > J may not be guaranteed as well, but
464 *> only the first such right-hand side is reported. If a small
465 *> componentwise error is not requested (PARAMS(3) = 0.0) then
466 *> the Jth right-hand side is the first with a normwise error
467 *> bound that is not guaranteed (the smallest J such
468 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
469 *> the Jth right-hand side is the first with either a normwise or
470 *> componentwise error bound that is not guaranteed (the smallest
471 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
472 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
473 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
474 *> about all of the right-hand sides check ERR_BNDS_NORM or
475 *> ERR_BNDS_COMP.
476 *> \endverbatim
477 *
478 * Authors:
479 * ========
480 *
481 *> \author Univ. of Tennessee
482 *> \author Univ. of California Berkeley
483 *> \author Univ. of Colorado Denver
484 *> \author NAG Ltd.
485 *
486 *> \ingroup complex16POsolve
487 *
488 * =====================================================================
489  SUBROUTINE zposvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
490  \$ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
491  \$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
492  \$ NPARAMS, PARAMS, WORK, RWORK, INFO )
493 *
494 * -- LAPACK driver routine --
495 * -- LAPACK is a software package provided by Univ. of Tennessee, --
496 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
497 *
498 * .. Scalar Arguments ..
499  CHARACTER EQUED, FACT, UPLO
500  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
501  \$ N_ERR_BNDS
502  DOUBLE PRECISION RCOND, RPVGRW
503 * ..
504 * .. Array Arguments ..
505  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
506  \$ WORK( * ), X( LDX, * )
507  DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
508  \$ err_bnds_norm( nrhs, * ),
509  \$ err_bnds_comp( nrhs, * )
510 * ..
511 *
512 * ==================================================================
513 *
514 * .. Parameters ..
515  DOUBLE PRECISION ZERO, ONE
516  PARAMETER ( ZERO = 0.0d+0, one = 1.0d+0 )
517  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
518  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
519  INTEGER CMP_ERR_I, PIV_GROWTH_I
520  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
521  \$ berr_i = 3 )
522  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
523  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
524  \$ piv_growth_i = 9 )
525 * ..
526 * .. Local Scalars ..
527  LOGICAL EQUIL, NOFACT, RCEQU
528  INTEGER INFEQU, J
529  DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
530 * ..
531 * .. External Functions ..
532  EXTERNAL lsame, dlamch, zla_porpvgrw
533  LOGICAL LSAME
534  DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW
535 * ..
536 * .. External Subroutines ..
537  EXTERNAL zpoequb, zpotrf, zpotrs, zlacpy,
539 * ..
540 * .. Intrinsic Functions ..
541  INTRINSIC max, min
542 * ..
543 * .. Executable Statements ..
544 *
545  info = 0
546  nofact = lsame( fact, 'N' )
547  equil = lsame( fact, 'E' )
548  smlnum = dlamch( 'Safe minimum' )
549  bignum = one / smlnum
550  IF( nofact .OR. equil ) THEN
551  equed = 'N'
552  rcequ = .false.
553  ELSE
554  rcequ = lsame( equed, 'Y' )
555  ENDIF
556 *
557 * Default is failure. If an input parameter is wrong or
558 * factorization fails, make everything look horrible. Only the
559 * pivot growth is set here, the rest is initialized in ZPORFSX.
560 *
561  rpvgrw = zero
562 *
563 * Test the input parameters. PARAMS is not tested until ZPORFSX.
564 *
565  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
566  \$ lsame( fact, 'F' ) ) THEN
567  info = -1
568  ELSE IF( .NOT.lsame( uplo, 'U' ) .AND.
569  \$ .NOT.lsame( uplo, 'L' ) ) THEN
570  info = -2
571  ELSE IF( n.LT.0 ) THEN
572  info = -3
573  ELSE IF( nrhs.LT.0 ) THEN
574  info = -4
575  ELSE IF( lda.LT.max( 1, n ) ) THEN
576  info = -6
577  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
578  info = -8
579  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
580  \$ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
581  info = -9
582  ELSE
583  IF ( rcequ ) THEN
584  smin = bignum
585  smax = zero
586  DO 10 j = 1, n
587  smin = min( smin, s( j ) )
588  smax = max( smax, s( j ) )
589  10 CONTINUE
590  IF( smin.LE.zero ) THEN
591  info = -10
592  ELSE IF( n.GT.0 ) THEN
593  scond = max( smin, smlnum ) / min( smax, bignum )
594  ELSE
595  scond = one
596  END IF
597  END IF
598  IF( info.EQ.0 ) THEN
599  IF( ldb.LT.max( 1, n ) ) THEN
600  info = -12
601  ELSE IF( ldx.LT.max( 1, n ) ) THEN
602  info = -14
603  END IF
604  END IF
605  END IF
606 *
607  IF( info.NE.0 ) THEN
608  CALL xerbla( 'ZPOSVXX', -info )
609  RETURN
610  END IF
611 *
612  IF( equil ) THEN
613 *
614 * Compute row and column scalings to equilibrate the matrix A.
615 *
616  CALL zpoequb( n, a, lda, s, scond, amax, infequ )
617  IF( infequ.EQ.0 ) THEN
618 *
619 * Equilibrate the matrix.
620 *
621  CALL zlaqhe( uplo, n, a, lda, s, scond, amax, equed )
622  rcequ = lsame( equed, 'Y' )
623  END IF
624  END IF
625 *
626 * Scale the right-hand side.
627 *
628  IF( rcequ ) CALL zlascl2( n, nrhs, s, b, ldb )
629 *
630  IF( nofact .OR. equil ) THEN
631 *
632 * Compute the Cholesky factorization of A.
633 *
634  CALL zlacpy( uplo, n, n, a, lda, af, ldaf )
635  CALL zpotrf( uplo, n, af, ldaf, info )
636 *
637 * Return if INFO is non-zero.
638 *
639  IF( info.GT.0 ) THEN
640 *
641 * Pivot in column INFO is exactly 0
642 * Compute the reciprocal pivot growth factor of the
643 * leading rank-deficient INFO columns of A.
644 *
645  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
646  RETURN
647  END IF
648  END IF
649 *
650 * Compute the reciprocal pivot growth factor RPVGRW.
651 *
652  rpvgrw = zla_porpvgrw( uplo, n, a, lda, af, ldaf, rwork )
653 *
654 * Compute the solution matrix X.
655 *
656  CALL zlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
657  CALL zpotrs( uplo, n, nrhs, af, ldaf, x, ldx, info )
658 *
659 * Use iterative refinement to improve the computed solution and
660 * compute error bounds and backward error estimates for it.
661 *
662  CALL zporfsx( uplo, equed, n, nrhs, a, lda, af, ldaf,
663  \$ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
664  \$ err_bnds_comp, nparams, params, work, rwork, info )
665
666 *
667 * Scale solutions.
668 *
669  IF ( rcequ ) THEN
670  CALL zlascl2( n, nrhs, s, x, ldx )
671  END IF
672 *
673  RETURN
674 *
675 * End of ZPOSVXX
676 *
677  END
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine zlaqhe(UPLO, N, A, LDA, S, SCOND, AMAX, EQUED)
ZLAQHE scales a Hermitian matrix.
Definition: zlaqhe.f:134
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 zlascl2(M, N, D, X, LDX)
ZLASCL2 performs diagonal scaling on a vector.
Definition: zlascl2.f:91
subroutine zporfsx(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)
ZPORFSX
Definition: zporfsx.f:393
double precision function zla_porpvgrw(UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
ZLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian...
Definition: zla_porpvgrw.f:107
subroutine zpoequb(N, A, LDA, S, SCOND, AMAX, INFO)
ZPOEQUB
Definition: zpoequb.f:119
subroutine zpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
ZPOTRS
Definition: zpotrs.f:110
subroutine zposvxx(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)
ZPOSVXX computes the solution to system of linear equations A * X = B for PO matrices
Definition: zposvxx.f:493
subroutine zpotrf(UPLO, N, A, LDA, INFO)
ZPOTRF VARIANT: right looking block version of the algorithm, calling Level 3 BLAS.
Definition: zpotrf.f:102