LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
dposvxx.f
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1 *> \brief <b> DPOSVXX 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 DPOSVXX( 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, IWORK, 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 * INTEGER IWORK( * )
34 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35 * \$ X( LDX, * ), WORK( * )
36 * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ),
37 * \$ ERR_BNDS_NORM( NRHS, * ),
38 * \$ ERR_BNDS_COMP( NRHS, * )
39 * ..
40 *
41 *
42 *> \par Purpose:
43 * =============
44 *>
45 *> \verbatim
46 *>
47 *> DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
48 *> to compute the solution to a double precision system of linear equations
49 *> A * X = B, where A is an N-by-N symmetric positive definite matrix
50 *> and X and B are N-by-NRHS matrices.
51 *>
52 *> If requested, both normwise and maximum componentwise error bounds
53 *> are returned. DPOSVXX will return a solution with a tiny
54 *> guaranteed error (O(eps) where eps is the working machine
55 *> precision) unless the matrix is very ill-conditioned, in which
56 *> case a warning is returned. Relevant condition numbers also are
57 *> calculated and returned.
58 *>
59 *> DPOSVXX accepts user-provided factorizations and equilibration
60 *> factors; see the definitions of the FACT and EQUED options.
61 *> Solving with refinement and using a factorization from a previous
62 *> DPOSVXX call will also produce a solution with either O(eps)
63 *> errors or warnings, but we cannot make that claim for general
64 *> user-provided factorizations and equilibration factors if they
65 *> differ from what DPOSVXX would itself produce.
66 *> \endverbatim
67 *
68 *> \par Description:
69 * =================
70 *>
71 *> \verbatim
72 *>
73 *> The following steps are performed:
74 *>
75 *> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate
76 *> the system:
77 *>
78 *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
79 *>
80 *> Whether or not the system will be equilibrated depends on the
81 *> scaling of the matrix A, but if equilibration is used, A is
82 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
83 *>
84 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
85 *> factor the matrix A (after equilibration if FACT = 'E') as
86 *> A = U**T* U, if UPLO = 'U', or
87 *> A = L * L**T, if UPLO = 'L',
88 *> where U is an upper triangular matrix and L is a lower triangular
89 *> matrix.
90 *>
91 *> 3. If the leading i-by-i principal minor is not positive definite,
92 *> then the routine returns with INFO = i. Otherwise, the factored
93 *> form of A is used to estimate the condition number of the matrix
94 *> A (see argument RCOND). If the reciprocal of the condition number
95 *> is less than machine precision, the routine still goes on to solve
96 *> for X and compute error bounds as described below.
97 *>
98 *> 4. The system of equations is solved for X using the factored form
99 *> of A.
100 *>
101 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
102 *> the routine will use iterative refinement to try to get a small
103 *> error and error bounds. Refinement calculates the residual to at
104 *> least twice the working precision.
105 *>
106 *> 6. If equilibration was used, the matrix X is premultiplied by
107 *> diag(S) so that it solves the original system before
108 *> equilibration.
109 *> \endverbatim
110 *
111 * Arguments:
112 * ==========
113 *
114 *> \verbatim
115 *> Some optional parameters are bundled in the PARAMS array. These
116 *> settings determine how refinement is performed, but often the
117 *> defaults are acceptable. If the defaults are acceptable, users
118 *> can pass NPARAMS = 0 which prevents the source code from accessing
119 *> the PARAMS argument.
120 *> \endverbatim
121 *>
122 *> \param[in] FACT
123 *> \verbatim
124 *> FACT is CHARACTER*1
125 *> Specifies whether or not the factored form of the matrix A is
126 *> supplied on entry, and if not, whether the matrix A should be
127 *> equilibrated before it is factored.
128 *> = 'F': On entry, AF contains the factored form of A.
129 *> If EQUED is not 'N', the matrix A has been
130 *> equilibrated with scaling factors given by S.
131 *> A and AF are not modified.
132 *> = 'N': The matrix A will be copied to AF and factored.
133 *> = 'E': The matrix A will be equilibrated if necessary, then
134 *> copied to AF and factored.
135 *> \endverbatim
136 *>
137 *> \param[in] UPLO
138 *> \verbatim
139 *> UPLO is CHARACTER*1
140 *> = 'U': Upper triangle of A is stored;
141 *> = 'L': Lower triangle of A is stored.
142 *> \endverbatim
143 *>
144 *> \param[in] N
145 *> \verbatim
146 *> N is INTEGER
147 *> The number of linear equations, i.e., the order of the
148 *> matrix A. N >= 0.
149 *> \endverbatim
150 *>
151 *> \param[in] NRHS
152 *> \verbatim
153 *> NRHS is INTEGER
154 *> The number of right hand sides, i.e., the number of columns
155 *> of the matrices B and X. NRHS >= 0.
156 *> \endverbatim
157 *>
158 *> \param[in,out] A
159 *> \verbatim
160 *> A is DOUBLE PRECISION array, dimension (LDA,N)
161 *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
162 *> 'Y', then A must contain the equilibrated matrix
163 *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
164 *> triangular part of A contains the upper triangular part of the
165 *> matrix A, and the strictly lower triangular part of A is not
166 *> referenced. If UPLO = 'L', the leading N-by-N lower triangular
167 *> part of A contains the lower triangular part of the matrix A, and
168 *> the strictly upper triangular part of A is not referenced. A is
169 *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
170 *> 'N' on exit.
171 *>
172 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
173 *> diag(S)*A*diag(S).
174 *> \endverbatim
175 *>
176 *> \param[in] LDA
177 *> \verbatim
178 *> LDA is INTEGER
179 *> The leading dimension of the array A. LDA >= max(1,N).
180 *> \endverbatim
181 *>
182 *> \param[in,out] AF
183 *> \verbatim
184 *> AF is DOUBLE PRECISION array, dimension (LDAF,N)
185 *> If FACT = 'F', then AF is an input argument and on entry
186 *> contains the triangular factor U or L from the Cholesky
187 *> factorization A = U**T*U or A = L*L**T, in the same storage
188 *> format as A. If EQUED .ne. 'N', then AF is the factored
189 *> form of the equilibrated matrix diag(S)*A*diag(S).
190 *>
191 *> If FACT = 'N', then AF is an output argument and on exit
192 *> returns the triangular factor U or L from the Cholesky
193 *> factorization A = U**T*U or A = L*L**T of the original
194 *> matrix A.
195 *>
196 *> If FACT = 'E', then AF is an output argument and on exit
197 *> returns the triangular factor U or L from the Cholesky
198 *> factorization A = U**T*U or A = L*L**T of the equilibrated
199 *> matrix A (see the description of A for the form of the
200 *> equilibrated matrix).
201 *> \endverbatim
202 *>
203 *> \param[in] LDAF
204 *> \verbatim
205 *> LDAF is INTEGER
206 *> The leading dimension of the array AF. LDAF >= max(1,N).
207 *> \endverbatim
208 *>
209 *> \param[in,out] EQUED
210 *> \verbatim
211 *> EQUED is CHARACTER*1
212 *> Specifies the form of equilibration that was done.
213 *> = 'N': No equilibration (always true if FACT = 'N').
214 *> = 'Y': Both row and column equilibration, i.e., A has been
215 *> replaced by diag(S) * A * diag(S).
216 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
217 *> output argument.
218 *> \endverbatim
219 *>
220 *> \param[in,out] S
221 *> \verbatim
222 *> S is DOUBLE PRECISION array, dimension (N)
223 *> The row scale factors for A. If EQUED = 'Y', A is multiplied on
224 *> the left and right by diag(S). S is an input argument if FACT =
225 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
226 *> = 'Y', each element of S must be positive. If S is output, each
227 *> element of S is a power of the radix. If S is input, each element
228 *> of S should be a power of the radix to ensure a reliable solution
229 *> and error estimates. Scaling by powers of the radix does not cause
230 *> rounding errors unless the result underflows or overflows.
231 *> Rounding errors during scaling lead to refining with a matrix that
232 *> is not equivalent to the input matrix, producing error estimates
233 *> that may not be reliable.
234 *> \endverbatim
235 *>
236 *> \param[in,out] B
237 *> \verbatim
238 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
239 *> On entry, the N-by-NRHS right hand side matrix B.
240 *> On exit,
241 *> if EQUED = 'N', B is not modified;
242 *> if EQUED = 'Y', B is overwritten by diag(S)*B;
243 *> \endverbatim
244 *>
245 *> \param[in] LDB
246 *> \verbatim
247 *> LDB is INTEGER
248 *> The leading dimension of the array B. LDB >= max(1,N).
249 *> \endverbatim
250 *>
251 *> \param[out] X
252 *> \verbatim
253 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
254 *> If INFO = 0, the N-by-NRHS solution matrix X to the original
255 *> system of equations. Note that A and B are modified on exit if
256 *> EQUED .ne. 'N', and the solution to the equilibrated system is
257 *> inv(diag(S))*X.
258 *> \endverbatim
259 *>
260 *> \param[in] LDX
261 *> \verbatim
262 *> LDX is INTEGER
263 *> The leading dimension of the array X. LDX >= max(1,N).
264 *> \endverbatim
265 *>
266 *> \param[out] RCOND
267 *> \verbatim
268 *> RCOND is DOUBLE PRECISION
269 *> Reciprocal scaled condition number. This is an estimate of the
270 *> reciprocal Skeel condition number of the matrix A after
271 *> equilibration (if done). If this is less than the machine
272 *> precision (in particular, if it is zero), the matrix is singular
273 *> to working precision. Note that the error may still be small even
274 *> if this number is very small and the matrix appears ill-
275 *> conditioned.
276 *> \endverbatim
277 *>
278 *> \param[out] RPVGRW
279 *> \verbatim
280 *> RPVGRW is DOUBLE PRECISION
281 *> Reciprocal pivot growth. On exit, this contains the reciprocal
282 *> pivot growth factor norm(A)/norm(U). The "max absolute element"
283 *> norm is used. If this is much less than 1, then the stability of
284 *> the LU factorization of the (equilibrated) matrix A could be poor.
285 *> This also means that the solution X, estimated condition numbers,
286 *> and error bounds could be unreliable. If factorization fails with
287 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
288 *> for the leading INFO columns of A.
289 *> \endverbatim
290 *>
291 *> \param[out] BERR
292 *> \verbatim
293 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
294 *> Componentwise relative backward error. This is the
295 *> componentwise relative backward error of each solution vector X(j)
296 *> (i.e., the smallest relative change in any element of A or B that
297 *> makes X(j) an exact solution).
298 *> \endverbatim
299 *>
300 *> \param[in] N_ERR_BNDS
301 *> \verbatim
302 *> N_ERR_BNDS is INTEGER
303 *> Number of error bounds to return for each right hand side
304 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
305 *> ERR_BNDS_COMP below.
306 *> \endverbatim
307 *>
308 *> \param[out] ERR_BNDS_NORM
309 *> \verbatim
310 *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
311 *> For each right-hand side, this array contains information about
312 *> various error bounds and condition numbers corresponding to the
313 *> normwise relative error, which is defined as follows:
314 *>
315 *> Normwise relative error in the ith solution vector:
316 *> max_j (abs(XTRUE(j,i) - X(j,i)))
317 *> ------------------------------
318 *> max_j abs(X(j,i))
319 *>
320 *> The array is indexed by the type of error information as described
321 *> below. There currently are up to three pieces of information
322 *> returned.
323 *>
324 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
325 *> right-hand side.
326 *>
327 *> The second index in ERR_BNDS_NORM(:,err) contains the following
328 *> three fields:
329 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
330 *> reciprocal condition number is less than the threshold
331 *> sqrt(n) * dlamch('Epsilon').
332 *>
333 *> err = 2 "Guaranteed" error bound: The estimated forward error,
334 *> almost certainly within a factor of 10 of the true error
335 *> so long as the next entry is greater than the threshold
336 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
337 *> be trusted if the previous boolean is true.
338 *>
339 *> err = 3 Reciprocal condition number: Estimated normwise
340 *> reciprocal condition number. Compared with the threshold
341 *> sqrt(n) * dlamch('Epsilon') to determine if the error
342 *> estimate is "guaranteed". These reciprocal condition
343 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
344 *> appropriately scaled matrix Z.
345 *> Let Z = S*A, where S scales each row by a power of the
346 *> radix so all absolute row sums of Z are approximately 1.
347 *>
348 *> See Lapack Working Note 165 for further details and extra
349 *> cautions.
350 *> \endverbatim
351 *>
352 *> \param[out] ERR_BNDS_COMP
353 *> \verbatim
354 *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS)
355 *> For each right-hand side, this array contains information about
356 *> various error bounds and condition numbers corresponding to the
357 *> componentwise relative error, which is defined as follows:
358 *>
359 *> Componentwise relative error in the ith solution vector:
360 *> abs(XTRUE(j,i) - X(j,i))
361 *> max_j ----------------------
362 *> abs(X(j,i))
363 *>
364 *> The array is indexed by the right-hand side i (on which the
365 *> componentwise relative error depends), and the type of error
366 *> information as described below. There currently are up to three
367 *> pieces of information returned for each right-hand side. If
368 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
369 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
370 *> the first (:,N_ERR_BNDS) entries are returned.
371 *>
372 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
373 *> right-hand side.
374 *>
375 *> The second index in ERR_BNDS_COMP(:,err) contains the following
376 *> three fields:
377 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
378 *> reciprocal condition number is less than the threshold
379 *> sqrt(n) * dlamch('Epsilon').
380 *>
381 *> err = 2 "Guaranteed" error bound: The estimated forward error,
382 *> almost certainly within a factor of 10 of the true error
383 *> so long as the next entry is greater than the threshold
384 *> sqrt(n) * dlamch('Epsilon'). This error bound should only
385 *> be trusted if the previous boolean is true.
386 *>
387 *> err = 3 Reciprocal condition number: Estimated componentwise
388 *> reciprocal condition number. Compared with the threshold
389 *> sqrt(n) * dlamch('Epsilon') to determine if the error
390 *> estimate is "guaranteed". These reciprocal condition
391 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
392 *> appropriately scaled matrix Z.
393 *> Let Z = S*(A*diag(x)), where x is the solution for the
394 *> current right-hand side and S scales each row of
395 *> A*diag(x) by a power of the radix so all absolute row
396 *> sums of Z are approximately 1.
397 *>
398 *> See Lapack Working Note 165 for further details and extra
399 *> cautions.
400 *> \endverbatim
401 *>
402 *> \param[in] NPARAMS
403 *> \verbatim
404 *> NPARAMS is INTEGER
405 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
406 *> PARAMS array is never referenced and default values are used.
407 *> \endverbatim
408 *>
409 *> \param[in,out] PARAMS
410 *> \verbatim
411 *> PARAMS is DOUBLE PRECISION array, dimension NPARAMS
412 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
413 *> that entry will be filled with default value used for that
414 *> parameter. Only positions up to NPARAMS are accessed; defaults
415 *> are used for higher-numbered parameters.
416 *>
417 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
418 *> refinement or not.
419 *> Default: 1.0D+0
420 *> = 0.0 : No refinement is performed, and no error bounds are
421 *> computed.
422 *> = 1.0 : Use the extra-precise refinement algorithm.
423 *> (other values are reserved for future use)
424 *>
425 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
426 *> computations allowed for refinement.
427 *> Default: 10
428 *> Aggressive: Set to 100 to permit convergence using approximate
429 *> factorizations or factorizations other than LU. If
430 *> the factorization uses a technique other than
431 *> Gaussian elimination, the guarantees in
432 *> err_bnds_norm and err_bnds_comp may no longer be
433 *> trustworthy.
434 *>
435 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
436 *> will attempt to find a solution with small componentwise
437 *> relative error in the double-precision algorithm. Positive
438 *> is true, 0.0 is false.
439 *> Default: 1.0 (attempt componentwise convergence)
440 *> \endverbatim
441 *>
442 *> \param[out] WORK
443 *> \verbatim
444 *> WORK is DOUBLE PRECISION array, dimension (4*N)
445 *> \endverbatim
446 *>
447 *> \param[out] IWORK
448 *> \verbatim
449 *> IWORK is INTEGER array, dimension (N)
450 *> \endverbatim
451 *>
452 *> \param[out] INFO
453 *> \verbatim
454 *> INFO is INTEGER
455 *> = 0: Successful exit. The solution to every right-hand side is
456 *> guaranteed.
457 *> < 0: If INFO = -i, the i-th argument had an illegal value
458 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
459 *> has been completed, but the factor U is exactly singular, so
460 *> the solution and error bounds could not be computed. RCOND = 0
461 *> is returned.
462 *> = N+J: The solution corresponding to the Jth right-hand side is
463 *> not guaranteed. The solutions corresponding to other right-
464 *> hand sides K with K > J may not be guaranteed as well, but
465 *> only the first such right-hand side is reported. If a small
466 *> componentwise error is not requested (PARAMS(3) = 0.0) then
467 *> the Jth right-hand side is the first with a normwise error
468 *> bound that is not guaranteed (the smallest J such
469 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
470 *> the Jth right-hand side is the first with either a normwise or
471 *> componentwise error bound that is not guaranteed (the smallest
472 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
473 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
474 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
475 *> about all of the right-hand sides check ERR_BNDS_NORM or
476 *> ERR_BNDS_COMP.
477 *> \endverbatim
478 *
479 * Authors:
480 * ========
481 *
482 *> \author Univ. of Tennessee
483 *> \author Univ. of California Berkeley
484 *> \author Univ. of Colorado Denver
485 *> \author NAG Ltd.
486 *
487 *> \date April 2012
488 *
489 *> \ingroup doublePOsolve
490 *
491 * =====================================================================
492  SUBROUTINE dposvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED,
493  \$ s, b, ldb, x, ldx, rcond, rpvgrw, berr,
494  \$ n_err_bnds, err_bnds_norm, err_bnds_comp,
495  \$ nparams, params, work, iwork, info )
496 *
497 * -- LAPACK driver routine (version 3.4.1) --
498 * -- LAPACK is a software package provided by Univ. of Tennessee, --
499 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
500 * April 2012
501 *
502 * .. Scalar Arguments ..
503  CHARACTER equed, fact, uplo
504  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
505  \$ n_err_bnds
506  DOUBLE PRECISION rcond, rpvgrw
507 * ..
508 * .. Array Arguments ..
509  INTEGER iwork( * )
510  DOUBLE PRECISION a( lda, * ), af( ldaf, * ), b( ldb, * ),
511  \$ x( ldx, * ), work( * )
512  DOUBLE PRECISION s( * ), params( * ), berr( * ),
513  \$ err_bnds_norm( nrhs, * ),
514  \$ err_bnds_comp( nrhs, * )
515 * ..
516 *
517 * ==================================================================
518 *
519 * .. Parameters ..
520  DOUBLE PRECISION zero, one
521  parameter( zero = 0.0d+0, one = 1.0d+0 )
522  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
523  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
524  INTEGER cmp_err_i, piv_growth_i
525  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
526  \$ berr_i = 3 )
527  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
528  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
529  \$ piv_growth_i = 9 )
530 * ..
531 * .. Local Scalars ..
532  LOGICAL equil, nofact, rcequ
533  INTEGER infequ, j
534  DOUBLE PRECISION amax, bignum, smin, smax,
535  \$ scond, smlnum
536 * ..
537 * .. External Functions ..
538  EXTERNAL lsame, dlamch, dla_porpvgrw
539  LOGICAL lsame
540  DOUBLE PRECISION dlamch, dla_porpvgrw
541 * ..
542 * .. External Subroutines ..
543  EXTERNAL dpoequb, dpotrf, dpotrs, dlacpy, dlaqsy,
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 = dlamch( '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 DPORFSX.
566 *
567  rpvgrw = zero
568 *
569 * Test the input parameters. PARAMS is not tested until DPORFSX.
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( 'DPOSVXX', -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 dpoequb( n, a, lda, s, scond, amax, infequ )
623  IF( infequ.EQ.0 ) THEN
624 *
625 * Equilibrate the matrix.
626 *
627  CALL dlaqsy( 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 dlascl2( n, nrhs, s, b, ldb )
635 *
636  IF( nofact .OR. equil ) THEN
637 *
638 * Compute the Cholesky factorization of A.
639 *
640  CALL dlacpy( uplo, n, n, a, lda, af, ldaf )
641  CALL dpotrf( uplo, n, af, ldaf, info )
642 *
643 * Return if INFO is non-zero.
644 *
645  IF( info.NE.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 = dla_porpvgrw( uplo, info, a, lda, af, ldaf, work )
652  return
653  ENDIF
654  END IF
655 *
656 * Compute the reciprocal growth factor RPVGRW.
657 *
658  rpvgrw = dla_porpvgrw( uplo, n, a, lda, af, ldaf, work )
659 *
660 * Compute the solution matrix X.
661 *
662  CALL dlacpy( 'Full', n, nrhs, b, ldb, x, ldx )
663  CALL dpotrs( 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 dporfsx( 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, iwork, info )
671
672 *
673 * Scale solutions.
674 *
675  IF ( rcequ ) THEN
676  CALL dlascl2( n, nrhs, s, x, ldx )
677  END IF
678 *
679  return
680 *
681 * End of DPOSVXX
682 *
683  END