LAPACK  3.5.0
LAPACK: Linear Algebra PACKage
 All Classes Files Functions Variables Typedefs Macros
csysvxx.f
Go to the documentation of this file.
1 *> \brief <b> CSYSVXX computes the solution to system of linear equations A * X = B for SY matrices</b>
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
9 *> Download CSYSVXX + dependencies
10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csysvxx.f">
11 *> [TGZ]</a>
12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csysvxx.f">
13 *> [ZIP]</a>
14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csysvxx.f">
15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
22 * EQUED, 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 * INTEGER IPIV( * )
34 * COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
35 * $ X( LDX, * ), WORK( * )
36 * REAL S( * ), PARAMS( * ), BERR( * ),
37 * $ ERR_BNDS_NORM( NRHS, * ),
38 * $ ERR_BNDS_COMP( NRHS, * ), RWORK( * )
39 * ..
40 *
41 *
42 *> \par Purpose:
43 * =============
44 *>
45 *> \verbatim
46 *>
47 *> CSYSVXX uses the diagonal pivoting factorization to compute the
48 *> solution to a complex system of linear equations A * X = B, where
49 *> A is an N-by-N symmetric matrix and X and B are N-by-NRHS
50 *> matrices.
51 *>
52 *> If requested, both normwise and maximum componentwise error bounds
53 *> are returned. CSYSVXX 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 *> CSYSVXX 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 *> CSYSVXX 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 CSYSVXX 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', real 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 LU decomposition is used to factor
85 *> the matrix A (after equilibration if FACT = 'E') as
86 *>
87 *> A = U * D * U**T, if UPLO = 'U', or
88 *> A = L * D * L**T, if UPLO = 'L',
89 *>
90 *> where U (or L) is a product of permutation and unit upper (lower)
91 *> triangular matrices, and D is symmetric and block diagonal with
92 *> 1-by-1 and 2-by-2 diagonal blocks.
93 *>
94 *> 3. If some D(i,i)=0, so that D is exactly singular, then the
95 *> routine returns with INFO = i. Otherwise, the factored form of A
96 *> is used to estimate the condition number of the matrix A (see
97 *> argument RCOND). If the reciprocal of the condition number is
98 *> less than machine precision, the routine still goes on to solve
99 *> for X and compute error bounds as described below.
100 *>
101 *> 4. The system of equations is solved for X using the factored form
102 *> of A.
103 *>
104 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
105 *> the routine will use iterative refinement to try to get a small
106 *> error and error bounds. Refinement calculates the residual to at
107 *> least twice the working precision.
108 *>
109 *> 6. If equilibration was used, the matrix X is premultiplied by
110 *> diag(R) so that it solves the original system before
111 *> equilibration.
112 *> \endverbatim
113 *
114 * Arguments:
115 * ==========
116 *
117 *> \verbatim
118 *> Some optional parameters are bundled in the PARAMS array. These
119 *> settings determine how refinement is performed, but often the
120 *> defaults are acceptable. If the defaults are acceptable, users
121 *> can pass NPARAMS = 0 which prevents the source code from accessing
122 *> the PARAMS argument.
123 *> \endverbatim
124 *>
125 *> \param[in] FACT
126 *> \verbatim
127 *> FACT is CHARACTER*1
128 *> Specifies whether or not the factored form of the matrix A is
129 *> supplied on entry, and if not, whether the matrix A should be
130 *> equilibrated before it is factored.
131 *> = 'F': On entry, AF and IPIV contain the factored form of A.
132 *> If EQUED is not 'N', the matrix A has been
133 *> equilibrated with scaling factors given by S.
134 *> A, AF, and IPIV are not modified.
135 *> = 'N': The matrix A will be copied to AF and factored.
136 *> = 'E': The matrix A will be equilibrated if necessary, then
137 *> copied to AF and factored.
138 *> \endverbatim
139 *>
140 *> \param[in] UPLO
141 *> \verbatim
142 *> UPLO is CHARACTER*1
143 *> = 'U': Upper triangle of A is stored;
144 *> = 'L': Lower triangle of A is stored.
145 *> \endverbatim
146 *>
147 *> \param[in] N
148 *> \verbatim
149 *> N is INTEGER
150 *> The number of linear equations, i.e., the order of the
151 *> matrix A. N >= 0.
152 *> \endverbatim
153 *>
154 *> \param[in] NRHS
155 *> \verbatim
156 *> NRHS is INTEGER
157 *> The number of right hand sides, i.e., the number of columns
158 *> of the matrices B and X. NRHS >= 0.
159 *> \endverbatim
160 *>
161 *> \param[in,out] A
162 *> \verbatim
163 *> A is COMPLEX array, dimension (LDA,N)
164 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N
165 *> upper triangular part of A contains the upper triangular
166 *> part of the matrix A, and the strictly lower triangular
167 *> part of A is not referenced. If UPLO = 'L', the leading
168 *> N-by-N lower triangular part of A contains the lower
169 *> triangular part of the matrix A, and the strictly upper
170 *> triangular part of A is not referenced.
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 COMPLEX array, dimension (LDAF,N)
185 *> If FACT = 'F', then AF is an input argument and on entry
186 *> contains the block diagonal matrix D and the multipliers
187 *> used to obtain the factor U or L from the factorization A =
188 *> U*D*U**T or A = L*D*L**T as computed by SSYTRF.
189 *>
190 *> If FACT = 'N', then AF is an output argument and on exit
191 *> returns the block diagonal matrix D and the multipliers
192 *> used to obtain the factor U or L from the factorization A =
193 *> U*D*U**T or A = L*D*L**T.
194 *> \endverbatim
195 *>
196 *> \param[in] LDAF
197 *> \verbatim
198 *> LDAF is INTEGER
199 *> The leading dimension of the array AF. LDAF >= max(1,N).
200 *> \endverbatim
201 *>
202 *> \param[in,out] IPIV
203 *> \verbatim
204 *> IPIV is INTEGER array, dimension (N)
205 *> If FACT = 'F', then IPIV is an input argument and on entry
206 *> contains details of the interchanges and the block
207 *> structure of D, as determined by SSYTRF. If IPIV(k) > 0,
208 *> then rows and columns k and IPIV(k) were interchanged and
209 *> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and
210 *> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and
211 *> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2
212 *> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0,
213 *> then rows and columns k+1 and -IPIV(k) were interchanged
214 *> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
215 *>
216 *> If FACT = 'N', then IPIV is an output argument and on exit
217 *> contains details of the interchanges and the block
218 *> structure of D, as determined by SSYTRF.
219 *> \endverbatim
220 *>
221 *> \param[in,out] EQUED
222 *> \verbatim
223 *> EQUED is CHARACTER*1
224 *> Specifies the form of equilibration that was done.
225 *> = 'N': No equilibration (always true if FACT = 'N').
226 *> = 'Y': Both row and column equilibration, i.e., A has been
227 *> replaced by diag(S) * A * diag(S).
228 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an
229 *> output argument.
230 *> \endverbatim
231 *>
232 *> \param[in,out] S
233 *> \verbatim
234 *> S is REAL array, dimension (N)
235 *> The scale factors for A. If EQUED = 'Y', A is multiplied on
236 *> the left and right by diag(S). S is an input argument if FACT =
237 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
238 *> = 'Y', each element of S must be positive. If S is output, each
239 *> element of S is a power of the radix. If S is input, each element
240 *> of S should be a power of the radix to ensure a reliable solution
241 *> and error estimates. Scaling by powers of the radix does not cause
242 *> rounding errors unless the result underflows or overflows.
243 *> Rounding errors during scaling lead to refining with a matrix that
244 *> is not equivalent to the input matrix, producing error estimates
245 *> that may not be reliable.
246 *> \endverbatim
247 *>
248 *> \param[in,out] B
249 *> \verbatim
250 *> B is COMPLEX array, dimension (LDB,NRHS)
251 *> On entry, the N-by-NRHS right hand side matrix B.
252 *> On exit,
253 *> if EQUED = 'N', B is not modified;
254 *> if EQUED = 'Y', B is overwritten by diag(S)*B;
255 *> \endverbatim
256 *>
257 *> \param[in] LDB
258 *> \verbatim
259 *> LDB is INTEGER
260 *> The leading dimension of the array B. LDB >= max(1,N).
261 *> \endverbatim
262 *>
263 *> \param[out] X
264 *> \verbatim
265 *> X is COMPLEX array, dimension (LDX,NRHS)
266 *> If INFO = 0, the N-by-NRHS solution matrix X to the original
267 *> system of equations. Note that A and B are modified on exit if
268 *> EQUED .ne. 'N', and the solution to the equilibrated system is
269 *> inv(diag(S))*X.
270 *> \endverbatim
271 *>
272 *> \param[in] LDX
273 *> \verbatim
274 *> LDX is INTEGER
275 *> The leading dimension of the array X. LDX >= max(1,N).
276 *> \endverbatim
277 *>
278 *> \param[out] RCOND
279 *> \verbatim
280 *> RCOND is REAL
281 *> Reciprocal scaled condition number. This is an estimate of the
282 *> reciprocal Skeel condition number of the matrix A after
283 *> equilibration (if done). If this is less than the machine
284 *> precision (in particular, if it is zero), the matrix is singular
285 *> to working precision. Note that the error may still be small even
286 *> if this number is very small and the matrix appears ill-
287 *> conditioned.
288 *> \endverbatim
289 *>
290 *> \param[out] RPVGRW
291 *> \verbatim
292 *> RPVGRW is REAL
293 *> Reciprocal pivot growth. On exit, this contains the reciprocal
294 *> pivot growth factor norm(A)/norm(U). The "max absolute element"
295 *> norm is used. If this is much less than 1, then the stability of
296 *> the LU factorization of the (equilibrated) matrix A could be poor.
297 *> This also means that the solution X, estimated condition numbers,
298 *> and error bounds could be unreliable. If factorization fails with
299 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor
300 *> for the leading INFO columns of A.
301 *> \endverbatim
302 *>
303 *> \param[out] BERR
304 *> \verbatim
305 *> BERR is REAL array, dimension (NRHS)
306 *> Componentwise relative backward error. This is the
307 *> componentwise relative backward error of each solution vector X(j)
308 *> (i.e., the smallest relative change in any element of A or B that
309 *> makes X(j) an exact solution).
310 *> \endverbatim
311 *>
312 *> \param[in] N_ERR_BNDS
313 *> \verbatim
314 *> N_ERR_BNDS is INTEGER
315 *> Number of error bounds to return for each right hand side
316 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and
317 *> ERR_BNDS_COMP below.
318 *> \endverbatim
319 *>
320 *> \param[out] ERR_BNDS_NORM
321 *> \verbatim
322 *> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
323 *> For each right-hand side, this array contains information about
324 *> various error bounds and condition numbers corresponding to the
325 *> normwise relative error, which is defined as follows:
326 *>
327 *> Normwise relative error in the ith solution vector:
328 *> max_j (abs(XTRUE(j,i) - X(j,i)))
329 *> ------------------------------
330 *> max_j abs(X(j,i))
331 *>
332 *> The array is indexed by the type of error information as described
333 *> below. There currently are up to three pieces of information
334 *> returned.
335 *>
336 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
337 *> right-hand side.
338 *>
339 *> The second index in ERR_BNDS_NORM(:,err) contains the following
340 *> three fields:
341 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
342 *> reciprocal condition number is less than the threshold
343 *> sqrt(n) * slamch('Epsilon').
344 *>
345 *> err = 2 "Guaranteed" error bound: The estimated forward error,
346 *> almost certainly within a factor of 10 of the true error
347 *> so long as the next entry is greater than the threshold
348 *> sqrt(n) * slamch('Epsilon'). This error bound should only
349 *> be trusted if the previous boolean is true.
350 *>
351 *> err = 3 Reciprocal condition number: Estimated normwise
352 *> reciprocal condition number. Compared with the threshold
353 *> sqrt(n) * slamch('Epsilon') to determine if the error
354 *> estimate is "guaranteed". These reciprocal condition
355 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
356 *> appropriately scaled matrix Z.
357 *> Let Z = S*A, where S scales each row by a power of the
358 *> radix so all absolute row sums of Z are approximately 1.
359 *>
360 *> See Lapack Working Note 165 for further details and extra
361 *> cautions.
362 *> \endverbatim
363 *>
364 *> \param[out] ERR_BNDS_COMP
365 *> \verbatim
366 *> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
367 *> For each right-hand side, this array contains information about
368 *> various error bounds and condition numbers corresponding to the
369 *> componentwise relative error, which is defined as follows:
370 *>
371 *> Componentwise relative error in the ith solution vector:
372 *> abs(XTRUE(j,i) - X(j,i))
373 *> max_j ----------------------
374 *> abs(X(j,i))
375 *>
376 *> The array is indexed by the right-hand side i (on which the
377 *> componentwise relative error depends), and the type of error
378 *> information as described below. There currently are up to three
379 *> pieces of information returned for each right-hand side. If
380 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
381 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
382 *> the first (:,N_ERR_BNDS) entries are returned.
383 *>
384 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
385 *> right-hand side.
386 *>
387 *> The second index in ERR_BNDS_COMP(:,err) contains the following
388 *> three fields:
389 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the
390 *> reciprocal condition number is less than the threshold
391 *> sqrt(n) * slamch('Epsilon').
392 *>
393 *> err = 2 "Guaranteed" error bound: The estimated forward error,
394 *> almost certainly within a factor of 10 of the true error
395 *> so long as the next entry is greater than the threshold
396 *> sqrt(n) * slamch('Epsilon'). This error bound should only
397 *> be trusted if the previous boolean is true.
398 *>
399 *> err = 3 Reciprocal condition number: Estimated componentwise
400 *> reciprocal condition number. Compared with the threshold
401 *> sqrt(n) * slamch('Epsilon') to determine if the error
402 *> estimate is "guaranteed". These reciprocal condition
403 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
404 *> appropriately scaled matrix Z.
405 *> Let Z = S*(A*diag(x)), where x is the solution for the
406 *> current right-hand side and S scales each row of
407 *> A*diag(x) by a power of the radix so all absolute row
408 *> sums of Z are approximately 1.
409 *>
410 *> See Lapack Working Note 165 for further details and extra
411 *> cautions.
412 *> \endverbatim
413 *>
414 *> \param[in] NPARAMS
415 *> \verbatim
416 *> NPARAMS is INTEGER
417 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the
418 *> PARAMS array is never referenced and default values are used.
419 *> \endverbatim
420 *>
421 *> \param[in,out] PARAMS
422 *> \verbatim
423 *> PARAMS is REAL array, dimension NPARAMS
424 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then
425 *> that entry will be filled with default value used for that
426 *> parameter. Only positions up to NPARAMS are accessed; defaults
427 *> are used for higher-numbered parameters.
428 *>
429 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
430 *> refinement or not.
431 *> Default: 1.0
432 *> = 0.0 : No refinement is performed, and no error bounds are
433 *> computed.
434 *> = 1.0 : Use the double-precision refinement algorithm,
435 *> possibly with doubled-single computations if the
436 *> compilation environment does not support DOUBLE
437 *> PRECISION.
438 *> (other values are reserved for future use)
439 *>
440 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
441 *> computations allowed for refinement.
442 *> Default: 10
443 *> Aggressive: Set to 100 to permit convergence using approximate
444 *> factorizations or factorizations other than LU. If
445 *> the factorization uses a technique other than
446 *> Gaussian elimination, the guarantees in
447 *> err_bnds_norm and err_bnds_comp may no longer be
448 *> trustworthy.
449 *>
450 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
451 *> will attempt to find a solution with small componentwise
452 *> relative error in the double-precision algorithm. Positive
453 *> is true, 0.0 is false.
454 *> Default: 1.0 (attempt componentwise convergence)
455 *> \endverbatim
456 *>
457 *> \param[out] WORK
458 *> \verbatim
459 *> WORK is COMPLEX array, dimension (2*N)
460 *> \endverbatim
461 *>
462 *> \param[out] RWORK
463 *> \verbatim
464 *> RWORK is REAL array, dimension (2*N)
465 *> \endverbatim
466 *>
467 *> \param[out] INFO
468 *> \verbatim
469 *> INFO is INTEGER
470 *> = 0: Successful exit. The solution to every right-hand side is
471 *> guaranteed.
472 *> < 0: If INFO = -i, the i-th argument had an illegal value
473 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
474 *> has been completed, but the factor U is exactly singular, so
475 *> the solution and error bounds could not be computed. RCOND = 0
476 *> is returned.
477 *> = N+J: The solution corresponding to the Jth right-hand side is
478 *> not guaranteed. The solutions corresponding to other right-
479 *> hand sides K with K > J may not be guaranteed as well, but
480 *> only the first such right-hand side is reported. If a small
481 *> componentwise error is not requested (PARAMS(3) = 0.0) then
482 *> the Jth right-hand side is the first with a normwise error
483 *> bound that is not guaranteed (the smallest J such
484 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
485 *> the Jth right-hand side is the first with either a normwise or
486 *> componentwise error bound that is not guaranteed (the smallest
487 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
488 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
489 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
490 *> about all of the right-hand sides check ERR_BNDS_NORM or
491 *> ERR_BNDS_COMP.
492 *> \endverbatim
493 *
494 * Authors:
495 * ========
496 *
497 *> \author Univ. of Tennessee
498 *> \author Univ. of California Berkeley
499 *> \author Univ. of Colorado Denver
500 *> \author NAG Ltd.
501 *
502 *> \date April 2012
503 *
504 *> \ingroup complexSYsolve
505 *
506 * =====================================================================
507  SUBROUTINE csysvxx( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
508  $ equed, s, b, ldb, x, ldx, rcond, rpvgrw, berr,
509  $ n_err_bnds, err_bnds_norm, err_bnds_comp,
510  $ nparams, params, work, rwork, info )
511 *
512 * -- LAPACK driver routine (version 3.4.1) --
513 * -- LAPACK is a software package provided by Univ. of Tennessee, --
514 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
515 * April 2012
516 *
517 * .. Scalar Arguments ..
518  CHARACTER equed, fact, uplo
519  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
520  $ n_err_bnds
521  REAL rcond, rpvgrw
522 * ..
523 * .. Array Arguments ..
524  INTEGER ipiv( * )
525  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
526  $ x( ldx, * ), work( * )
527  REAL s( * ), params( * ), berr( * ),
528  $ err_bnds_norm( nrhs, * ),
529  $ err_bnds_comp( nrhs, * ), rwork( * )
530 * ..
531 *
532 * ==================================================================
533 *
534 * .. Parameters ..
535  REAL zero, one
536  parameter( zero = 0.0e+0, one = 1.0e+0 )
537  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
538  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
539  INTEGER cmp_err_i, piv_growth_i
540  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
541  $ berr_i = 3 )
542  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
543  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
544  $ piv_growth_i = 9 )
545 * ..
546 * .. Local Scalars ..
547  LOGICAL equil, nofact, rcequ
548  INTEGER infequ, j
549  REAL amax, bignum, smin, smax, scond, smlnum
550 * ..
551 * .. External Functions ..
552  EXTERNAL lsame, slamch, cla_syrpvgrw
553  LOGICAL lsame
554  REAL slamch, cla_syrpvgrw
555 * ..
556 * .. External Subroutines ..
557  EXTERNAL csycon, csyequb, csytrf, csytrs, clacpy,
559 * ..
560 * .. Intrinsic Functions ..
561  INTRINSIC max, min
562 * ..
563 * .. Executable Statements ..
564 *
565  info = 0
566  nofact = lsame( fact, 'N' )
567  equil = lsame( fact, 'E' )
568  smlnum = slamch( 'Safe minimum' )
569  bignum = one / smlnum
570  IF( nofact .OR. equil ) THEN
571  equed = 'N'
572  rcequ = .false.
573  ELSE
574  rcequ = lsame( equed, 'Y' )
575  ENDIF
576 *
577 * Default is failure. If an input parameter is wrong or
578 * factorization fails, make everything look horrible. Only the
579 * pivot growth is set here, the rest is initialized in CSYRFSX.
580 *
581  rpvgrw = zero
582 *
583 * Test the input parameters. PARAMS is not tested until CSYRFSX.
584 *
585  IF( .NOT.nofact .AND. .NOT.equil .AND. .NOT.
586  $ lsame( fact, 'F' ) ) THEN
587  info = -1
588  ELSE IF( .NOT.lsame(uplo, 'U') .AND.
589  $ .NOT.lsame(uplo, 'L') ) THEN
590  info = -2
591  ELSE IF( n.LT.0 ) THEN
592  info = -3
593  ELSE IF( nrhs.LT.0 ) THEN
594  info = -4
595  ELSE IF( lda.LT.max( 1, n ) ) THEN
596  info = -6
597  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
598  info = -8
599  ELSE IF( lsame( fact, 'F' ) .AND. .NOT.
600  $ ( rcequ .OR. lsame( equed, 'N' ) ) ) THEN
601  info = -9
602  ELSE
603  IF ( rcequ ) THEN
604  smin = bignum
605  smax = zero
606  DO 10 j = 1, n
607  smin = min( smin, s( j ) )
608  smax = max( smax, s( j ) )
609  10 CONTINUE
610  IF( smin.LE.zero ) THEN
611  info = -10
612  ELSE IF( n.GT.0 ) THEN
613  scond = max( smin, smlnum ) / min( smax, bignum )
614  ELSE
615  scond = one
616  END IF
617  END IF
618  IF( info.EQ.0 ) THEN
619  IF( ldb.LT.max( 1, n ) ) THEN
620  info = -12
621  ELSE IF( ldx.LT.max( 1, n ) ) THEN
622  info = -14
623  END IF
624  END IF
625  END IF
626 *
627  IF( info.NE.0 ) THEN
628  CALL xerbla( 'CSYSVXX', -info )
629  RETURN
630  END IF
631 *
632  IF( equil ) THEN
633 *
634 * Compute row and column scalings to equilibrate the matrix A.
635 *
636  CALL csyequb( uplo, n, a, lda, s, scond, amax, work, infequ )
637  IF( infequ.EQ.0 ) THEN
638 *
639 * Equilibrate the matrix.
640 *
641  CALL claqsy( uplo, n, a, lda, s, scond, amax, equed )
642  rcequ = lsame( equed, 'Y' )
643  END IF
644 
645  END IF
646 *
647 * Scale the right hand-side.
648 *
649  IF( rcequ ) CALL clascl2( n, nrhs, s, b, ldb )
650 *
651  IF( nofact .OR. equil ) THEN
652 *
653 * Compute the LDL^T or UDU^T factorization of A.
654 *
655  CALL clacpy( uplo, n, n, a, lda, af, ldaf )
656  CALL csytrf( uplo, n, af, ldaf, ipiv, work, 5*max(1,n), info )
657 *
658 * Return if INFO is non-zero.
659 *
660  IF( info.GT.0 ) THEN
661 *
662 * Pivot in column INFO is exactly 0
663 * Compute the reciprocal pivot growth factor of the
664 * leading rank-deficient INFO columns of A.
665 *
666  IF ( n.GT.0 )
667  $ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af,
668  $ ldaf, ipiv, rwork )
669  RETURN
670  END IF
671  END IF
672 *
673 * Compute the reciprocal pivot growth factor RPVGRW.
674 *
675  IF ( n.GT.0 )
676  $ rpvgrw = cla_syrpvgrw( uplo, n, info, a, lda, af, ldaf,
677  $ ipiv, rwork )
678 *
679 * Compute the solution matrix X.
680 *
681  CALL clacpy( 'Full', n, nrhs, b, ldb, x, ldx )
682  CALL csytrs( uplo, n, nrhs, af, ldaf, ipiv, x, ldx, info )
683 *
684 * Use iterative refinement to improve the computed solution and
685 * compute error bounds and backward error estimates for it.
686 *
687  CALL csyrfsx( uplo, equed, n, nrhs, a, lda, af, ldaf, ipiv,
688  $ s, b, ldb, x, ldx, rcond, berr, n_err_bnds, err_bnds_norm,
689  $ err_bnds_comp, nparams, params, work, rwork, info )
690 *
691 * Scale solutions.
692 *
693  IF ( rcequ ) THEN
694  CALL clascl2(n, nrhs, s, x, ldx )
695  END IF
696 *
697  RETURN
698 *
699 * End of CSYSVXX
700 *
701  END