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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices</b> 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download SPOSVXX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sposvxx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sposvxx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sposvxx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 00022 * S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 00023 * N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 00024 * NPARAMS, PARAMS, WORK, IWORK, INFO ) 00025 * 00026 * .. Scalar Arguments .. 00027 * CHARACTER EQUED, FACT, UPLO 00028 * INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 00029 * $ N_ERR_BNDS 00030 * REAL RCOND, RPVGRW 00031 * .. 00032 * .. Array Arguments .. 00033 * INTEGER IWORK( * ) 00034 * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00035 * $ X( LDX, * ), WORK( * ) 00036 * REAL S( * ), PARAMS( * ), BERR( * ), 00037 * $ ERR_BNDS_NORM( NRHS, * ), 00038 * $ ERR_BNDS_COMP( NRHS, * ) 00039 * .. 00040 * 00041 * 00042 *> \par Purpose: 00043 * ============= 00044 *> 00045 *> \verbatim 00046 *> 00047 *> SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T 00048 *> to compute the solution to a real system of linear equations 00049 *> A * X = B, where A is an N-by-N symmetric positive definite matrix 00050 *> and X and B are N-by-NRHS matrices. 00051 *> 00052 *> If requested, both normwise and maximum componentwise error bounds 00053 *> are returned. SPOSVXX will return a solution with a tiny 00054 *> guaranteed error (O(eps) where eps is the working machine 00055 *> precision) unless the matrix is very ill-conditioned, in which 00056 *> case a warning is returned. Relevant condition numbers also are 00057 *> calculated and returned. 00058 *> 00059 *> SPOSVXX accepts user-provided factorizations and equilibration 00060 *> factors; see the definitions of the FACT and EQUED options. 00061 *> Solving with refinement and using a factorization from a previous 00062 *> SPOSVXX call will also produce a solution with either O(eps) 00063 *> errors or warnings, but we cannot make that claim for general 00064 *> user-provided factorizations and equilibration factors if they 00065 *> differ from what SPOSVXX would itself produce. 00066 *> \endverbatim 00067 * 00068 *> \par Description: 00069 * ================= 00070 *> 00071 *> \verbatim 00072 *> 00073 *> The following steps are performed: 00074 *> 00075 *> 1. If FACT = 'E', real scaling factors are computed to equilibrate 00076 *> the system: 00077 *> 00078 *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 00079 *> 00080 *> Whether or not the system will be equilibrated depends on the 00081 *> scaling of the matrix A, but if equilibration is used, A is 00082 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00083 *> 00084 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00085 *> factor the matrix A (after equilibration if FACT = 'E') as 00086 *> A = U**T* U, if UPLO = 'U', or 00087 *> A = L * L**T, if UPLO = 'L', 00088 *> where U is an upper triangular matrix and L is a lower triangular 00089 *> matrix. 00090 *> 00091 *> 3. If the leading i-by-i principal minor is not positive definite, 00092 *> then the routine returns with INFO = i. Otherwise, the factored 00093 *> form of A is used to estimate the condition number of the matrix 00094 *> A (see argument RCOND). If the reciprocal of the condition number 00095 *> is less than machine precision, the routine still goes on to solve 00096 *> for X and compute error bounds as described below. 00097 *> 00098 *> 4. The system of equations is solved for X using the factored form 00099 *> of A. 00100 *> 00101 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00102 *> the routine will use iterative refinement to try to get a small 00103 *> error and error bounds. Refinement calculates the residual to at 00104 *> least twice the working precision. 00105 *> 00106 *> 6. If equilibration was used, the matrix X is premultiplied by 00107 *> diag(S) so that it solves the original system before 00108 *> equilibration. 00109 *> \endverbatim 00110 * 00111 * Arguments: 00112 * ========== 00113 * 00114 *> \verbatim 00115 *> Some optional parameters are bundled in the PARAMS array. These 00116 *> settings determine how refinement is performed, but often the 00117 *> defaults are acceptable. If the defaults are acceptable, users 00118 *> can pass NPARAMS = 0 which prevents the source code from accessing 00119 *> the PARAMS argument. 00120 *> \endverbatim 00121 *> 00122 *> \param[in] FACT 00123 *> \verbatim 00124 *> FACT is CHARACTER*1 00125 *> Specifies whether or not the factored form of the matrix A is 00126 *> supplied on entry, and if not, whether the matrix A should be 00127 *> equilibrated before it is factored. 00128 *> = 'F': On entry, AF contains the factored form of A. 00129 *> If EQUED is not 'N', the matrix A has been 00130 *> equilibrated with scaling factors given by S. 00131 *> A and AF are not modified. 00132 *> = 'N': The matrix A will be copied to AF and factored. 00133 *> = 'E': The matrix A will be equilibrated if necessary, then 00134 *> copied to AF and factored. 00135 *> \endverbatim 00136 *> 00137 *> \param[in] UPLO 00138 *> \verbatim 00139 *> UPLO is CHARACTER*1 00140 *> = 'U': Upper triangle of A is stored; 00141 *> = 'L': Lower triangle of A is stored. 00142 *> \endverbatim 00143 *> 00144 *> \param[in] N 00145 *> \verbatim 00146 *> N is INTEGER 00147 *> The number of linear equations, i.e., the order of the 00148 *> matrix A. N >= 0. 00149 *> \endverbatim 00150 *> 00151 *> \param[in] NRHS 00152 *> \verbatim 00153 *> NRHS is INTEGER 00154 *> The number of right hand sides, i.e., the number of columns 00155 *> of the matrices B and X. NRHS >= 0. 00156 *> \endverbatim 00157 *> 00158 *> \param[in,out] A 00159 *> \verbatim 00160 *> A is REAL array, dimension (LDA,N) 00161 *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 00162 *> 'Y', then A must contain the equilibrated matrix 00163 *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper 00164 *> triangular part of A contains the upper triangular part of the 00165 *> matrix A, and the strictly lower triangular part of A is not 00166 *> referenced. If UPLO = 'L', the leading N-by-N lower triangular 00167 *> part of A contains the lower triangular part of the matrix A, and 00168 *> the strictly upper triangular part of A is not referenced. A is 00169 *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 00170 *> 'N' on exit. 00171 *> 00172 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00173 *> diag(S)*A*diag(S). 00174 *> \endverbatim 00175 *> 00176 *> \param[in] LDA 00177 *> \verbatim 00178 *> LDA is INTEGER 00179 *> The leading dimension of the array A. LDA >= max(1,N). 00180 *> \endverbatim 00181 *> 00182 *> \param[in,out] AF 00183 *> \verbatim 00184 *> AF is REAL array, dimension (LDAF,N) 00185 *> If FACT = 'F', then AF is an input argument and on entry 00186 *> contains the triangular factor U or L from the Cholesky 00187 *> factorization A = U**T*U or A = L*L**T, in the same storage 00188 *> format as A. If EQUED .ne. 'N', then AF is the factored 00189 *> form of the equilibrated matrix diag(S)*A*diag(S). 00190 *> 00191 *> If FACT = 'N', then AF is an output argument and on exit 00192 *> returns the triangular factor U or L from the Cholesky 00193 *> factorization A = U**T*U or A = L*L**T of the original 00194 *> matrix A. 00195 *> 00196 *> If FACT = 'E', then AF is an output argument and on exit 00197 *> returns the triangular factor U or L from the Cholesky 00198 *> factorization A = U**T*U or A = L*L**T of the equilibrated 00199 *> matrix A (see the description of A for the form of the 00200 *> equilibrated matrix). 00201 *> \endverbatim 00202 *> 00203 *> \param[in] LDAF 00204 *> \verbatim 00205 *> LDAF is INTEGER 00206 *> The leading dimension of the array AF. LDAF >= max(1,N). 00207 *> \endverbatim 00208 *> 00209 *> \param[in,out] EQUED 00210 *> \verbatim 00211 *> EQUED is CHARACTER*1 00212 *> Specifies the form of equilibration that was done. 00213 *> = 'N': No equilibration (always true if FACT = 'N'). 00214 *> = 'Y': Both row and column equilibration, i.e., A has been 00215 *> replaced by diag(S) * A * diag(S). 00216 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an 00217 *> output argument. 00218 *> \endverbatim 00219 *> 00220 *> \param[in,out] S 00221 *> \verbatim 00222 *> S is REAL array, dimension (N) 00223 *> The row scale factors for A. If EQUED = 'Y', A is multiplied on 00224 *> the left and right by diag(S). S is an input argument if FACT = 00225 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00226 *> = 'Y', each element of S must be positive. If S is output, each 00227 *> element of S is a power of the radix. If S is input, each element 00228 *> of S should be a power of the radix to ensure a reliable solution 00229 *> and error estimates. Scaling by powers of the radix does not cause 00230 *> rounding errors unless the result underflows or overflows. 00231 *> Rounding errors during scaling lead to refining with a matrix that 00232 *> is not equivalent to the input matrix, producing error estimates 00233 *> that may not be reliable. 00234 *> \endverbatim 00235 *> 00236 *> \param[in,out] B 00237 *> \verbatim 00238 *> B is REAL array, dimension (LDB,NRHS) 00239 *> On entry, the N-by-NRHS right hand side matrix B. 00240 *> On exit, 00241 *> if EQUED = 'N', B is not modified; 00242 *> if EQUED = 'Y', B is overwritten by diag(S)*B; 00243 *> \endverbatim 00244 *> 00245 *> \param[in] LDB 00246 *> \verbatim 00247 *> LDB is INTEGER 00248 *> The leading dimension of the array B. LDB >= max(1,N). 00249 *> \endverbatim 00250 *> 00251 *> \param[out] X 00252 *> \verbatim 00253 *> X is REAL array, dimension (LDX,NRHS) 00254 *> If INFO = 0, the N-by-NRHS solution matrix X to the original 00255 *> system of equations. Note that A and B are modified on exit if 00256 *> EQUED .ne. 'N', and the solution to the equilibrated system is 00257 *> inv(diag(S))*X. 00258 *> \endverbatim 00259 *> 00260 *> \param[in] LDX 00261 *> \verbatim 00262 *> LDX is INTEGER 00263 *> The leading dimension of the array X. LDX >= max(1,N). 00264 *> \endverbatim 00265 *> 00266 *> \param[out] RCOND 00267 *> \verbatim 00268 *> RCOND is REAL 00269 *> Reciprocal scaled condition number. This is an estimate of the 00270 *> reciprocal Skeel condition number of the matrix A after 00271 *> equilibration (if done). If this is less than the machine 00272 *> precision (in particular, if it is zero), the matrix is singular 00273 *> to working precision. Note that the error may still be small even 00274 *> if this number is very small and the matrix appears ill- 00275 *> conditioned. 00276 *> \endverbatim 00277 *> 00278 *> \param[out] RPVGRW 00279 *> \verbatim 00280 *> RPVGRW is REAL 00281 *> Reciprocal pivot growth. On exit, this contains the reciprocal 00282 *> pivot growth factor norm(A)/norm(U). The "max absolute element" 00283 *> norm is used. If this is much less than 1, then the stability of 00284 *> the LU factorization of the (equilibrated) matrix A could be poor. 00285 *> This also means that the solution X, estimated condition numbers, 00286 *> and error bounds could be unreliable. If factorization fails with 00287 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor 00288 *> for the leading INFO columns of A. 00289 *> \endverbatim 00290 *> 00291 *> \param[out] BERR 00292 *> \verbatim 00293 *> BERR is REAL array, dimension (NRHS) 00294 *> Componentwise relative backward error. This is the 00295 *> componentwise relative backward error of each solution vector X(j) 00296 *> (i.e., the smallest relative change in any element of A or B that 00297 *> makes X(j) an exact solution). 00298 *> \endverbatim 00299 *> 00300 *> \param[in] N_ERR_BNDS 00301 *> \verbatim 00302 *> N_ERR_BNDS is INTEGER 00303 *> Number of error bounds to return for each right hand side 00304 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and 00305 *> ERR_BNDS_COMP below. 00306 *> \endverbatim 00307 *> 00308 *> \param[out] ERR_BNDS_NORM 00309 *> \verbatim 00310 *> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 00311 *> For each right-hand side, this array contains information about 00312 *> various error bounds and condition numbers corresponding to the 00313 *> normwise relative error, which is defined as follows: 00314 *> 00315 *> Normwise relative error in the ith solution vector: 00316 *> max_j (abs(XTRUE(j,i) - X(j,i))) 00317 *> ------------------------------ 00318 *> max_j abs(X(j,i)) 00319 *> 00320 *> The array is indexed by the type of error information as described 00321 *> below. There currently are up to three pieces of information 00322 *> returned. 00323 *> 00324 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00325 *> right-hand side. 00326 *> 00327 *> The second index in ERR_BNDS_NORM(:,err) contains the following 00328 *> three fields: 00329 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00330 *> reciprocal condition number is less than the threshold 00331 *> sqrt(n) * slamch('Epsilon'). 00332 *> 00333 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00334 *> almost certainly within a factor of 10 of the true error 00335 *> so long as the next entry is greater than the threshold 00336 *> sqrt(n) * slamch('Epsilon'). This error bound should only 00337 *> be trusted if the previous boolean is true. 00338 *> 00339 *> err = 3 Reciprocal condition number: Estimated normwise 00340 *> reciprocal condition number. Compared with the threshold 00341 *> sqrt(n) * slamch('Epsilon') to determine if the error 00342 *> estimate is "guaranteed". These reciprocal condition 00343 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00344 *> appropriately scaled matrix Z. 00345 *> Let Z = S*A, where S scales each row by a power of the 00346 *> radix so all absolute row sums of Z are approximately 1. 00347 *> 00348 *> See Lapack Working Note 165 for further details and extra 00349 *> cautions. 00350 *> \endverbatim 00351 *> 00352 *> \param[out] ERR_BNDS_COMP 00353 *> \verbatim 00354 *> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 00355 *> For each right-hand side, this array contains information about 00356 *> various error bounds and condition numbers corresponding to the 00357 *> componentwise relative error, which is defined as follows: 00358 *> 00359 *> Componentwise relative error in the ith solution vector: 00360 *> abs(XTRUE(j,i) - X(j,i)) 00361 *> max_j ---------------------- 00362 *> abs(X(j,i)) 00363 *> 00364 *> The array is indexed by the right-hand side i (on which the 00365 *> componentwise relative error depends), and the type of error 00366 *> information as described below. There currently are up to three 00367 *> pieces of information returned for each right-hand side. If 00368 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00369 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00370 *> the first (:,N_ERR_BNDS) entries are returned. 00371 *> 00372 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00373 *> right-hand side. 00374 *> 00375 *> The second index in ERR_BNDS_COMP(:,err) contains the following 00376 *> three fields: 00377 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00378 *> reciprocal condition number is less than the threshold 00379 *> sqrt(n) * slamch('Epsilon'). 00380 *> 00381 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00382 *> almost certainly within a factor of 10 of the true error 00383 *> so long as the next entry is greater than the threshold 00384 *> sqrt(n) * slamch('Epsilon'). This error bound should only 00385 *> be trusted if the previous boolean is true. 00386 *> 00387 *> err = 3 Reciprocal condition number: Estimated componentwise 00388 *> reciprocal condition number. Compared with the threshold 00389 *> sqrt(n) * slamch('Epsilon') to determine if the error 00390 *> estimate is "guaranteed". These reciprocal condition 00391 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00392 *> appropriately scaled matrix Z. 00393 *> Let Z = S*(A*diag(x)), where x is the solution for the 00394 *> current right-hand side and S scales each row of 00395 *> A*diag(x) by a power of the radix so all absolute row 00396 *> sums of Z are approximately 1. 00397 *> 00398 *> See Lapack Working Note 165 for further details and extra 00399 *> cautions. 00400 *> \endverbatim 00401 *> 00402 *> \param[in] NPARAMS 00403 *> \verbatim 00404 *> NPARAMS is INTEGER 00405 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the 00406 *> PARAMS array is never referenced and default values are used. 00407 *> \endverbatim 00408 *> 00409 *> \param[in,out] PARAMS 00410 *> \verbatim 00411 *> PARAMS is / output) REAL array, dimension NPARAMS 00412 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then 00413 *> that entry will be filled with default value used for that 00414 *> parameter. Only positions up to NPARAMS are accessed; defaults 00415 *> are used for higher-numbered parameters. 00416 *> 00417 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00418 *> refinement or not. 00419 *> Default: 1.0 00420 *> = 0.0 : No refinement is performed, and no error bounds are 00421 *> computed. 00422 *> = 1.0 : Use the double-precision refinement algorithm, 00423 *> possibly with doubled-single computations if the 00424 *> compilation environment does not support DOUBLE 00425 *> PRECISION. 00426 *> (other values are reserved for future use) 00427 *> 00428 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00429 *> computations allowed for refinement. 00430 *> Default: 10 00431 *> Aggressive: Set to 100 to permit convergence using approximate 00432 *> factorizations or factorizations other than LU. If 00433 *> the factorization uses a technique other than 00434 *> Gaussian elimination, the guarantees in 00435 *> err_bnds_norm and err_bnds_comp may no longer be 00436 *> trustworthy. 00437 *> 00438 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00439 *> will attempt to find a solution with small componentwise 00440 *> relative error in the double-precision algorithm. Positive 00441 *> is true, 0.0 is false. 00442 *> Default: 1.0 (attempt componentwise convergence) 00443 *> \endverbatim 00444 *> 00445 *> \param[out] WORK 00446 *> \verbatim 00447 *> WORK is REAL array, dimension (4*N) 00448 *> \endverbatim 00449 *> 00450 *> \param[out] IWORK 00451 *> \verbatim 00452 *> IWORK is INTEGER array, dimension (N) 00453 *> \endverbatim 00454 *> 00455 *> \param[out] INFO 00456 *> \verbatim 00457 *> INFO is INTEGER 00458 *> = 0: Successful exit. The solution to every right-hand side is 00459 *> guaranteed. 00460 *> < 0: If INFO = -i, the i-th argument had an illegal value 00461 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00462 *> has been completed, but the factor U is exactly singular, so 00463 *> the solution and error bounds could not be computed. RCOND = 0 00464 *> is returned. 00465 *> = N+J: The solution corresponding to the Jth right-hand side is 00466 *> not guaranteed. The solutions corresponding to other right- 00467 *> hand sides K with K > J may not be guaranteed as well, but 00468 *> only the first such right-hand side is reported. If a small 00469 *> componentwise error is not requested (PARAMS(3) = 0.0) then 00470 *> the Jth right-hand side is the first with a normwise error 00471 *> bound that is not guaranteed (the smallest J such 00472 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00473 *> the Jth right-hand side is the first with either a normwise or 00474 *> componentwise error bound that is not guaranteed (the smallest 00475 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00476 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00477 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00478 *> about all of the right-hand sides check ERR_BNDS_NORM or 00479 *> ERR_BNDS_COMP. 00480 *> \endverbatim 00481 * 00482 * Authors: 00483 * ======== 00484 * 00485 *> \author Univ. of Tennessee 00486 *> \author Univ. of California Berkeley 00487 *> \author Univ. of Colorado Denver 00488 *> \author NAG Ltd. 00489 * 00490 *> \date April 2012 00491 * 00492 *> \ingroup realPOsolve 00493 * 00494 * ===================================================================== 00495 SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 00496 $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 00497 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 00498 $ NPARAMS, PARAMS, WORK, IWORK, INFO ) 00499 * 00500 * -- LAPACK driver routine (version 3.4.1) -- 00501 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00502 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00503 * April 2012 00504 * 00505 * .. Scalar Arguments .. 00506 CHARACTER EQUED, FACT, UPLO 00507 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 00508 $ N_ERR_BNDS 00509 REAL RCOND, RPVGRW 00510 * .. 00511 * .. Array Arguments .. 00512 INTEGER IWORK( * ) 00513 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00514 $ X( LDX, * ), WORK( * ) 00515 REAL S( * ), PARAMS( * ), BERR( * ), 00516 $ ERR_BNDS_NORM( NRHS, * ), 00517 $ ERR_BNDS_COMP( NRHS, * ) 00518 * .. 00519 * 00520 * ================================================================== 00521 * 00522 * .. Parameters .. 00523 REAL ZERO, ONE 00524 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00525 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00526 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00527 INTEGER CMP_ERR_I, PIV_GROWTH_I 00528 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00529 $ BERR_I = 3 ) 00530 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00531 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00532 $ PIV_GROWTH_I = 9 ) 00533 * .. 00534 * .. Local Scalars .. 00535 LOGICAL EQUIL, NOFACT, RCEQU 00536 INTEGER INFEQU, J 00537 REAL AMAX, BIGNUM, SMIN, SMAX, 00538 $ SCOND, SMLNUM 00539 * .. 00540 * .. External Functions .. 00541 EXTERNAL LSAME, SLAMCH, SLA_PORPVGRW 00542 LOGICAL LSAME 00543 REAL SLAMCH, SLA_PORPVGRW 00544 * .. 00545 * .. External Subroutines .. 00546 EXTERNAL SPOEQUB, SPOTRF, SPOTRS, SLACPY, SLAQSY, 00547 $ XERBLA, SLASCL2, SPORFSX 00548 * .. 00549 * .. Intrinsic Functions .. 00550 INTRINSIC MAX, MIN 00551 * .. 00552 * .. Executable Statements .. 00553 * 00554 INFO = 0 00555 NOFACT = LSAME( FACT, 'N' ) 00556 EQUIL = LSAME( FACT, 'E' ) 00557 SMLNUM = SLAMCH( 'Safe minimum' ) 00558 BIGNUM = ONE / SMLNUM 00559 IF( NOFACT .OR. EQUIL ) THEN 00560 EQUED = 'N' 00561 RCEQU = .FALSE. 00562 ELSE 00563 RCEQU = LSAME( EQUED, 'Y' ) 00564 ENDIF 00565 * 00566 * Default is failure. If an input parameter is wrong or 00567 * factorization fails, make everything look horrible. Only the 00568 * pivot growth is set here, the rest is initialized in SPORFSX. 00569 * 00570 RPVGRW = ZERO 00571 * 00572 * Test the input parameters. PARAMS is not tested until SPORFSX. 00573 * 00574 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00575 $ LSAME( FACT, 'F' ) ) THEN 00576 INFO = -1 00577 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. 00578 $ .NOT.LSAME( UPLO, 'L' ) ) THEN 00579 INFO = -2 00580 ELSE IF( N.LT.0 ) THEN 00581 INFO = -3 00582 ELSE IF( NRHS.LT.0 ) THEN 00583 INFO = -4 00584 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00585 INFO = -6 00586 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00587 INFO = -8 00588 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00589 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00590 INFO = -9 00591 ELSE 00592 IF ( RCEQU ) THEN 00593 SMIN = BIGNUM 00594 SMAX = ZERO 00595 DO 10 J = 1, N 00596 SMIN = MIN( SMIN, S( J ) ) 00597 SMAX = MAX( SMAX, S( J ) ) 00598 10 CONTINUE 00599 IF( SMIN.LE.ZERO ) THEN 00600 INFO = -10 00601 ELSE IF( N.GT.0 ) THEN 00602 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00603 ELSE 00604 SCOND = ONE 00605 END IF 00606 END IF 00607 IF( INFO.EQ.0 ) THEN 00608 IF( LDB.LT.MAX( 1, N ) ) THEN 00609 INFO = -12 00610 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00611 INFO = -14 00612 END IF 00613 END IF 00614 END IF 00615 * 00616 IF( INFO.NE.0 ) THEN 00617 CALL XERBLA( 'SPOSVXX', -INFO ) 00618 RETURN 00619 END IF 00620 * 00621 IF( EQUIL ) THEN 00622 * 00623 * Compute row and column scalings to equilibrate the matrix A. 00624 * 00625 CALL SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) 00626 IF( INFEQU.EQ.0 ) THEN 00627 * 00628 * Equilibrate the matrix. 00629 * 00630 CALL SLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00631 RCEQU = LSAME( EQUED, 'Y' ) 00632 END IF 00633 END IF 00634 * 00635 * Scale the right-hand side. 00636 * 00637 IF( RCEQU ) CALL SLASCL2( N, NRHS, S, B, LDB ) 00638 * 00639 IF( NOFACT .OR. EQUIL ) THEN 00640 * 00641 * Compute the Cholesky factorization of A. 00642 * 00643 CALL SLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00644 CALL SPOTRF( UPLO, N, AF, LDAF, INFO ) 00645 * 00646 * Return if INFO is non-zero. 00647 * 00648 IF( INFO.NE.0 ) THEN 00649 * 00650 * Pivot in column INFO is exactly 0 00651 * Compute the reciprocal pivot growth factor of the 00652 * leading rank-deficient INFO columns of A. 00653 * 00654 RPVGRW = SLA_PORPVGRW( UPLO, INFO, A, LDA, AF, LDAF, WORK ) 00655 RETURN 00656 ENDIF 00657 END IF 00658 * 00659 * Compute the reciprocal growth factor RPVGRW. 00660 * 00661 RPVGRW = SLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK ) 00662 * 00663 * Compute the solution matrix X. 00664 * 00665 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00666 CALL SPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 00667 * 00668 * Use iterative refinement to improve the computed solution and 00669 * compute error bounds and backward error estimates for it. 00670 * 00671 CALL SPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, 00672 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00673 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) 00674 00675 * 00676 * Scale solutions. 00677 * 00678 IF ( RCEQU ) THEN 00679 CALL SLASCL2 ( N, NRHS, S, X, LDX ) 00680 END IF 00681 * 00682 RETURN 00683 * 00684 * End of SPOSVXX 00685 * 00686 END