![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief <b> ZPOSVXX 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 ZPOSVXX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zposvxx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zposvxx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zposvxx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZPOSVXX( 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, RWORK, 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 * DOUBLE PRECISION RCOND, RPVGRW 00031 * .. 00032 * .. Array Arguments .. 00033 * COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00034 * $ WORK( * ), X( LDX, * ) 00035 * DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), 00036 * $ ERR_BNDS_NORM( NRHS, * ), 00037 * $ ERR_BNDS_COMP( NRHS, * ) 00038 * .. 00039 * 00040 * 00041 *> \par Purpose: 00042 * ============= 00043 *> 00044 *> \verbatim 00045 *> 00046 *> ZPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T 00047 *> to compute the solution to a complex*16 system of linear equations 00048 *> A * X = B, where A is an N-by-N symmetric positive definite matrix 00049 *> and X and B are N-by-NRHS matrices. 00050 *> 00051 *> If requested, both normwise and maximum componentwise error bounds 00052 *> are returned. ZPOSVXX will return a solution with a tiny 00053 *> guaranteed error (O(eps) where eps is the working machine 00054 *> precision) unless the matrix is very ill-conditioned, in which 00055 *> case a warning is returned. Relevant condition numbers also are 00056 *> calculated and returned. 00057 *> 00058 *> ZPOSVXX accepts user-provided factorizations and equilibration 00059 *> factors; see the definitions of the FACT and EQUED options. 00060 *> Solving with refinement and using a factorization from a previous 00061 *> ZPOSVXX call will also produce a solution with either O(eps) 00062 *> errors or warnings, but we cannot make that claim for general 00063 *> user-provided factorizations and equilibration factors if they 00064 *> differ from what ZPOSVXX would itself produce. 00065 *> \endverbatim 00066 * 00067 *> \par Description: 00068 * ================= 00069 *> 00070 *> \verbatim 00071 *> 00072 *> The following steps are performed: 00073 *> 00074 *> 1. If FACT = 'E', double precision scaling factors are computed to equilibrate 00075 *> the system: 00076 *> 00077 *> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 00078 *> 00079 *> Whether or not the system will be equilibrated depends on the 00080 *> scaling of the matrix A, but if equilibration is used, A is 00081 *> overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00082 *> 00083 *> 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00084 *> factor the matrix A (after equilibration if FACT = 'E') as 00085 *> A = U**T* U, if UPLO = 'U', or 00086 *> A = L * L**T, if UPLO = 'L', 00087 *> where U is an upper triangular matrix and L is a lower triangular 00088 *> matrix. 00089 *> 00090 *> 3. If the leading i-by-i principal minor is not positive definite, 00091 *> then the routine returns with INFO = i. Otherwise, the factored 00092 *> form of A is used to estimate the condition number of the matrix 00093 *> A (see argument RCOND). If the reciprocal of the condition number 00094 *> is less than machine precision, the routine still goes on to solve 00095 *> for X and compute error bounds as described below. 00096 *> 00097 *> 4. The system of equations is solved for X using the factored form 00098 *> of A. 00099 *> 00100 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00101 *> the routine will use iterative refinement to try to get a small 00102 *> error and error bounds. Refinement calculates the residual to at 00103 *> least twice the working precision. 00104 *> 00105 *> 6. If equilibration was used, the matrix X is premultiplied by 00106 *> diag(S) so that it solves the original system before 00107 *> equilibration. 00108 *> \endverbatim 00109 * 00110 * Arguments: 00111 * ========== 00112 * 00113 *> \verbatim 00114 *> Some optional parameters are bundled in the PARAMS array. These 00115 *> settings determine how refinement is performed, but often the 00116 *> defaults are acceptable. If the defaults are acceptable, users 00117 *> can pass NPARAMS = 0 which prevents the source code from accessing 00118 *> the PARAMS argument. 00119 *> \endverbatim 00120 *> 00121 *> \param[in] FACT 00122 *> \verbatim 00123 *> FACT is CHARACTER*1 00124 *> Specifies whether or not the factored form of the matrix A is 00125 *> supplied on entry, and if not, whether the matrix A should be 00126 *> equilibrated before it is factored. 00127 *> = 'F': On entry, AF contains the factored form of A. 00128 *> If EQUED is not 'N', the matrix A has been 00129 *> equilibrated with scaling factors given by S. 00130 *> A and AF are not modified. 00131 *> = 'N': The matrix A will be copied to AF and factored. 00132 *> = 'E': The matrix A will be equilibrated if necessary, then 00133 *> copied to AF and factored. 00134 *> \endverbatim 00135 *> 00136 *> \param[in] UPLO 00137 *> \verbatim 00138 *> UPLO is CHARACTER*1 00139 *> = 'U': Upper triangle of A is stored; 00140 *> = 'L': Lower triangle of A is stored. 00141 *> \endverbatim 00142 *> 00143 *> \param[in] N 00144 *> \verbatim 00145 *> N is INTEGER 00146 *> The number of linear equations, i.e., the order of the 00147 *> matrix A. N >= 0. 00148 *> \endverbatim 00149 *> 00150 *> \param[in] NRHS 00151 *> \verbatim 00152 *> NRHS is INTEGER 00153 *> The number of right hand sides, i.e., the number of columns 00154 *> of the matrices B and X. NRHS >= 0. 00155 *> \endverbatim 00156 *> 00157 *> \param[in,out] A 00158 *> \verbatim 00159 *> A is COMPLEX*16 array, dimension (LDA,N) 00160 *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 00161 *> 'Y', then A must contain the equilibrated matrix 00162 *> diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper 00163 *> triangular part of A contains the upper triangular part of the 00164 *> matrix A, and the strictly lower triangular part of A is not 00165 *> referenced. If UPLO = 'L', the leading N-by-N lower triangular 00166 *> part of A contains the lower triangular part of the matrix A, and 00167 *> the strictly upper triangular part of A is not referenced. A is 00168 *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 00169 *> 'N' on exit. 00170 *> 00171 *> On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00172 *> diag(S)*A*diag(S). 00173 *> \endverbatim 00174 *> 00175 *> \param[in] LDA 00176 *> \verbatim 00177 *> LDA is INTEGER 00178 *> The leading dimension of the array A. LDA >= max(1,N). 00179 *> \endverbatim 00180 *> 00181 *> \param[in,out] AF 00182 *> \verbatim 00183 *> AF is COMPLEX*16 array, dimension (LDAF,N) 00184 *> If FACT = 'F', then AF is an input argument and on entry 00185 *> contains the triangular factor U or L from the Cholesky 00186 *> factorization A = U**T*U or A = L*L**T, in the same storage 00187 *> format as A. If EQUED .ne. 'N', then AF is the factored 00188 *> form of the equilibrated matrix diag(S)*A*diag(S). 00189 *> 00190 *> If FACT = 'N', then AF is an output argument and on exit 00191 *> returns the triangular factor U or L from the Cholesky 00192 *> factorization A = U**T*U or A = L*L**T of the original 00193 *> matrix A. 00194 *> 00195 *> If FACT = 'E', then AF is an output argument and on exit 00196 *> returns the triangular factor U or L from the Cholesky 00197 *> factorization A = U**T*U or A = L*L**T of the equilibrated 00198 *> matrix A (see the description of A for the form of the 00199 *> equilibrated matrix). 00200 *> \endverbatim 00201 *> 00202 *> \param[in] LDAF 00203 *> \verbatim 00204 *> LDAF is INTEGER 00205 *> The leading dimension of the array AF. LDAF >= max(1,N). 00206 *> \endverbatim 00207 *> 00208 *> \param[in,out] EQUED 00209 *> \verbatim 00210 *> EQUED is CHARACTER*1 00211 *> Specifies the form of equilibration that was done. 00212 *> = 'N': No equilibration (always true if FACT = 'N'). 00213 *> = 'Y': Both row and column equilibration, i.e., A has been 00214 *> replaced by diag(S) * A * diag(S). 00215 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an 00216 *> output argument. 00217 *> \endverbatim 00218 *> 00219 *> \param[in,out] S 00220 *> \verbatim 00221 *> S is DOUBLE PRECISION array, dimension (N) 00222 *> The row scale factors for A. If EQUED = 'Y', A is multiplied on 00223 *> the left and right by diag(S). S is an input argument if FACT = 00224 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00225 *> = 'Y', each element of S must be positive. If S is output, each 00226 *> element of S is a power of the radix. If S is input, each element 00227 *> of S should be a power of the radix to ensure a reliable solution 00228 *> and error estimates. Scaling by powers of the radix does not cause 00229 *> rounding errors unless the result underflows or overflows. 00230 *> Rounding errors during scaling lead to refining with a matrix that 00231 *> is not equivalent to the input matrix, producing error estimates 00232 *> that may not be reliable. 00233 *> \endverbatim 00234 *> 00235 *> \param[in,out] B 00236 *> \verbatim 00237 *> B is COMPLEX*16 array, dimension (LDB,NRHS) 00238 *> On entry, the N-by-NRHS right hand side matrix B. 00239 *> On exit, 00240 *> if EQUED = 'N', B is not modified; 00241 *> if EQUED = 'Y', B is overwritten by diag(S)*B; 00242 *> \endverbatim 00243 *> 00244 *> \param[in] LDB 00245 *> \verbatim 00246 *> LDB is INTEGER 00247 *> The leading dimension of the array B. LDB >= max(1,N). 00248 *> \endverbatim 00249 *> 00250 *> \param[out] X 00251 *> \verbatim 00252 *> X is COMPLEX*16 array, dimension (LDX,NRHS) 00253 *> If INFO = 0, the N-by-NRHS solution matrix X to the original 00254 *> system of equations. Note that A and B are modified on exit if 00255 *> EQUED .ne. 'N', and the solution to the equilibrated system is 00256 *> inv(diag(S))*X. 00257 *> \endverbatim 00258 *> 00259 *> \param[in] LDX 00260 *> \verbatim 00261 *> LDX is INTEGER 00262 *> The leading dimension of the array X. LDX >= max(1,N). 00263 *> \endverbatim 00264 *> 00265 *> \param[out] RCOND 00266 *> \verbatim 00267 *> RCOND is DOUBLE PRECISION 00268 *> Reciprocal scaled condition number. This is an estimate of the 00269 *> reciprocal Skeel condition number of the matrix A after 00270 *> equilibration (if done). If this is less than the machine 00271 *> precision (in particular, if it is zero), the matrix is singular 00272 *> to working precision. Note that the error may still be small even 00273 *> if this number is very small and the matrix appears ill- 00274 *> conditioned. 00275 *> \endverbatim 00276 *> 00277 *> \param[out] RPVGRW 00278 *> \verbatim 00279 *> RPVGRW is DOUBLE PRECISION 00280 *> Reciprocal pivot growth. On exit, this contains the reciprocal 00281 *> pivot growth factor norm(A)/norm(U). The "max absolute element" 00282 *> norm is used. If this is much less than 1, then the stability of 00283 *> the LU factorization of the (equilibrated) matrix A could be poor. 00284 *> This also means that the solution X, estimated condition numbers, 00285 *> and error bounds could be unreliable. If factorization fails with 00286 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor 00287 *> for the leading INFO columns of A. 00288 *> \endverbatim 00289 *> 00290 *> \param[out] BERR 00291 *> \verbatim 00292 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00293 *> Componentwise relative backward error. This is the 00294 *> componentwise relative backward error of each solution vector X(j) 00295 *> (i.e., the smallest relative change in any element of A or B that 00296 *> makes X(j) an exact solution). 00297 *> \endverbatim 00298 *> 00299 *> \param[in] N_ERR_BNDS 00300 *> \verbatim 00301 *> N_ERR_BNDS is INTEGER 00302 *> Number of error bounds to return for each right hand side 00303 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and 00304 *> ERR_BNDS_COMP below. 00305 *> \endverbatim 00306 *> 00307 *> \param[out] ERR_BNDS_NORM 00308 *> \verbatim 00309 *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00310 *> For each right-hand side, this array contains information about 00311 *> various error bounds and condition numbers corresponding to the 00312 *> normwise relative error, which is defined as follows: 00313 *> 00314 *> Normwise relative error in the ith solution vector: 00315 *> max_j (abs(XTRUE(j,i) - X(j,i))) 00316 *> ------------------------------ 00317 *> max_j abs(X(j,i)) 00318 *> 00319 *> The array is indexed by the type of error information as described 00320 *> below. There currently are up to three pieces of information 00321 *> returned. 00322 *> 00323 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00324 *> right-hand side. 00325 *> 00326 *> The second index in ERR_BNDS_NORM(:,err) contains the following 00327 *> three fields: 00328 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00329 *> reciprocal condition number is less than the threshold 00330 *> sqrt(n) * dlamch('Epsilon'). 00331 *> 00332 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00333 *> almost certainly within a factor of 10 of the true error 00334 *> so long as the next entry is greater than the threshold 00335 *> sqrt(n) * dlamch('Epsilon'). This error bound should only 00336 *> be trusted if the previous boolean is true. 00337 *> 00338 *> err = 3 Reciprocal condition number: Estimated normwise 00339 *> reciprocal condition number. Compared with the threshold 00340 *> sqrt(n) * dlamch('Epsilon') to determine if the error 00341 *> estimate is "guaranteed". These reciprocal condition 00342 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00343 *> appropriately scaled matrix Z. 00344 *> Let Z = S*A, where S scales each row by a power of the 00345 *> radix so all absolute row sums of Z are approximately 1. 00346 *> 00347 *> See Lapack Working Note 165 for further details and extra 00348 *> cautions. 00349 *> \endverbatim 00350 *> 00351 *> \param[out] ERR_BNDS_COMP 00352 *> \verbatim 00353 *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00354 *> For each right-hand side, this array contains information about 00355 *> various error bounds and condition numbers corresponding to the 00356 *> componentwise relative error, which is defined as follows: 00357 *> 00358 *> Componentwise relative error in the ith solution vector: 00359 *> abs(XTRUE(j,i) - X(j,i)) 00360 *> max_j ---------------------- 00361 *> abs(X(j,i)) 00362 *> 00363 *> The array is indexed by the right-hand side i (on which the 00364 *> componentwise relative error depends), and the type of error 00365 *> information as described below. There currently are up to three 00366 *> pieces of information returned for each right-hand side. If 00367 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00368 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00369 *> the first (:,N_ERR_BNDS) entries are returned. 00370 *> 00371 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00372 *> right-hand side. 00373 *> 00374 *> The second index in ERR_BNDS_COMP(:,err) contains the following 00375 *> three fields: 00376 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00377 *> reciprocal condition number is less than the threshold 00378 *> sqrt(n) * dlamch('Epsilon'). 00379 *> 00380 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00381 *> almost certainly within a factor of 10 of the true error 00382 *> so long as the next entry is greater than the threshold 00383 *> sqrt(n) * dlamch('Epsilon'). This error bound should only 00384 *> be trusted if the previous boolean is true. 00385 *> 00386 *> err = 3 Reciprocal condition number: Estimated componentwise 00387 *> reciprocal condition number. Compared with the threshold 00388 *> sqrt(n) * dlamch('Epsilon') to determine if the error 00389 *> estimate is "guaranteed". These reciprocal condition 00390 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00391 *> appropriately scaled matrix Z. 00392 *> Let Z = S*(A*diag(x)), where x is the solution for the 00393 *> current right-hand side and S scales each row of 00394 *> A*diag(x) by a power of the radix so all absolute row 00395 *> sums of Z are approximately 1. 00396 *> 00397 *> See Lapack Working Note 165 for further details and extra 00398 *> cautions. 00399 *> \endverbatim 00400 *> 00401 *> \param[in] NPARAMS 00402 *> \verbatim 00403 *> NPARAMS is INTEGER 00404 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the 00405 *> PARAMS array is never referenced and default values are used. 00406 *> \endverbatim 00407 *> 00408 *> \param[in,out] PARAMS 00409 *> \verbatim 00410 *> PARAMS is / output) DOUBLE PRECISION array, dimension NPARAMS 00411 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then 00412 *> that entry will be filled with default value used for that 00413 *> parameter. Only positions up to NPARAMS are accessed; defaults 00414 *> are used for higher-numbered parameters. 00415 *> 00416 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00417 *> refinement or not. 00418 *> Default: 1.0D+0 00419 *> = 0.0 : No refinement is performed, and no error bounds are 00420 *> computed. 00421 *> = 1.0 : Use the extra-precise refinement algorithm. 00422 *> (other values are reserved for future use) 00423 *> 00424 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00425 *> computations allowed for refinement. 00426 *> Default: 10 00427 *> Aggressive: Set to 100 to permit convergence using approximate 00428 *> factorizations or factorizations other than LU. If 00429 *> the factorization uses a technique other than 00430 *> Gaussian elimination, the guarantees in 00431 *> err_bnds_norm and err_bnds_comp may no longer be 00432 *> trustworthy. 00433 *> 00434 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00435 *> will attempt to find a solution with small componentwise 00436 *> relative error in the double-precision algorithm. Positive 00437 *> is true, 0.0 is false. 00438 *> Default: 1.0 (attempt componentwise convergence) 00439 *> \endverbatim 00440 *> 00441 *> \param[out] WORK 00442 *> \verbatim 00443 *> WORK is COMPLEX*16 array, dimension (2*N) 00444 *> \endverbatim 00445 *> 00446 *> \param[out] RWORK 00447 *> \verbatim 00448 *> RWORK is DOUBLE PRECISION array, dimension (2*N) 00449 *> \endverbatim 00450 *> 00451 *> \param[out] INFO 00452 *> \verbatim 00453 *> INFO is INTEGER 00454 *> = 0: Successful exit. The solution to every right-hand side is 00455 *> guaranteed. 00456 *> < 0: If INFO = -i, the i-th argument had an illegal value 00457 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00458 *> has been completed, but the factor U is exactly singular, so 00459 *> the solution and error bounds could not be computed. RCOND = 0 00460 *> is returned. 00461 *> = N+J: The solution corresponding to the Jth right-hand side is 00462 *> not guaranteed. The solutions corresponding to other right- 00463 *> hand sides K with K > J may not be guaranteed as well, but 00464 *> only the first such right-hand side is reported. If a small 00465 *> componentwise error is not requested (PARAMS(3) = 0.0) then 00466 *> the Jth right-hand side is the first with a normwise error 00467 *> bound that is not guaranteed (the smallest J such 00468 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00469 *> the Jth right-hand side is the first with either a normwise or 00470 *> componentwise error bound that is not guaranteed (the smallest 00471 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00472 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00473 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00474 *> about all of the right-hand sides check ERR_BNDS_NORM or 00475 *> ERR_BNDS_COMP. 00476 *> \endverbatim 00477 * 00478 * Authors: 00479 * ======== 00480 * 00481 *> \author Univ. of Tennessee 00482 *> \author Univ. of California Berkeley 00483 *> \author Univ. of Colorado Denver 00484 *> \author NAG Ltd. 00485 * 00486 *> \date April 2012 00487 * 00488 *> \ingroup complex16POsolve 00489 * 00490 * ===================================================================== 00491 SUBROUTINE ZPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 00492 $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 00493 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 00494 $ NPARAMS, PARAMS, WORK, RWORK, INFO ) 00495 * 00496 * -- LAPACK driver routine (version 3.4.1) -- 00497 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00498 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00499 * April 2012 00500 * 00501 * .. Scalar Arguments .. 00502 CHARACTER EQUED, FACT, UPLO 00503 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 00504 $ N_ERR_BNDS 00505 DOUBLE PRECISION RCOND, RPVGRW 00506 * .. 00507 * .. Array Arguments .. 00508 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00509 $ WORK( * ), X( LDX, * ) 00510 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), 00511 $ ERR_BNDS_NORM( NRHS, * ), 00512 $ ERR_BNDS_COMP( NRHS, * ) 00513 * .. 00514 * 00515 * ================================================================== 00516 * 00517 * .. Parameters .. 00518 DOUBLE PRECISION ZERO, ONE 00519 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00520 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00521 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00522 INTEGER CMP_ERR_I, PIV_GROWTH_I 00523 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00524 $ BERR_I = 3 ) 00525 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00526 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00527 $ PIV_GROWTH_I = 9 ) 00528 * .. 00529 * .. Local Scalars .. 00530 LOGICAL EQUIL, NOFACT, RCEQU 00531 INTEGER INFEQU, J 00532 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 00533 * .. 00534 * .. External Functions .. 00535 EXTERNAL LSAME, DLAMCH, ZLA_PORPVGRW 00536 LOGICAL LSAME 00537 DOUBLE PRECISION DLAMCH, ZLA_PORPVGRW 00538 * .. 00539 * .. External Subroutines .. 00540 EXTERNAL ZPOCON, ZPOEQUB, ZPOTRF, ZPOTRS, ZLACPY, 00541 $ ZLAQHE, XERBLA, ZLASCL2, ZPORFSX 00542 * .. 00543 * .. Intrinsic Functions .. 00544 INTRINSIC MAX, MIN 00545 * .. 00546 * .. Executable Statements .. 00547 * 00548 INFO = 0 00549 NOFACT = LSAME( FACT, 'N' ) 00550 EQUIL = LSAME( FACT, 'E' ) 00551 SMLNUM = DLAMCH( 'Safe minimum' ) 00552 BIGNUM = ONE / SMLNUM 00553 IF( NOFACT .OR. EQUIL ) THEN 00554 EQUED = 'N' 00555 RCEQU = .FALSE. 00556 ELSE 00557 RCEQU = LSAME( EQUED, 'Y' ) 00558 ENDIF 00559 * 00560 * Default is failure. If an input parameter is wrong or 00561 * factorization fails, make everything look horrible. Only the 00562 * pivot growth is set here, the rest is initialized in ZPORFSX. 00563 * 00564 RPVGRW = ZERO 00565 * 00566 * Test the input parameters. PARAMS is not tested until ZPORFSX. 00567 * 00568 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00569 $ LSAME( FACT, 'F' ) ) THEN 00570 INFO = -1 00571 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. 00572 $ .NOT.LSAME( UPLO, 'L' ) ) THEN 00573 INFO = -2 00574 ELSE IF( N.LT.0 ) THEN 00575 INFO = -3 00576 ELSE IF( NRHS.LT.0 ) THEN 00577 INFO = -4 00578 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00579 INFO = -6 00580 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00581 INFO = -8 00582 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00583 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00584 INFO = -9 00585 ELSE 00586 IF ( RCEQU ) THEN 00587 SMIN = BIGNUM 00588 SMAX = ZERO 00589 DO 10 J = 1, N 00590 SMIN = MIN( SMIN, S( J ) ) 00591 SMAX = MAX( SMAX, S( J ) ) 00592 10 CONTINUE 00593 IF( SMIN.LE.ZERO ) THEN 00594 INFO = -10 00595 ELSE IF( N.GT.0 ) THEN 00596 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00597 ELSE 00598 SCOND = ONE 00599 END IF 00600 END IF 00601 IF( INFO.EQ.0 ) THEN 00602 IF( LDB.LT.MAX( 1, N ) ) THEN 00603 INFO = -12 00604 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00605 INFO = -14 00606 END IF 00607 END IF 00608 END IF 00609 * 00610 IF( INFO.NE.0 ) THEN 00611 CALL XERBLA( 'ZPOSVXX', -INFO ) 00612 RETURN 00613 END IF 00614 * 00615 IF( EQUIL ) THEN 00616 * 00617 * Compute row and column scalings to equilibrate the matrix A. 00618 * 00619 CALL ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) 00620 IF( INFEQU.EQ.0 ) THEN 00621 * 00622 * Equilibrate the matrix. 00623 * 00624 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00625 RCEQU = LSAME( EQUED, 'Y' ) 00626 END IF 00627 END IF 00628 * 00629 * Scale the right-hand side. 00630 * 00631 IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB ) 00632 * 00633 IF( NOFACT .OR. EQUIL ) THEN 00634 * 00635 * Compute the Cholesky factorization of A. 00636 * 00637 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00638 CALL ZPOTRF( UPLO, N, AF, LDAF, INFO ) 00639 * 00640 * Return if INFO is non-zero. 00641 * 00642 IF( INFO.GT.0 ) THEN 00643 * 00644 * Pivot in column INFO is exactly 0 00645 * Compute the reciprocal pivot growth factor of the 00646 * leading rank-deficient INFO columns of A. 00647 * 00648 RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 00649 RETURN 00650 END IF 00651 END IF 00652 * 00653 * Compute the reciprocal pivot growth factor RPVGRW. 00654 * 00655 RPVGRW = ZLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, RWORK ) 00656 * 00657 * Compute the solution matrix X. 00658 * 00659 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00660 CALL ZPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 00661 * 00662 * Use iterative refinement to improve the computed solution and 00663 * compute error bounds and backward error estimates for it. 00664 * 00665 CALL ZPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, 00666 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00667 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 00668 00669 * 00670 * Scale solutions. 00671 * 00672 IF ( RCEQU ) THEN 00673 CALL ZLASCL2( N, NRHS, S, X, LDX ) 00674 END IF 00675 * 00676 RETURN 00677 * 00678 * End of ZPOSVXX 00679 * 00680 END