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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DSYSVXX 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download DSYSVXX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsysvxx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsysvxx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsysvxx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 00022 * EQUED, 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 * DOUBLE PRECISION RCOND, RPVGRW 00031 * .. 00032 * .. Array Arguments .. 00033 * INTEGER IPIV( * ), IWORK( * ) 00034 * DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00035 * $ X( LDX, * ), WORK( * ) 00036 * DOUBLE PRECISION 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 *> DSYSVXX uses the diagonal pivoting factorization to compute the 00048 *> solution to a double precision system of linear equations A * X = B, where A 00049 *> is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. 00050 *> 00051 *> If requested, both normwise and maximum componentwise error bounds 00052 *> are returned. DSYSVXX 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 *> DSYSVXX 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 *> DSYSVXX 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 DSYSVXX 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 LU decomposition is used to factor 00084 *> the matrix A (after equilibration if FACT = 'E') as 00085 *> 00086 *> A = U * D * U**T, if UPLO = 'U', or 00087 *> A = L * D * L**T, if UPLO = 'L', 00088 *> 00089 *> where U (or L) is a product of permutation and unit upper (lower) 00090 *> triangular matrices, and D is symmetric and block diagonal with 00091 *> 1-by-1 and 2-by-2 diagonal blocks. 00092 *> 00093 *> 3. If some D(i,i)=0, so that D is exactly singular, then the 00094 *> routine returns with INFO = i. Otherwise, the factored form of A 00095 *> is used to estimate the condition number of the matrix A (see 00096 *> argument RCOND). If the reciprocal of the condition number is 00097 *> less than machine precision, the routine still goes on to solve 00098 *> for X and compute error bounds as described below. 00099 *> 00100 *> 4. The system of equations is solved for X using the factored form 00101 *> of A. 00102 *> 00103 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00104 *> the routine will use iterative refinement to try to get a small 00105 *> error and error bounds. Refinement calculates the residual to at 00106 *> least twice the working precision. 00107 *> 00108 *> 6. If equilibration was used, the matrix X is premultiplied by 00109 *> diag(R) so that it solves the original system before 00110 *> equilibration. 00111 *> \endverbatim 00112 * 00113 * Arguments: 00114 * ========== 00115 * 00116 *> \verbatim 00117 *> Some optional parameters are bundled in the PARAMS array. These 00118 *> settings determine how refinement is performed, but often the 00119 *> defaults are acceptable. If the defaults are acceptable, users 00120 *> can pass NPARAMS = 0 which prevents the source code from accessing 00121 *> the PARAMS argument. 00122 *> \endverbatim 00123 *> 00124 *> \param[in] FACT 00125 *> \verbatim 00126 *> FACT is CHARACTER*1 00127 *> Specifies whether or not the factored form of the matrix A is 00128 *> supplied on entry, and if not, whether the matrix A should be 00129 *> equilibrated before it is factored. 00130 *> = 'F': On entry, AF and IPIV contain the factored form of A. 00131 *> If EQUED is not 'N', the matrix A has been 00132 *> equilibrated with scaling factors given by S. 00133 *> A, AF, and IPIV are not modified. 00134 *> = 'N': The matrix A will be copied to AF and factored. 00135 *> = 'E': The matrix A will be equilibrated if necessary, then 00136 *> copied to AF and factored. 00137 *> \endverbatim 00138 *> 00139 *> \param[in] UPLO 00140 *> \verbatim 00141 *> UPLO is CHARACTER*1 00142 *> = 'U': Upper triangle of A is stored; 00143 *> = 'L': Lower triangle of A is stored. 00144 *> \endverbatim 00145 *> 00146 *> \param[in] N 00147 *> \verbatim 00148 *> N is INTEGER 00149 *> The number of linear equations, i.e., the order of the 00150 *> matrix A. N >= 0. 00151 *> \endverbatim 00152 *> 00153 *> \param[in] NRHS 00154 *> \verbatim 00155 *> NRHS is INTEGER 00156 *> The number of right hand sides, i.e., the number of columns 00157 *> of the matrices B and X. NRHS >= 0. 00158 *> \endverbatim 00159 *> 00160 *> \param[in,out] A 00161 *> \verbatim 00162 *> A is DOUBLE PRECISION array, dimension (LDA,N) 00163 *> The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00164 *> upper triangular part of A contains the upper triangular 00165 *> part of the matrix A, and the strictly lower triangular 00166 *> part of A is not referenced. If UPLO = 'L', the leading 00167 *> N-by-N lower triangular part of A contains the lower 00168 *> triangular part of the matrix A, and the strictly upper 00169 *> triangular part of A is not referenced. 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 DOUBLE PRECISION array, dimension (LDAF,N) 00184 *> If FACT = 'F', then AF is an input argument and on entry 00185 *> contains the block diagonal matrix D and the multipliers 00186 *> used to obtain the factor U or L from the factorization A = 00187 *> U*D*U**T or A = L*D*L**T as computed by DSYTRF. 00188 *> 00189 *> If FACT = 'N', then AF is an output argument and on exit 00190 *> returns the block diagonal matrix D and the multipliers 00191 *> used to obtain the factor U or L from the factorization A = 00192 *> U*D*U**T or A = L*D*L**T. 00193 *> \endverbatim 00194 *> 00195 *> \param[in] LDAF 00196 *> \verbatim 00197 *> LDAF is INTEGER 00198 *> The leading dimension of the array AF. LDAF >= max(1,N). 00199 *> \endverbatim 00200 *> 00201 *> \param[in,out] IPIV 00202 *> \verbatim 00203 *> IPIV is INTEGER array, dimension (N) 00204 *> If FACT = 'F', then IPIV is an input argument and on entry 00205 *> contains details of the interchanges and the block 00206 *> structure of D, as determined by DSYTRF. If IPIV(k) > 0, 00207 *> then rows and columns k and IPIV(k) were interchanged and 00208 *> D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and 00209 *> IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and 00210 *> -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 00211 *> diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, 00212 *> then rows and columns k+1 and -IPIV(k) were interchanged 00213 *> and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00214 *> 00215 *> If FACT = 'N', then IPIV is an output argument and on exit 00216 *> contains details of the interchanges and the block 00217 *> structure of D, as determined by DSYTRF. 00218 *> \endverbatim 00219 *> 00220 *> \param[in,out] EQUED 00221 *> \verbatim 00222 *> EQUED is CHARACTER*1 00223 *> Specifies the form of equilibration that was done. 00224 *> = 'N': No equilibration (always true if FACT = 'N'). 00225 *> = 'Y': Both row and column equilibration, i.e., A has been 00226 *> replaced by diag(S) * A * diag(S). 00227 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an 00228 *> output argument. 00229 *> \endverbatim 00230 *> 00231 *> \param[in,out] S 00232 *> \verbatim 00233 *> S is DOUBLE PRECISION array, dimension (N) 00234 *> The scale factors for A. If EQUED = 'Y', A is multiplied on 00235 *> the left and right by diag(S). S is an input argument if FACT = 00236 *> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00237 *> = 'Y', each element of S must be positive. If S is output, each 00238 *> element of S is a power of the radix. If S is input, each element 00239 *> of S should be a power of the radix to ensure a reliable solution 00240 *> and error estimates. Scaling by powers of the radix does not cause 00241 *> rounding errors unless the result underflows or overflows. 00242 *> Rounding errors during scaling lead to refining with a matrix that 00243 *> is not equivalent to the input matrix, producing error estimates 00244 *> that may not be reliable. 00245 *> \endverbatim 00246 *> 00247 *> \param[in,out] B 00248 *> \verbatim 00249 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00250 *> On entry, the N-by-NRHS right hand side matrix B. 00251 *> On exit, 00252 *> if EQUED = 'N', B is not modified; 00253 *> if EQUED = 'Y', B is overwritten by diag(S)*B; 00254 *> \endverbatim 00255 *> 00256 *> \param[in] LDB 00257 *> \verbatim 00258 *> LDB is INTEGER 00259 *> The leading dimension of the array B. LDB >= max(1,N). 00260 *> \endverbatim 00261 *> 00262 *> \param[out] X 00263 *> \verbatim 00264 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00265 *> If INFO = 0, the N-by-NRHS solution matrix X to the original 00266 *> system of equations. Note that A and B are modified on exit if 00267 *> EQUED .ne. 'N', and the solution to the equilibrated system is 00268 *> inv(diag(S))*X. 00269 *> \endverbatim 00270 *> 00271 *> \param[in] LDX 00272 *> \verbatim 00273 *> LDX is INTEGER 00274 *> The leading dimension of the array X. LDX >= max(1,N). 00275 *> \endverbatim 00276 *> 00277 *> \param[out] RCOND 00278 *> \verbatim 00279 *> RCOND is DOUBLE PRECISION 00280 *> Reciprocal scaled condition number. This is an estimate of the 00281 *> reciprocal Skeel condition number of the matrix A after 00282 *> equilibration (if done). If this is less than the machine 00283 *> precision (in particular, if it is zero), the matrix is singular 00284 *> to working precision. Note that the error may still be small even 00285 *> if this number is very small and the matrix appears ill- 00286 *> conditioned. 00287 *> \endverbatim 00288 *> 00289 *> \param[out] RPVGRW 00290 *> \verbatim 00291 *> RPVGRW is DOUBLE PRECISION 00292 *> Reciprocal pivot growth. On exit, this contains the reciprocal 00293 *> pivot growth factor norm(A)/norm(U). The "max absolute element" 00294 *> norm is used. If this is much less than 1, then the stability of 00295 *> the LU factorization of the (equilibrated) matrix A could be poor. 00296 *> This also means that the solution X, estimated condition numbers, 00297 *> and error bounds could be unreliable. If factorization fails with 00298 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor 00299 *> for the leading INFO columns of A. 00300 *> \endverbatim 00301 *> 00302 *> \param[out] BERR 00303 *> \verbatim 00304 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00305 *> Componentwise relative backward error. This is the 00306 *> componentwise relative backward error of each solution vector X(j) 00307 *> (i.e., the smallest relative change in any element of A or B that 00308 *> makes X(j) an exact solution). 00309 *> \endverbatim 00310 *> 00311 *> \param[in] N_ERR_BNDS 00312 *> \verbatim 00313 *> N_ERR_BNDS is INTEGER 00314 *> Number of error bounds to return for each right hand side 00315 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and 00316 *> ERR_BNDS_COMP below. 00317 *> \endverbatim 00318 *> 00319 *> \param[out] ERR_BNDS_NORM 00320 *> \verbatim 00321 *> ERR_BNDS_NORM is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00322 *> For each right-hand side, this array contains information about 00323 *> various error bounds and condition numbers corresponding to the 00324 *> normwise relative error, which is defined as follows: 00325 *> 00326 *> Normwise relative error in the ith solution vector: 00327 *> max_j (abs(XTRUE(j,i) - X(j,i))) 00328 *> ------------------------------ 00329 *> max_j abs(X(j,i)) 00330 *> 00331 *> The array is indexed by the type of error information as described 00332 *> below. There currently are up to three pieces of information 00333 *> returned. 00334 *> 00335 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00336 *> right-hand side. 00337 *> 00338 *> The second index in ERR_BNDS_NORM(:,err) contains the following 00339 *> three fields: 00340 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00341 *> reciprocal condition number is less than the threshold 00342 *> sqrt(n) * dlamch('Epsilon'). 00343 *> 00344 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00345 *> almost certainly within a factor of 10 of the true error 00346 *> so long as the next entry is greater than the threshold 00347 *> sqrt(n) * dlamch('Epsilon'). This error bound should only 00348 *> be trusted if the previous boolean is true. 00349 *> 00350 *> err = 3 Reciprocal condition number: Estimated normwise 00351 *> reciprocal condition number. Compared with the threshold 00352 *> sqrt(n) * dlamch('Epsilon') to determine if the error 00353 *> estimate is "guaranteed". These reciprocal condition 00354 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00355 *> appropriately scaled matrix Z. 00356 *> Let Z = S*A, where S scales each row by a power of the 00357 *> radix so all absolute row sums of Z are approximately 1. 00358 *> 00359 *> See Lapack Working Note 165 for further details and extra 00360 *> cautions. 00361 *> \endverbatim 00362 *> 00363 *> \param[out] ERR_BNDS_COMP 00364 *> \verbatim 00365 *> ERR_BNDS_COMP is DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00366 *> For each right-hand side, this array contains information about 00367 *> various error bounds and condition numbers corresponding to the 00368 *> componentwise relative error, which is defined as follows: 00369 *> 00370 *> Componentwise relative error in the ith solution vector: 00371 *> abs(XTRUE(j,i) - X(j,i)) 00372 *> max_j ---------------------- 00373 *> abs(X(j,i)) 00374 *> 00375 *> The array is indexed by the right-hand side i (on which the 00376 *> componentwise relative error depends), and the type of error 00377 *> information as described below. There currently are up to three 00378 *> pieces of information returned for each right-hand side. If 00379 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00380 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00381 *> the first (:,N_ERR_BNDS) entries are returned. 00382 *> 00383 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00384 *> right-hand side. 00385 *> 00386 *> The second index in ERR_BNDS_COMP(:,err) contains the following 00387 *> three fields: 00388 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00389 *> reciprocal condition number is less than the threshold 00390 *> sqrt(n) * dlamch('Epsilon'). 00391 *> 00392 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00393 *> almost certainly within a factor of 10 of the true error 00394 *> so long as the next entry is greater than the threshold 00395 *> sqrt(n) * dlamch('Epsilon'). This error bound should only 00396 *> be trusted if the previous boolean is true. 00397 *> 00398 *> err = 3 Reciprocal condition number: Estimated componentwise 00399 *> reciprocal condition number. Compared with the threshold 00400 *> sqrt(n) * dlamch('Epsilon') to determine if the error 00401 *> estimate is "guaranteed". These reciprocal condition 00402 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00403 *> appropriately scaled matrix Z. 00404 *> Let Z = S*(A*diag(x)), where x is the solution for the 00405 *> current right-hand side and S scales each row of 00406 *> A*diag(x) by a power of the radix so all absolute row 00407 *> sums of Z are approximately 1. 00408 *> 00409 *> See Lapack Working Note 165 for further details and extra 00410 *> cautions. 00411 *> \endverbatim 00412 *> 00413 *> \param[in] NPARAMS 00414 *> \verbatim 00415 *> NPARAMS is INTEGER 00416 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the 00417 *> PARAMS array is never referenced and default values are used. 00418 *> \endverbatim 00419 *> 00420 *> \param[in,out] PARAMS 00421 *> \verbatim 00422 *> PARAMS is / output) DOUBLE PRECISION array, dimension (NPARAMS) 00423 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then 00424 *> that entry will be filled with default value used for that 00425 *> parameter. Only positions up to NPARAMS are accessed; defaults 00426 *> are used for higher-numbered parameters. 00427 *> 00428 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00429 *> refinement or not. 00430 *> Default: 1.0D+0 00431 *> = 0.0 : No refinement is performed, and no error bounds are 00432 *> computed. 00433 *> = 1.0 : Use the extra-precise refinement algorithm. 00434 *> (other values are reserved for future use) 00435 *> 00436 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00437 *> computations allowed for refinement. 00438 *> Default: 10 00439 *> Aggressive: Set to 100 to permit convergence using approximate 00440 *> factorizations or factorizations other than LU. If 00441 *> the factorization uses a technique other than 00442 *> Gaussian elimination, the guarantees in 00443 *> err_bnds_norm and err_bnds_comp may no longer be 00444 *> trustworthy. 00445 *> 00446 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00447 *> will attempt to find a solution with small componentwise 00448 *> relative error in the double-precision algorithm. Positive 00449 *> is true, 0.0 is false. 00450 *> Default: 1.0 (attempt componentwise convergence) 00451 *> \endverbatim 00452 *> 00453 *> \param[out] WORK 00454 *> \verbatim 00455 *> WORK is DOUBLE PRECISION array, dimension (4*N) 00456 *> \endverbatim 00457 *> 00458 *> \param[out] IWORK 00459 *> \verbatim 00460 *> IWORK is INTEGER array, dimension (N) 00461 *> \endverbatim 00462 *> 00463 *> \param[out] INFO 00464 *> \verbatim 00465 *> INFO is INTEGER 00466 *> = 0: Successful exit. The solution to every right-hand side is 00467 *> guaranteed. 00468 *> < 0: If INFO = -i, the i-th argument had an illegal value 00469 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00470 *> has been completed, but the factor U is exactly singular, so 00471 *> the solution and error bounds could not be computed. RCOND = 0 00472 *> is returned. 00473 *> = N+J: The solution corresponding to the Jth right-hand side is 00474 *> not guaranteed. The solutions corresponding to other right- 00475 *> hand sides K with K > J may not be guaranteed as well, but 00476 *> only the first such right-hand side is reported. If a small 00477 *> componentwise error is not requested (PARAMS(3) = 0.0) then 00478 *> the Jth right-hand side is the first with a normwise error 00479 *> bound that is not guaranteed (the smallest J such 00480 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00481 *> the Jth right-hand side is the first with either a normwise or 00482 *> componentwise error bound that is not guaranteed (the smallest 00483 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00484 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00485 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00486 *> about all of the right-hand sides check ERR_BNDS_NORM or 00487 *> ERR_BNDS_COMP. 00488 *> \endverbatim 00489 * 00490 * Authors: 00491 * ======== 00492 * 00493 *> \author Univ. of Tennessee 00494 *> \author Univ. of California Berkeley 00495 *> \author Univ. of Colorado Denver 00496 *> \author NAG Ltd. 00497 * 00498 *> \date April 2012 00499 * 00500 *> \ingroup doubleSYcomputational 00501 * 00502 * ===================================================================== 00503 SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 00504 $ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 00505 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 00506 $ NPARAMS, PARAMS, WORK, IWORK, INFO ) 00507 * 00508 * -- LAPACK computational routine (version 3.4.1) -- 00509 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00510 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00511 * April 2012 00512 * 00513 * .. Scalar Arguments .. 00514 CHARACTER EQUED, FACT, UPLO 00515 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 00516 $ N_ERR_BNDS 00517 DOUBLE PRECISION RCOND, RPVGRW 00518 * .. 00519 * .. Array Arguments .. 00520 INTEGER IPIV( * ), IWORK( * ) 00521 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00522 $ X( LDX, * ), WORK( * ) 00523 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), 00524 $ ERR_BNDS_NORM( NRHS, * ), 00525 $ ERR_BNDS_COMP( NRHS, * ) 00526 * .. 00527 * 00528 * ================================================================== 00529 * 00530 * .. Parameters .. 00531 DOUBLE PRECISION ZERO, ONE 00532 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00533 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00534 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00535 INTEGER CMP_ERR_I, PIV_GROWTH_I 00536 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00537 $ BERR_I = 3 ) 00538 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00539 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00540 $ PIV_GROWTH_I = 9 ) 00541 * .. 00542 * .. Local Scalars .. 00543 LOGICAL EQUIL, NOFACT, RCEQU 00544 INTEGER INFEQU, J 00545 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 00546 * .. 00547 * .. External Functions .. 00548 EXTERNAL LSAME, DLAMCH, DLA_SYRPVGRW 00549 LOGICAL LSAME 00550 DOUBLE PRECISION DLAMCH, DLA_SYRPVGRW 00551 * .. 00552 * .. External Subroutines .. 00553 EXTERNAL DSYCON, DSYEQUB, DSYTRF, DSYTRS, 00554 $ DLACPY, DLAQSY, XERBLA, DLASCL2, DSYRFSX 00555 * .. 00556 * .. Intrinsic Functions .. 00557 INTRINSIC MAX, MIN 00558 * .. 00559 * .. Executable Statements .. 00560 * 00561 INFO = 0 00562 NOFACT = LSAME( FACT, 'N' ) 00563 EQUIL = LSAME( FACT, 'E' ) 00564 SMLNUM = DLAMCH( 'Safe minimum' ) 00565 BIGNUM = ONE / SMLNUM 00566 IF( NOFACT .OR. EQUIL ) THEN 00567 EQUED = 'N' 00568 RCEQU = .FALSE. 00569 ELSE 00570 RCEQU = LSAME( EQUED, 'Y' ) 00571 ENDIF 00572 * 00573 * Default is failure. If an input parameter is wrong or 00574 * factorization fails, make everything look horrible. Only the 00575 * pivot growth is set here, the rest is initialized in DSYRFSX. 00576 * 00577 RPVGRW = ZERO 00578 * 00579 * Test the input parameters. PARAMS is not tested until DSYRFSX. 00580 * 00581 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00582 $ LSAME( FACT, 'F' ) ) THEN 00583 INFO = -1 00584 ELSE IF( .NOT.LSAME(UPLO, 'U') .AND. 00585 $ .NOT.LSAME(UPLO, 'L') ) THEN 00586 INFO = -2 00587 ELSE IF( N.LT.0 ) THEN 00588 INFO = -3 00589 ELSE IF( NRHS.LT.0 ) THEN 00590 INFO = -4 00591 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00592 INFO = -6 00593 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00594 INFO = -8 00595 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00596 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00597 INFO = -9 00598 ELSE 00599 IF ( RCEQU ) THEN 00600 SMIN = BIGNUM 00601 SMAX = ZERO 00602 DO 10 J = 1, N 00603 SMIN = MIN( SMIN, S( J ) ) 00604 SMAX = MAX( SMAX, S( J ) ) 00605 10 CONTINUE 00606 IF( SMIN.LE.ZERO ) THEN 00607 INFO = -10 00608 ELSE IF( N.GT.0 ) THEN 00609 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00610 ELSE 00611 SCOND = ONE 00612 END IF 00613 END IF 00614 IF( INFO.EQ.0 ) THEN 00615 IF( LDB.LT.MAX( 1, N ) ) THEN 00616 INFO = -12 00617 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00618 INFO = -14 00619 END IF 00620 END IF 00621 END IF 00622 * 00623 IF( INFO.NE.0 ) THEN 00624 CALL XERBLA( 'DSYSVXX', -INFO ) 00625 RETURN 00626 END IF 00627 * 00628 IF( EQUIL ) THEN 00629 * 00630 * Compute row and column scalings to equilibrate the matrix A. 00631 * 00632 CALL DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU ) 00633 IF( INFEQU.EQ.0 ) THEN 00634 * 00635 * Equilibrate the matrix. 00636 * 00637 CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00638 RCEQU = LSAME( EQUED, 'Y' ) 00639 END IF 00640 END IF 00641 * 00642 * Scale the right-hand side. 00643 * 00644 IF( RCEQU ) CALL DLASCL2( N, NRHS, S, B, LDB ) 00645 * 00646 IF( NOFACT .OR. EQUIL ) THEN 00647 * 00648 * Compute the LDL^T or UDU^T factorization of A. 00649 * 00650 CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00651 CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO ) 00652 * 00653 * Return if INFO is non-zero. 00654 * 00655 IF( INFO.GT.0 ) THEN 00656 * 00657 * Pivot in column INFO is exactly 0 00658 * Compute the reciprocal pivot growth factor of the 00659 * leading rank-deficient INFO columns of A. 00660 * 00661 IF ( N.GT.0 ) 00662 $ RPVGRW = DLA_SYRPVGRW(UPLO, N, INFO, A, LDA, AF, 00663 $ LDAF, IPIV, WORK ) 00664 RETURN 00665 END IF 00666 END IF 00667 * 00668 * Compute the reciprocal pivot growth factor RPVGRW. 00669 * 00670 IF ( N.GT.0 ) 00671 $ RPVGRW = DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, 00672 $ IPIV, WORK ) 00673 * 00674 * Compute the solution matrix X. 00675 * 00676 CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00677 CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 00678 * 00679 * Use iterative refinement to improve the computed solution and 00680 * compute error bounds and backward error estimates for it. 00681 * 00682 CALL DSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, 00683 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00684 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) 00685 * 00686 * Scale solutions. 00687 * 00688 IF ( RCEQU ) THEN 00689 CALL DLASCL2 ( N, NRHS, S, X, LDX ) 00690 END IF 00691 * 00692 RETURN 00693 * 00694 * End of DSYSVXX 00695 * 00696 END