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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief <b> SGBSVXX computes the solution to system of linear equations A * X = B for GB 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 SGBSVXX + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbsvxx.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgbsvxx.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgbsvxx.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE SGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 00022 * LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 00023 * RCOND, RPVGRW, BERR, N_ERR_BNDS, 00024 * ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 00025 * WORK, IWORK, INFO ) 00026 * 00027 * .. Scalar Arguments .. 00028 * CHARACTER EQUED, FACT, TRANS 00029 * INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, 00030 * $ N_ERR_BNDS 00031 * REAL RCOND, RPVGRW 00032 * .. 00033 * .. Array Arguments .. 00034 * INTEGER IPIV( * ), IWORK( * ) 00035 * REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 00036 * $ X( LDX , * ),WORK( * ) 00037 * REAL R( * ), C( * ), PARAMS( * ), BERR( * ), 00038 * $ ERR_BNDS_NORM( NRHS, * ), 00039 * $ ERR_BNDS_COMP( NRHS, * ) 00040 * .. 00041 * 00042 * 00043 *> \par Purpose: 00044 * ============= 00045 *> 00046 *> \verbatim 00047 *> 00048 *> SGBSVXX uses the LU factorization to compute the solution to a 00049 *> real system of linear equations A * X = B, where A is an 00050 *> N-by-N matrix and X and B are N-by-NRHS matrices. 00051 *> 00052 *> If requested, both normwise and maximum componentwise error bounds 00053 *> are returned. SGBSVXX 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 *> SGBSVXX 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 *> SGBSVXX 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 SGBSVXX 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 *> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B 00079 *> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B 00080 *> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B 00081 *> 00082 *> Whether or not the system will be equilibrated depends on the 00083 *> scaling of the matrix A, but if equilibration is used, A is 00084 *> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') 00085 *> or diag(C)*B (if TRANS = 'T' or 'C'). 00086 *> 00087 *> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor 00088 *> the matrix A (after equilibration if FACT = 'E') as 00089 *> 00090 *> A = P * L * U, 00091 *> 00092 *> where P is a permutation matrix, L is a unit lower triangular 00093 *> matrix, and U is upper triangular. 00094 *> 00095 *> 3. If some U(i,i)=0, so that U is exactly singular, then the 00096 *> routine returns with INFO = i. Otherwise, the factored form of A 00097 *> is used to estimate the condition number of the matrix A (see 00098 *> argument RCOND). If the reciprocal of the condition number is less 00099 *> than machine precision, the routine still goes on to solve for X 00100 *> and compute error bounds as described below. 00101 *> 00102 *> 4. The system of equations is solved for X using the factored form 00103 *> of A. 00104 *> 00105 *> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00106 *> the routine will use iterative refinement to try to get a small 00107 *> error and error bounds. Refinement calculates the residual to at 00108 *> least twice the working precision. 00109 *> 00110 *> 6. If equilibration was used, the matrix X is premultiplied by 00111 *> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so 00112 *> that it solves the original system before equilibration. 00113 *> \endverbatim 00114 * 00115 * Arguments: 00116 * ========== 00117 * 00118 *> \verbatim 00119 *> Some optional parameters are bundled in the PARAMS array. These 00120 *> settings determine how refinement is performed, but often the 00121 *> defaults are acceptable. If the defaults are acceptable, users 00122 *> can pass NPARAMS = 0 which prevents the source code from accessing 00123 *> the PARAMS argument. 00124 *> \endverbatim 00125 *> 00126 *> \param[in] FACT 00127 *> \verbatim 00128 *> FACT is CHARACTER*1 00129 *> Specifies whether or not the factored form of the matrix A is 00130 *> supplied on entry, and if not, whether the matrix A should be 00131 *> equilibrated before it is factored. 00132 *> = 'F': On entry, AF and IPIV contain the factored form of A. 00133 *> If EQUED is not 'N', the matrix A has been 00134 *> equilibrated with scaling factors given by R and C. 00135 *> A, AF, and IPIV are not modified. 00136 *> = 'N': The matrix A will be copied to AF and factored. 00137 *> = 'E': The matrix A will be equilibrated if necessary, then 00138 *> copied to AF and factored. 00139 *> \endverbatim 00140 *> 00141 *> \param[in] TRANS 00142 *> \verbatim 00143 *> TRANS is CHARACTER*1 00144 *> Specifies the form of the system of equations: 00145 *> = 'N': A * X = B (No transpose) 00146 *> = 'T': A**T * X = B (Transpose) 00147 *> = 'C': A**H * X = B (Conjugate Transpose = Transpose) 00148 *> \endverbatim 00149 *> 00150 *> \param[in] N 00151 *> \verbatim 00152 *> N is INTEGER 00153 *> The number of linear equations, i.e., the order of the 00154 *> matrix A. N >= 0. 00155 *> \endverbatim 00156 *> 00157 *> \param[in] KL 00158 *> \verbatim 00159 *> KL is INTEGER 00160 *> The number of subdiagonals within the band of A. KL >= 0. 00161 *> \endverbatim 00162 *> 00163 *> \param[in] KU 00164 *> \verbatim 00165 *> KU is INTEGER 00166 *> The number of superdiagonals within the band of A. KU >= 0. 00167 *> \endverbatim 00168 *> 00169 *> \param[in] NRHS 00170 *> \verbatim 00171 *> NRHS is INTEGER 00172 *> The number of right hand sides, i.e., the number of columns 00173 *> of the matrices B and X. NRHS >= 0. 00174 *> \endverbatim 00175 *> 00176 *> \param[in,out] AB 00177 *> \verbatim 00178 *> AB is REAL array, dimension (LDAB,N) 00179 *> On entry, the matrix A in band storage, in rows 1 to KL+KU+1. 00180 *> The j-th column of A is stored in the j-th column of the 00181 *> array AB as follows: 00182 *> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) 00183 *> 00184 *> If FACT = 'F' and EQUED is not 'N', then AB must have been 00185 *> equilibrated by the scaling factors in R and/or C. AB is not 00186 *> modified if FACT = 'F' or 'N', or if FACT = 'E' and 00187 *> EQUED = 'N' on exit. 00188 *> 00189 *> On exit, if EQUED .ne. 'N', A is scaled as follows: 00190 *> EQUED = 'R': A := diag(R) * A 00191 *> EQUED = 'C': A := A * diag(C) 00192 *> EQUED = 'B': A := diag(R) * A * diag(C). 00193 *> \endverbatim 00194 *> 00195 *> \param[in] LDAB 00196 *> \verbatim 00197 *> LDAB is INTEGER 00198 *> The leading dimension of the array AB. LDAB >= KL+KU+1. 00199 *> \endverbatim 00200 *> 00201 *> \param[in,out] AFB 00202 *> \verbatim 00203 *> AFB is REAL array, dimension (LDAFB,N) 00204 *> If FACT = 'F', then AFB is an input argument and on entry 00205 *> contains details of the LU factorization of the band matrix 00206 *> A, as computed by SGBTRF. U is stored as an upper triangular 00207 *> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, 00208 *> and the multipliers used during the factorization are stored 00209 *> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is 00210 *> the factored form of the equilibrated matrix A. 00211 *> 00212 *> If FACT = 'N', then AF is an output argument and on exit 00213 *> returns the factors L and U from the factorization A = P*L*U 00214 *> of the original matrix A. 00215 *> 00216 *> If FACT = 'E', then AF is an output argument and on exit 00217 *> returns the factors L and U from the factorization A = P*L*U 00218 *> of the equilibrated matrix A (see the description of A for 00219 *> the form of the equilibrated matrix). 00220 *> \endverbatim 00221 *> 00222 *> \param[in] LDAFB 00223 *> \verbatim 00224 *> LDAFB is INTEGER 00225 *> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. 00226 *> \endverbatim 00227 *> 00228 *> \param[in,out] IPIV 00229 *> \verbatim 00230 *> IPIV is INTEGER array, dimension (N) 00231 *> If FACT = 'F', then IPIV is an input argument and on entry 00232 *> contains the pivot indices from the factorization A = P*L*U 00233 *> as computed by SGETRF; row i of the matrix was interchanged 00234 *> with row IPIV(i). 00235 *> 00236 *> If FACT = 'N', then IPIV is an output argument and on exit 00237 *> contains the pivot indices from the factorization A = P*L*U 00238 *> of the original matrix A. 00239 *> 00240 *> If FACT = 'E', then IPIV is an output argument and on exit 00241 *> contains the pivot indices from the factorization A = P*L*U 00242 *> of the equilibrated matrix A. 00243 *> \endverbatim 00244 *> 00245 *> \param[in,out] EQUED 00246 *> \verbatim 00247 *> EQUED is CHARACTER*1 00248 *> Specifies the form of equilibration that was done. 00249 *> = 'N': No equilibration (always true if FACT = 'N'). 00250 *> = 'R': Row equilibration, i.e., A has been premultiplied by 00251 *> diag(R). 00252 *> = 'C': Column equilibration, i.e., A has been postmultiplied 00253 *> by diag(C). 00254 *> = 'B': Both row and column equilibration, i.e., A has been 00255 *> replaced by diag(R) * A * diag(C). 00256 *> EQUED is an input argument if FACT = 'F'; otherwise, it is an 00257 *> output argument. 00258 *> \endverbatim 00259 *> 00260 *> \param[in,out] R 00261 *> \verbatim 00262 *> R is REAL array, dimension (N) 00263 *> The row scale factors for A. If EQUED = 'R' or 'B', A is 00264 *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R 00265 *> is not accessed. R is an input argument if FACT = 'F'; 00266 *> otherwise, R is an output argument. If FACT = 'F' and 00267 *> EQUED = 'R' or 'B', each element of R must be positive. 00268 *> If R is output, each element of R is a power of the radix. 00269 *> If R is input, each element of R should be a power of the radix 00270 *> to ensure a reliable solution and error estimates. Scaling by 00271 *> powers of the radix does not cause rounding errors unless the 00272 *> result underflows or overflows. Rounding errors during scaling 00273 *> lead to refining with a matrix that is not equivalent to the 00274 *> input matrix, producing error estimates that may not be 00275 *> reliable. 00276 *> \endverbatim 00277 *> 00278 *> \param[in,out] C 00279 *> \verbatim 00280 *> C is REAL array, dimension (N) 00281 *> The column scale factors for A. If EQUED = 'C' or 'B', A is 00282 *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C 00283 *> is not accessed. C is an input argument if FACT = 'F'; 00284 *> otherwise, C is an output argument. If FACT = 'F' and 00285 *> EQUED = 'C' or 'B', each element of C must be positive. 00286 *> If C is output, each element of C is a power of the radix. 00287 *> If C is input, each element of C should be a power of the radix 00288 *> to ensure a reliable solution and error estimates. Scaling by 00289 *> powers of the radix does not cause rounding errors unless the 00290 *> result underflows or overflows. Rounding errors during scaling 00291 *> lead to refining with a matrix that is not equivalent to the 00292 *> input matrix, producing error estimates that may not be 00293 *> reliable. 00294 *> \endverbatim 00295 *> 00296 *> \param[in,out] B 00297 *> \verbatim 00298 *> B is REAL array, dimension (LDB,NRHS) 00299 *> On entry, the N-by-NRHS right hand side matrix B. 00300 *> On exit, 00301 *> if EQUED = 'N', B is not modified; 00302 *> if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by 00303 *> diag(R)*B; 00304 *> if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is 00305 *> overwritten by diag(C)*B. 00306 *> \endverbatim 00307 *> 00308 *> \param[in] LDB 00309 *> \verbatim 00310 *> LDB is INTEGER 00311 *> The leading dimension of the array B. LDB >= max(1,N). 00312 *> \endverbatim 00313 *> 00314 *> \param[out] X 00315 *> \verbatim 00316 *> X is REAL array, dimension (LDX,NRHS) 00317 *> If INFO = 0, the N-by-NRHS solution matrix X to the original 00318 *> system of equations. Note that A and B are modified on exit 00319 *> if EQUED .ne. 'N', and the solution to the equilibrated system is 00320 *> inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or 00321 *> inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. 00322 *> \endverbatim 00323 *> 00324 *> \param[in] LDX 00325 *> \verbatim 00326 *> LDX is INTEGER 00327 *> The leading dimension of the array X. LDX >= max(1,N). 00328 *> \endverbatim 00329 *> 00330 *> \param[out] RCOND 00331 *> \verbatim 00332 *> RCOND is REAL 00333 *> Reciprocal scaled condition number. This is an estimate of the 00334 *> reciprocal Skeel condition number of the matrix A after 00335 *> equilibration (if done). If this is less than the machine 00336 *> precision (in particular, if it is zero), the matrix is singular 00337 *> to working precision. Note that the error may still be small even 00338 *> if this number is very small and the matrix appears ill- 00339 *> conditioned. 00340 *> \endverbatim 00341 *> 00342 *> \param[out] RPVGRW 00343 *> \verbatim 00344 *> RPVGRW is REAL 00345 *> Reciprocal pivot growth. On exit, this contains the reciprocal 00346 *> pivot growth factor norm(A)/norm(U). The "max absolute element" 00347 *> norm is used. If this is much less than 1, then the stability of 00348 *> the LU factorization of the (equilibrated) matrix A could be poor. 00349 *> This also means that the solution X, estimated condition numbers, 00350 *> and error bounds could be unreliable. If factorization fails with 00351 *> 0<INFO<=N, then this contains the reciprocal pivot growth factor 00352 *> for the leading INFO columns of A. In SGESVX, this quantity is 00353 *> returned in WORK(1). 00354 *> \endverbatim 00355 *> 00356 *> \param[out] BERR 00357 *> \verbatim 00358 *> BERR is REAL array, dimension (NRHS) 00359 *> Componentwise relative backward error. This is the 00360 *> componentwise relative backward error of each solution vector X(j) 00361 *> (i.e., the smallest relative change in any element of A or B that 00362 *> makes X(j) an exact solution). 00363 *> \endverbatim 00364 *> 00365 *> \param[in] N_ERR_BNDS 00366 *> \verbatim 00367 *> N_ERR_BNDS is INTEGER 00368 *> Number of error bounds to return for each right hand side 00369 *> and each type (normwise or componentwise). See ERR_BNDS_NORM and 00370 *> ERR_BNDS_COMP below. 00371 *> \endverbatim 00372 *> 00373 *> \param[out] ERR_BNDS_NORM 00374 *> \verbatim 00375 *> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS) 00376 *> For each right-hand side, this array contains information about 00377 *> various error bounds and condition numbers corresponding to the 00378 *> normwise relative error, which is defined as follows: 00379 *> 00380 *> Normwise relative error in the ith solution vector: 00381 *> max_j (abs(XTRUE(j,i) - X(j,i))) 00382 *> ------------------------------ 00383 *> max_j abs(X(j,i)) 00384 *> 00385 *> The array is indexed by the type of error information as described 00386 *> below. There currently are up to three pieces of information 00387 *> returned. 00388 *> 00389 *> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00390 *> right-hand side. 00391 *> 00392 *> The second index in ERR_BNDS_NORM(:,err) contains the following 00393 *> three fields: 00394 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00395 *> reciprocal condition number is less than the threshold 00396 *> sqrt(n) * slamch('Epsilon'). 00397 *> 00398 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00399 *> almost certainly within a factor of 10 of the true error 00400 *> so long as the next entry is greater than the threshold 00401 *> sqrt(n) * slamch('Epsilon'). This error bound should only 00402 *> be trusted if the previous boolean is true. 00403 *> 00404 *> err = 3 Reciprocal condition number: Estimated normwise 00405 *> reciprocal condition number. Compared with the threshold 00406 *> sqrt(n) * slamch('Epsilon') to determine if the error 00407 *> estimate is "guaranteed". These reciprocal condition 00408 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00409 *> appropriately scaled matrix Z. 00410 *> Let Z = S*A, where S scales each row by a power of the 00411 *> radix so all absolute row sums of Z are approximately 1. 00412 *> 00413 *> See Lapack Working Note 165 for further details and extra 00414 *> cautions. 00415 *> \endverbatim 00416 *> 00417 *> \param[out] ERR_BNDS_COMP 00418 *> \verbatim 00419 *> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS) 00420 *> For each right-hand side, this array contains information about 00421 *> various error bounds and condition numbers corresponding to the 00422 *> componentwise relative error, which is defined as follows: 00423 *> 00424 *> Componentwise relative error in the ith solution vector: 00425 *> abs(XTRUE(j,i) - X(j,i)) 00426 *> max_j ---------------------- 00427 *> abs(X(j,i)) 00428 *> 00429 *> The array is indexed by the right-hand side i (on which the 00430 *> componentwise relative error depends), and the type of error 00431 *> information as described below. There currently are up to three 00432 *> pieces of information returned for each right-hand side. If 00433 *> componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00434 *> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00435 *> the first (:,N_ERR_BNDS) entries are returned. 00436 *> 00437 *> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00438 *> right-hand side. 00439 *> 00440 *> The second index in ERR_BNDS_COMP(:,err) contains the following 00441 *> three fields: 00442 *> err = 1 "Trust/don't trust" boolean. Trust the answer if the 00443 *> reciprocal condition number is less than the threshold 00444 *> sqrt(n) * slamch('Epsilon'). 00445 *> 00446 *> err = 2 "Guaranteed" error bound: The estimated forward error, 00447 *> almost certainly within a factor of 10 of the true error 00448 *> so long as the next entry is greater than the threshold 00449 *> sqrt(n) * slamch('Epsilon'). This error bound should only 00450 *> be trusted if the previous boolean is true. 00451 *> 00452 *> err = 3 Reciprocal condition number: Estimated componentwise 00453 *> reciprocal condition number. Compared with the threshold 00454 *> sqrt(n) * slamch('Epsilon') to determine if the error 00455 *> estimate is "guaranteed". These reciprocal condition 00456 *> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00457 *> appropriately scaled matrix Z. 00458 *> Let Z = S*(A*diag(x)), where x is the solution for the 00459 *> current right-hand side and S scales each row of 00460 *> A*diag(x) by a power of the radix so all absolute row 00461 *> sums of Z are approximately 1. 00462 *> 00463 *> See Lapack Working Note 165 for further details and extra 00464 *> cautions. 00465 *> \endverbatim 00466 *> 00467 *> \param[in] NPARAMS 00468 *> \verbatim 00469 *> NPARAMS is INTEGER 00470 *> Specifies the number of parameters set in PARAMS. If .LE. 0, the 00471 *> PARAMS array is never referenced and default values are used. 00472 *> \endverbatim 00473 *> 00474 *> \param[in,out] PARAMS 00475 *> \verbatim 00476 *> PARAMS is / output) REAL array, dimension NPARAMS 00477 *> Specifies algorithm parameters. If an entry is .LT. 0.0, then 00478 *> that entry will be filled with default value used for that 00479 *> parameter. Only positions up to NPARAMS are accessed; defaults 00480 *> are used for higher-numbered parameters. 00481 *> 00482 *> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00483 *> refinement or not. 00484 *> Default: 1.0 00485 *> = 0.0 : No refinement is performed, and no error bounds are 00486 *> computed. 00487 *> = 1.0 : Use the double-precision refinement algorithm, 00488 *> possibly with doubled-single computations if the 00489 *> compilation environment does not support DOUBLE 00490 *> PRECISION. 00491 *> (other values are reserved for future use) 00492 *> 00493 *> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00494 *> computations allowed for refinement. 00495 *> Default: 10 00496 *> Aggressive: Set to 100 to permit convergence using approximate 00497 *> factorizations or factorizations other than LU. If 00498 *> the factorization uses a technique other than 00499 *> Gaussian elimination, the guarantees in 00500 *> err_bnds_norm and err_bnds_comp may no longer be 00501 *> trustworthy. 00502 *> 00503 *> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00504 *> will attempt to find a solution with small componentwise 00505 *> relative error in the double-precision algorithm. Positive 00506 *> is true, 0.0 is false. 00507 *> Default: 1.0 (attempt componentwise convergence) 00508 *> \endverbatim 00509 *> 00510 *> \param[out] WORK 00511 *> \verbatim 00512 *> WORK is REAL array, dimension (4*N) 00513 *> \endverbatim 00514 *> 00515 *> \param[out] IWORK 00516 *> \verbatim 00517 *> IWORK is INTEGER array, dimension (N) 00518 *> \endverbatim 00519 *> 00520 *> \param[out] INFO 00521 *> \verbatim 00522 *> INFO is INTEGER 00523 *> = 0: Successful exit. The solution to every right-hand side is 00524 *> guaranteed. 00525 *> < 0: If INFO = -i, the i-th argument had an illegal value 00526 *> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00527 *> has been completed, but the factor U is exactly singular, so 00528 *> the solution and error bounds could not be computed. RCOND = 0 00529 *> is returned. 00530 *> = N+J: The solution corresponding to the Jth right-hand side is 00531 *> not guaranteed. The solutions corresponding to other right- 00532 *> hand sides K with K > J may not be guaranteed as well, but 00533 *> only the first such right-hand side is reported. If a small 00534 *> componentwise error is not requested (PARAMS(3) = 0.0) then 00535 *> the Jth right-hand side is the first with a normwise error 00536 *> bound that is not guaranteed (the smallest J such 00537 *> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00538 *> the Jth right-hand side is the first with either a normwise or 00539 *> componentwise error bound that is not guaranteed (the smallest 00540 *> J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00541 *> ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00542 *> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00543 *> about all of the right-hand sides check ERR_BNDS_NORM or 00544 *> ERR_BNDS_COMP. 00545 *> \endverbatim 00546 * 00547 * Authors: 00548 * ======== 00549 * 00550 *> \author Univ. of Tennessee 00551 *> \author Univ. of California Berkeley 00552 *> \author Univ. of Colorado Denver 00553 *> \author NAG Ltd. 00554 * 00555 *> \date April 2012 00556 * 00557 *> \ingroup realGBsolve 00558 * 00559 * ===================================================================== 00560 SUBROUTINE SGBSVXX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, 00561 $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, 00562 $ RCOND, RPVGRW, BERR, N_ERR_BNDS, 00563 $ ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 00564 $ WORK, IWORK, INFO ) 00565 * 00566 * -- LAPACK driver routine (version 3.4.1) -- 00567 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00568 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00569 * April 2012 00570 * 00571 * .. Scalar Arguments .. 00572 CHARACTER EQUED, FACT, TRANS 00573 INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, NRHS, NPARAMS, 00574 $ N_ERR_BNDS 00575 REAL RCOND, RPVGRW 00576 * .. 00577 * .. Array Arguments .. 00578 INTEGER IPIV( * ), IWORK( * ) 00579 REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), 00580 $ X( LDX , * ),WORK( * ) 00581 REAL R( * ), C( * ), PARAMS( * ), BERR( * ), 00582 $ ERR_BNDS_NORM( NRHS, * ), 00583 $ ERR_BNDS_COMP( NRHS, * ) 00584 * .. 00585 * 00586 * ================================================================== 00587 * 00588 * .. Parameters .. 00589 REAL ZERO, ONE 00590 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00591 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00592 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00593 INTEGER CMP_ERR_I, PIV_GROWTH_I 00594 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00595 $ BERR_I = 3 ) 00596 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00597 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00598 $ PIV_GROWTH_I = 9 ) 00599 * .. 00600 * .. Local Scalars .. 00601 LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU 00602 INTEGER INFEQU, I, J, KL, KU 00603 REAL AMAX, BIGNUM, COLCND, RCMAX, RCMIN, 00604 $ ROWCND, SMLNUM 00605 * .. 00606 * .. External Functions .. 00607 EXTERNAL LSAME, SLAMCH, SLA_GBRPVGRW 00608 LOGICAL LSAME 00609 REAL SLAMCH, SLA_GBRPVGRW 00610 * .. 00611 * .. External Subroutines .. 00612 EXTERNAL SGBEQUB, SGBTRF, SGBTRS, SLACPY, SLAQGB, 00613 $ XERBLA, SLASCL2, SGBRFSX 00614 * .. 00615 * .. Intrinsic Functions .. 00616 INTRINSIC MAX, MIN 00617 * .. 00618 * .. Executable Statements .. 00619 * 00620 INFO = 0 00621 NOFACT = LSAME( FACT, 'N' ) 00622 EQUIL = LSAME( FACT, 'E' ) 00623 NOTRAN = LSAME( TRANS, 'N' ) 00624 SMLNUM = SLAMCH( 'Safe minimum' ) 00625 BIGNUM = ONE / SMLNUM 00626 IF( NOFACT .OR. EQUIL ) THEN 00627 EQUED = 'N' 00628 ROWEQU = .FALSE. 00629 COLEQU = .FALSE. 00630 ELSE 00631 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00632 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00633 END IF 00634 * 00635 * Default is failure. If an input parameter is wrong or 00636 * factorization fails, make everything look horrible. Only the 00637 * pivot growth is set here, the rest is initialized in SGBRFSX. 00638 * 00639 RPVGRW = ZERO 00640 * 00641 * Test the input parameters. PARAMS is not tested until SGBRFSX. 00642 * 00643 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00644 $ LSAME( FACT, 'F' ) ) THEN 00645 INFO = -1 00646 ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT. 00647 $ LSAME( TRANS, 'C' ) ) THEN 00648 INFO = -2 00649 ELSE IF( N.LT.0 ) THEN 00650 INFO = -3 00651 ELSE IF( KL.LT.0 ) THEN 00652 INFO = -4 00653 ELSE IF( KU.LT.0 ) THEN 00654 INFO = -5 00655 ELSE IF( NRHS.LT.0 ) THEN 00656 INFO = -6 00657 ELSE IF( LDAB.LT.KL+KU+1 ) THEN 00658 INFO = -8 00659 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN 00660 INFO = -10 00661 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00662 $ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00663 INFO = -12 00664 ELSE 00665 IF( ROWEQU ) THEN 00666 RCMIN = BIGNUM 00667 RCMAX = ZERO 00668 DO 10 J = 1, N 00669 RCMIN = MIN( RCMIN, R( J ) ) 00670 RCMAX = MAX( RCMAX, R( J ) ) 00671 10 CONTINUE 00672 IF( RCMIN.LE.ZERO ) THEN 00673 INFO = -13 00674 ELSE IF( N.GT.0 ) THEN 00675 ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00676 ELSE 00677 ROWCND = ONE 00678 END IF 00679 END IF 00680 IF( COLEQU .AND. INFO.EQ.0 ) THEN 00681 RCMIN = BIGNUM 00682 RCMAX = ZERO 00683 DO 20 J = 1, N 00684 RCMIN = MIN( RCMIN, C( J ) ) 00685 RCMAX = MAX( RCMAX, C( J ) ) 00686 20 CONTINUE 00687 IF( RCMIN.LE.ZERO ) THEN 00688 INFO = -14 00689 ELSE IF( N.GT.0 ) THEN 00690 COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM ) 00691 ELSE 00692 COLCND = ONE 00693 END IF 00694 END IF 00695 IF( INFO.EQ.0 ) THEN 00696 IF( LDB.LT.MAX( 1, N ) ) THEN 00697 INFO = -15 00698 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00699 INFO = -16 00700 END IF 00701 END IF 00702 END IF 00703 * 00704 IF( INFO.NE.0 ) THEN 00705 CALL XERBLA( 'SGBSVXX', -INFO ) 00706 RETURN 00707 END IF 00708 * 00709 IF( EQUIL ) THEN 00710 * 00711 * Compute row and column scalings to equilibrate the matrix A. 00712 * 00713 CALL SGBEQUB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 00714 $ AMAX, INFEQU ) 00715 IF( INFEQU.EQ.0 ) THEN 00716 * 00717 * Equilibrate the matrix. 00718 * 00719 CALL SLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND, 00720 $ AMAX, EQUED ) 00721 ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) 00722 COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) 00723 END IF 00724 * 00725 * If the scaling factors are not applied, set them to 1.0. 00726 * 00727 IF ( .NOT.ROWEQU ) THEN 00728 DO J = 1, N 00729 R( J ) = 1.0 00730 END DO 00731 END IF 00732 IF ( .NOT.COLEQU ) THEN 00733 DO J = 1, N 00734 C( J ) = 1.0 00735 END DO 00736 END IF 00737 END IF 00738 * 00739 * Scale the right hand side. 00740 * 00741 IF( NOTRAN ) THEN 00742 IF( ROWEQU ) CALL SLASCL2(N, NRHS, R, B, LDB) 00743 ELSE 00744 IF( COLEQU ) CALL SLASCL2(N, NRHS, C, B, LDB) 00745 END IF 00746 * 00747 IF( NOFACT .OR. EQUIL ) THEN 00748 * 00749 * Compute the LU factorization of A. 00750 * 00751 DO 40, J = 1, N 00752 DO 30, I = KL+1, 2*KL+KU+1 00753 AFB( I, J ) = AB( I-KL, J ) 00754 30 CONTINUE 00755 40 CONTINUE 00756 CALL SGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO ) 00757 * 00758 * Return if INFO is non-zero. 00759 * 00760 IF( INFO.GT.0 ) THEN 00761 * 00762 * Pivot in column INFO is exactly 0 00763 * Compute the reciprocal pivot growth factor of the 00764 * leading rank-deficient INFO columns of A. 00765 * 00766 RPVGRW = SLA_GBRPVGRW( N, KL, KU, INFO, AB, LDAB, AFB, 00767 $ LDAFB ) 00768 RETURN 00769 END IF 00770 END IF 00771 * 00772 * Compute the reciprocal pivot growth factor RPVGRW. 00773 * 00774 RPVGRW = SLA_GBRPVGRW( N, KL, KU, N, AB, LDAB, AFB, LDAFB ) 00775 * 00776 * Compute the solution matrix X. 00777 * 00778 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00779 CALL SGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX, 00780 $ INFO ) 00781 * 00782 * Use iterative refinement to improve the computed solution and 00783 * compute error bounds and backward error estimates for it. 00784 * 00785 CALL SGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, 00786 $ IPIV, R, C, B, LDB, X, LDX, RCOND, BERR, 00787 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, 00788 $ WORK, IWORK, INFO ) 00789 * 00790 * Scale solutions. 00791 * 00792 IF ( COLEQU .AND. NOTRAN ) THEN 00793 CALL SLASCL2 ( N, NRHS, C, X, LDX ) 00794 ELSE IF ( ROWEQU .AND. .NOT.NOTRAN ) THEN 00795 CALL SLASCL2 ( N, NRHS, R, X, LDX ) 00796 END IF 00797 * 00798 RETURN 00799 * 00800 * End of SGBSVXX 00801 * 00802 END