00001 SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, 00002 $ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, 00003 $ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, 00004 $ NPARAMS, PARAMS, WORK, IWORK, INFO ) 00005 * 00006 * -- LAPACK driver routine (version 3.2.2) -- 00007 * -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- 00008 * -- Jason Riedy of Univ. of California Berkeley. -- 00009 * -- June 2010 -- 00010 * 00011 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00012 * -- Univ. of California Berkeley and NAG Ltd. -- 00013 * 00014 IMPLICIT NONE 00015 * .. 00016 * .. Scalar Arguments .. 00017 CHARACTER EQUED, FACT, UPLO 00018 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, 00019 $ N_ERR_BNDS 00020 REAL RCOND, RPVGRW 00021 * .. 00022 * .. Array Arguments .. 00023 INTEGER IWORK( * ) 00024 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00025 $ X( LDX, * ), WORK( * ) 00026 REAL S( * ), PARAMS( * ), BERR( * ), 00027 $ ERR_BNDS_NORM( NRHS, * ), 00028 $ ERR_BNDS_COMP( NRHS, * ) 00029 * .. 00030 * 00031 * Purpose 00032 * ======= 00033 * 00034 * SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T 00035 * to compute the solution to a real system of linear equations 00036 * A * X = B, where A is an N-by-N symmetric positive definite matrix 00037 * and X and B are N-by-NRHS matrices. 00038 * 00039 * If requested, both normwise and maximum componentwise error bounds 00040 * are returned. SPOSVXX will return a solution with a tiny 00041 * guaranteed error (O(eps) where eps is the working machine 00042 * precision) unless the matrix is very ill-conditioned, in which 00043 * case a warning is returned. Relevant condition numbers also are 00044 * calculated and returned. 00045 * 00046 * SPOSVXX accepts user-provided factorizations and equilibration 00047 * factors; see the definitions of the FACT and EQUED options. 00048 * Solving with refinement and using a factorization from a previous 00049 * SPOSVXX call will also produce a solution with either O(eps) 00050 * errors or warnings, but we cannot make that claim for general 00051 * user-provided factorizations and equilibration factors if they 00052 * differ from what SPOSVXX would itself produce. 00053 * 00054 * Description 00055 * =========== 00056 * 00057 * The following steps are performed: 00058 * 00059 * 1. If FACT = 'E', real scaling factors are computed to equilibrate 00060 * the system: 00061 * 00062 * diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 00063 * 00064 * Whether or not the system will be equilibrated depends on the 00065 * scaling of the matrix A, but if equilibration is used, A is 00066 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00067 * 00068 * 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to 00069 * factor the matrix A (after equilibration if FACT = 'E') as 00070 * A = U**T* U, if UPLO = 'U', or 00071 * A = L * L**T, if UPLO = 'L', 00072 * where U is an upper triangular matrix and L is a lower triangular 00073 * matrix. 00074 * 00075 * 3. If the leading i-by-i principal minor is not positive definite, 00076 * then the routine returns with INFO = i. Otherwise, the factored 00077 * form of A is used to estimate the condition number of the matrix 00078 * A (see argument RCOND). If the reciprocal of the condition number 00079 * is less than machine precision, the routine still goes on to solve 00080 * for X and compute error bounds as described below. 00081 * 00082 * 4. The system of equations is solved for X using the factored form 00083 * of A. 00084 * 00085 * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00086 * the routine will use iterative refinement to try to get a small 00087 * error and error bounds. Refinement calculates the residual to at 00088 * least twice the working precision. 00089 * 00090 * 6. If equilibration was used, the matrix X is premultiplied by 00091 * diag(S) so that it solves the original system before 00092 * equilibration. 00093 * 00094 * Arguments 00095 * ========= 00096 * 00097 * Some optional parameters are bundled in the PARAMS array. These 00098 * settings determine how refinement is performed, but often the 00099 * defaults are acceptable. If the defaults are acceptable, users 00100 * can pass NPARAMS = 0 which prevents the source code from accessing 00101 * the PARAMS argument. 00102 * 00103 * FACT (input) CHARACTER*1 00104 * Specifies whether or not the factored form of the matrix A is 00105 * supplied on entry, and if not, whether the matrix A should be 00106 * equilibrated before it is factored. 00107 * = 'F': On entry, AF contains the factored form of A. 00108 * If EQUED is not 'N', the matrix A has been 00109 * equilibrated with scaling factors given by S. 00110 * A and AF are not modified. 00111 * = 'N': The matrix A will be copied to AF and factored. 00112 * = 'E': The matrix A will be equilibrated if necessary, then 00113 * copied to AF and factored. 00114 * 00115 * UPLO (input) CHARACTER*1 00116 * = 'U': Upper triangle of A is stored; 00117 * = 'L': Lower triangle of A is stored. 00118 * 00119 * N (input) INTEGER 00120 * The number of linear equations, i.e., the order of the 00121 * matrix A. N >= 0. 00122 * 00123 * NRHS (input) INTEGER 00124 * The number of right hand sides, i.e., the number of columns 00125 * of the matrices B and X. NRHS >= 0. 00126 * 00127 * A (input/output) REAL array, dimension (LDA,N) 00128 * On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = 00129 * 'Y', then A must contain the equilibrated matrix 00130 * diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper 00131 * triangular part of A contains the upper triangular part of the 00132 * matrix A, and the strictly lower triangular part of A is not 00133 * referenced. If UPLO = 'L', the leading N-by-N lower triangular 00134 * part of A contains the lower triangular part of the matrix A, and 00135 * the strictly upper triangular part of A is not referenced. A is 00136 * not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 00137 * 'N' on exit. 00138 * 00139 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00140 * diag(S)*A*diag(S). 00141 * 00142 * LDA (input) INTEGER 00143 * The leading dimension of the array A. LDA >= max(1,N). 00144 * 00145 * AF (input or output) REAL array, dimension (LDAF,N) 00146 * If FACT = 'F', then AF is an input argument and on entry 00147 * contains the triangular factor U or L from the Cholesky 00148 * factorization A = U**T*U or A = L*L**T, in the same storage 00149 * format as A. If EQUED .ne. 'N', then AF is the factored 00150 * form of the equilibrated matrix diag(S)*A*diag(S). 00151 * 00152 * If FACT = 'N', then AF is an output argument and on exit 00153 * returns the triangular factor U or L from the Cholesky 00154 * factorization A = U**T*U or A = L*L**T of the original 00155 * matrix A. 00156 * 00157 * If FACT = 'E', then AF is an output argument and on exit 00158 * returns the triangular factor U or L from the Cholesky 00159 * factorization A = U**T*U or A = L*L**T of the equilibrated 00160 * matrix A (see the description of A for the form of the 00161 * equilibrated matrix). 00162 * 00163 * LDAF (input) INTEGER 00164 * The leading dimension of the array AF. LDAF >= max(1,N). 00165 * 00166 * EQUED (input or output) CHARACTER*1 00167 * Specifies the form of equilibration that was done. 00168 * = 'N': No equilibration (always true if FACT = 'N'). 00169 * = 'Y': Both row and column equilibration, i.e., A has been 00170 * replaced by diag(S) * A * diag(S). 00171 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00172 * output argument. 00173 * 00174 * S (input or output) REAL array, dimension (N) 00175 * The row scale factors for A. If EQUED = 'Y', A is multiplied on 00176 * the left and right by diag(S). S is an input argument if FACT = 00177 * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00178 * = 'Y', each element of S must be positive. If S is output, each 00179 * element of S is a power of the radix. If S is input, each element 00180 * of S should be a power of the radix to ensure a reliable solution 00181 * and error estimates. Scaling by powers of the radix does not cause 00182 * rounding errors unless the result underflows or overflows. 00183 * Rounding errors during scaling lead to refining with a matrix that 00184 * is not equivalent to the input matrix, producing error estimates 00185 * that may not be reliable. 00186 * 00187 * B (input/output) REAL array, dimension (LDB,NRHS) 00188 * On entry, the N-by-NRHS right hand side matrix B. 00189 * On exit, 00190 * if EQUED = 'N', B is not modified; 00191 * if EQUED = 'Y', B is overwritten by diag(S)*B; 00192 * 00193 * LDB (input) INTEGER 00194 * The leading dimension of the array B. LDB >= max(1,N). 00195 * 00196 * X (output) REAL array, dimension (LDX,NRHS) 00197 * If INFO = 0, the N-by-NRHS solution matrix X to the original 00198 * system of equations. Note that A and B are modified on exit if 00199 * EQUED .ne. 'N', and the solution to the equilibrated system is 00200 * inv(diag(S))*X. 00201 * 00202 * LDX (input) INTEGER 00203 * The leading dimension of the array X. LDX >= max(1,N). 00204 * 00205 * RCOND (output) REAL 00206 * Reciprocal scaled condition number. This is an estimate of the 00207 * reciprocal Skeel condition number of the matrix A after 00208 * equilibration (if done). If this is less than the machine 00209 * precision (in particular, if it is zero), the matrix is singular 00210 * to working precision. Note that the error may still be small even 00211 * if this number is very small and the matrix appears ill- 00212 * conditioned. 00213 * 00214 * RPVGRW (output) REAL 00215 * Reciprocal pivot growth. On exit, this contains the reciprocal 00216 * pivot growth factor norm(A)/norm(U). The "max absolute element" 00217 * norm is used. If this is much less than 1, then the stability of 00218 * the LU factorization of the (equilibrated) matrix A could be poor. 00219 * This also means that the solution X, estimated condition numbers, 00220 * and error bounds could be unreliable. If factorization fails with 00221 * 0<INFO<=N, then this contains the reciprocal pivot growth factor 00222 * for the leading INFO columns of A. 00223 * 00224 * BERR (output) REAL array, dimension (NRHS) 00225 * Componentwise relative backward error. This is the 00226 * componentwise relative backward error of each solution vector X(j) 00227 * (i.e., the smallest relative change in any element of A or B that 00228 * makes X(j) an exact solution). 00229 * 00230 * N_ERR_BNDS (input) INTEGER 00231 * Number of error bounds to return for each right hand side 00232 * and each type (normwise or componentwise). See ERR_BNDS_NORM and 00233 * ERR_BNDS_COMP below. 00234 * 00235 * ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS) 00236 * For each right-hand side, this array contains information about 00237 * various error bounds and condition numbers corresponding to the 00238 * normwise relative error, which is defined as follows: 00239 * 00240 * Normwise relative error in the ith solution vector: 00241 * max_j (abs(XTRUE(j,i) - X(j,i))) 00242 * ------------------------------ 00243 * max_j abs(X(j,i)) 00244 * 00245 * The array is indexed by the type of error information as described 00246 * below. There currently are up to three pieces of information 00247 * returned. 00248 * 00249 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00250 * right-hand side. 00251 * 00252 * The second index in ERR_BNDS_NORM(:,err) contains the following 00253 * three fields: 00254 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00255 * reciprocal condition number is less than the threshold 00256 * sqrt(n) * slamch('Epsilon'). 00257 * 00258 * err = 2 "Guaranteed" error bound: The estimated forward error, 00259 * almost certainly within a factor of 10 of the true error 00260 * so long as the next entry is greater than the threshold 00261 * sqrt(n) * slamch('Epsilon'). This error bound should only 00262 * be trusted if the previous boolean is true. 00263 * 00264 * err = 3 Reciprocal condition number: Estimated normwise 00265 * reciprocal condition number. Compared with the threshold 00266 * sqrt(n) * slamch('Epsilon') to determine if the error 00267 * estimate is "guaranteed". These reciprocal condition 00268 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00269 * appropriately scaled matrix Z. 00270 * Let Z = S*A, where S scales each row by a power of the 00271 * radix so all absolute row sums of Z are approximately 1. 00272 * 00273 * See Lapack Working Note 165 for further details and extra 00274 * cautions. 00275 * 00276 * ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS) 00277 * For each right-hand side, this array contains information about 00278 * various error bounds and condition numbers corresponding to the 00279 * componentwise relative error, which is defined as follows: 00280 * 00281 * Componentwise relative error in the ith solution vector: 00282 * abs(XTRUE(j,i) - X(j,i)) 00283 * max_j ---------------------- 00284 * abs(X(j,i)) 00285 * 00286 * The array is indexed by the right-hand side i (on which the 00287 * componentwise relative error depends), and the type of error 00288 * information as described below. There currently are up to three 00289 * pieces of information returned for each right-hand side. If 00290 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00291 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00292 * the first (:,N_ERR_BNDS) entries are returned. 00293 * 00294 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00295 * right-hand side. 00296 * 00297 * The second index in ERR_BNDS_COMP(:,err) contains the following 00298 * three fields: 00299 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00300 * reciprocal condition number is less than the threshold 00301 * sqrt(n) * slamch('Epsilon'). 00302 * 00303 * err = 2 "Guaranteed" error bound: The estimated forward error, 00304 * almost certainly within a factor of 10 of the true error 00305 * so long as the next entry is greater than the threshold 00306 * sqrt(n) * slamch('Epsilon'). This error bound should only 00307 * be trusted if the previous boolean is true. 00308 * 00309 * err = 3 Reciprocal condition number: Estimated componentwise 00310 * reciprocal condition number. Compared with the threshold 00311 * sqrt(n) * slamch('Epsilon') to determine if the error 00312 * estimate is "guaranteed". These reciprocal condition 00313 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00314 * appropriately scaled matrix Z. 00315 * Let Z = S*(A*diag(x)), where x is the solution for the 00316 * current right-hand side and S scales each row of 00317 * A*diag(x) by a power of the radix so all absolute row 00318 * sums of Z are approximately 1. 00319 * 00320 * See Lapack Working Note 165 for further details and extra 00321 * cautions. 00322 * 00323 * NPARAMS (input) INTEGER 00324 * Specifies the number of parameters set in PARAMS. If .LE. 0, the 00325 * PARAMS array is never referenced and default values are used. 00326 * 00327 * PARAMS (input / output) REAL array, dimension NPARAMS 00328 * Specifies algorithm parameters. If an entry is .LT. 0.0, then 00329 * that entry will be filled with default value used for that 00330 * parameter. Only positions up to NPARAMS are accessed; defaults 00331 * are used for higher-numbered parameters. 00332 * 00333 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00334 * refinement or not. 00335 * Default: 1.0 00336 * = 0.0 : No refinement is performed, and no error bounds are 00337 * computed. 00338 * = 1.0 : Use the double-precision refinement algorithm, 00339 * possibly with doubled-single computations if the 00340 * compilation environment does not support DOUBLE 00341 * PRECISION. 00342 * (other values are reserved for future use) 00343 * 00344 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00345 * computations allowed for refinement. 00346 * Default: 10 00347 * Aggressive: Set to 100 to permit convergence using approximate 00348 * factorizations or factorizations other than LU. If 00349 * the factorization uses a technique other than 00350 * Gaussian elimination, the guarantees in 00351 * err_bnds_norm and err_bnds_comp may no longer be 00352 * trustworthy. 00353 * 00354 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00355 * will attempt to find a solution with small componentwise 00356 * relative error in the double-precision algorithm. Positive 00357 * is true, 0.0 is false. 00358 * Default: 1.0 (attempt componentwise convergence) 00359 * 00360 * WORK (workspace) REAL array, dimension (4*N) 00361 * 00362 * IWORK (workspace) INTEGER array, dimension (N) 00363 * 00364 * INFO (output) INTEGER 00365 * = 0: Successful exit. The solution to every right-hand side is 00366 * guaranteed. 00367 * < 0: If INFO = -i, the i-th argument had an illegal value 00368 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00369 * has been completed, but the factor U is exactly singular, so 00370 * the solution and error bounds could not be computed. RCOND = 0 00371 * is returned. 00372 * = N+J: The solution corresponding to the Jth right-hand side is 00373 * not guaranteed. The solutions corresponding to other right- 00374 * hand sides K with K > J may not be guaranteed as well, but 00375 * only the first such right-hand side is reported. If a small 00376 * componentwise error is not requested (PARAMS(3) = 0.0) then 00377 * the Jth right-hand side is the first with a normwise error 00378 * bound that is not guaranteed (the smallest J such 00379 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00380 * the Jth right-hand side is the first with either a normwise or 00381 * componentwise error bound that is not guaranteed (the smallest 00382 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00383 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00384 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00385 * about all of the right-hand sides check ERR_BNDS_NORM or 00386 * ERR_BNDS_COMP. 00387 * 00388 * ================================================================== 00389 * 00390 * .. Parameters .. 00391 REAL ZERO, ONE 00392 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00393 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00394 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00395 INTEGER CMP_ERR_I, PIV_GROWTH_I 00396 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00397 $ BERR_I = 3 ) 00398 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00399 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00400 $ PIV_GROWTH_I = 9 ) 00401 * .. 00402 * .. Local Scalars .. 00403 LOGICAL EQUIL, NOFACT, RCEQU 00404 INTEGER INFEQU, J 00405 REAL AMAX, BIGNUM, SMIN, SMAX, 00406 $ SCOND, SMLNUM 00407 * .. 00408 * .. External Functions .. 00409 EXTERNAL LSAME, SLAMCH, SLA_PORPVGRW 00410 LOGICAL LSAME 00411 REAL SLAMCH, SLA_PORPVGRW 00412 * .. 00413 * .. External Subroutines .. 00414 EXTERNAL SPOEQUB, SPOTRF, SPOTRS, SLACPY, SLAQSY, 00415 $ XERBLA, SLASCL2, SPORFSX 00416 * .. 00417 * .. Intrinsic Functions .. 00418 INTRINSIC MAX, MIN 00419 * .. 00420 * .. Executable Statements .. 00421 * 00422 INFO = 0 00423 NOFACT = LSAME( FACT, 'N' ) 00424 EQUIL = LSAME( FACT, 'E' ) 00425 SMLNUM = SLAMCH( 'Safe minimum' ) 00426 BIGNUM = ONE / SMLNUM 00427 IF( NOFACT .OR. EQUIL ) THEN 00428 EQUED = 'N' 00429 RCEQU = .FALSE. 00430 ELSE 00431 RCEQU = LSAME( EQUED, 'Y' ) 00432 ENDIF 00433 * 00434 * Default is failure. If an input parameter is wrong or 00435 * factorization fails, make everything look horrible. Only the 00436 * pivot growth is set here, the rest is initialized in SPORFSX. 00437 * 00438 RPVGRW = ZERO 00439 * 00440 * Test the input parameters. PARAMS is not tested until SPORFSX. 00441 * 00442 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00443 $ LSAME( FACT, 'F' ) ) THEN 00444 INFO = -1 00445 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. 00446 $ .NOT.LSAME( UPLO, 'L' ) ) THEN 00447 INFO = -2 00448 ELSE IF( N.LT.0 ) THEN 00449 INFO = -3 00450 ELSE IF( NRHS.LT.0 ) THEN 00451 INFO = -4 00452 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00453 INFO = -6 00454 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00455 INFO = -8 00456 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00457 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00458 INFO = -9 00459 ELSE 00460 IF ( RCEQU ) THEN 00461 SMIN = BIGNUM 00462 SMAX = ZERO 00463 DO 10 J = 1, N 00464 SMIN = MIN( SMIN, S( J ) ) 00465 SMAX = MAX( SMAX, S( J ) ) 00466 10 CONTINUE 00467 IF( SMIN.LE.ZERO ) THEN 00468 INFO = -10 00469 ELSE IF( N.GT.0 ) THEN 00470 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00471 ELSE 00472 SCOND = ONE 00473 END IF 00474 END IF 00475 IF( INFO.EQ.0 ) THEN 00476 IF( LDB.LT.MAX( 1, N ) ) THEN 00477 INFO = -12 00478 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00479 INFO = -14 00480 END IF 00481 END IF 00482 END IF 00483 * 00484 IF( INFO.NE.0 ) THEN 00485 CALL XERBLA( 'SPOSVXX', -INFO ) 00486 RETURN 00487 END IF 00488 * 00489 IF( EQUIL ) THEN 00490 * 00491 * Compute row and column scalings to equilibrate the matrix A. 00492 * 00493 CALL SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) 00494 IF( INFEQU.EQ.0 ) THEN 00495 * 00496 * Equilibrate the matrix. 00497 * 00498 CALL SLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00499 RCEQU = LSAME( EQUED, 'Y' ) 00500 END IF 00501 END IF 00502 * 00503 * Scale the right-hand side. 00504 * 00505 IF( RCEQU ) CALL SLASCL2( N, NRHS, S, B, LDB ) 00506 * 00507 IF( NOFACT .OR. EQUIL ) THEN 00508 * 00509 * Compute the Cholesky factorization of A. 00510 * 00511 CALL SLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00512 CALL SPOTRF( UPLO, N, AF, LDAF, INFO ) 00513 * 00514 * Return if INFO is non-zero. 00515 * 00516 IF( INFO.NE.0 ) THEN 00517 * 00518 * Pivot in column INFO is exactly 0 00519 * Compute the reciprocal pivot growth factor of the 00520 * leading rank-deficient INFO columns of A. 00521 * 00522 RPVGRW = SLA_PORPVGRW( UPLO, INFO, A, LDA, AF, LDAF, WORK ) 00523 RETURN 00524 ENDIF 00525 END IF 00526 * 00527 * Compute the reciprocal growth factor RPVGRW. 00528 * 00529 RPVGRW = SLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK ) 00530 * 00531 * Compute the solution matrix X. 00532 * 00533 CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00534 CALL SPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) 00535 * 00536 * Use iterative refinement to improve the computed solution and 00537 * compute error bounds and backward error estimates for it. 00538 * 00539 CALL SPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, 00540 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00541 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) 00542 00543 * 00544 * Scale solutions. 00545 * 00546 IF ( RCEQU ) THEN 00547 CALL SLASCL2 ( N, NRHS, S, X, LDX ) 00548 END IF 00549 * 00550 RETURN 00551 * 00552 * End of SPOSVXX 00553 * 00554 END
1.7.2