00001 SUBROUTINE DSYSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, 00002 $ EQUED, 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 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 DOUBLE PRECISION RCOND, RPVGRW 00021 * .. 00022 * .. Array Arguments .. 00023 INTEGER IPIV( * ), IWORK( * ) 00024 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00025 $ X( LDX, * ), WORK( * ) 00026 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), 00027 $ ERR_BNDS_NORM( NRHS, * ), 00028 $ ERR_BNDS_COMP( NRHS, * ) 00029 * .. 00030 * 00031 * Purpose 00032 * ======= 00033 * 00034 * DSYSVXX uses the diagonal pivoting factorization to compute the 00035 * solution to a double precision system of linear equations A * X = B, where A 00036 * is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. 00037 * 00038 * If requested, both normwise and maximum componentwise error bounds 00039 * are returned. DSYSVXX will return a solution with a tiny 00040 * guaranteed error (O(eps) where eps is the working machine 00041 * precision) unless the matrix is very ill-conditioned, in which 00042 * case a warning is returned. Relevant condition numbers also are 00043 * calculated and returned. 00044 * 00045 * DSYSVXX accepts user-provided factorizations and equilibration 00046 * factors; see the definitions of the FACT and EQUED options. 00047 * Solving with refinement and using a factorization from a previous 00048 * DSYSVXX call will also produce a solution with either O(eps) 00049 * errors or warnings, but we cannot make that claim for general 00050 * user-provided factorizations and equilibration factors if they 00051 * differ from what DSYSVXX would itself produce. 00052 * 00053 * Description 00054 * =========== 00055 * 00056 * The following steps are performed: 00057 * 00058 * 1. If FACT = 'E', double precision scaling factors are computed to equilibrate 00059 * the system: 00060 * 00061 * diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B 00062 * 00063 * Whether or not the system will be equilibrated depends on the 00064 * scaling of the matrix A, but if equilibration is used, A is 00065 * overwritten by diag(S)*A*diag(S) and B by diag(S)*B. 00066 * 00067 * 2. If FACT = 'N' or 'E', the LU decomposition is used to factor 00068 * the matrix A (after equilibration if FACT = 'E') as 00069 * 00070 * A = U * D * U**T, if UPLO = 'U', or 00071 * A = L * D * L**T, if UPLO = 'L', 00072 * 00073 * where U (or L) is a product of permutation and unit upper (lower) 00074 * triangular matrices, and D is symmetric and block diagonal with 00075 * 1-by-1 and 2-by-2 diagonal blocks. 00076 * 00077 * 3. If some D(i,i)=0, so that D is exactly singular, then the 00078 * routine returns with INFO = i. Otherwise, the factored form of A 00079 * is used to estimate the condition number of the matrix A (see 00080 * argument RCOND). If the reciprocal of the condition number is 00081 * less than machine precision, the routine still goes on to solve 00082 * for X and compute error bounds as described below. 00083 * 00084 * 4. The system of equations is solved for X using the factored form 00085 * of A. 00086 * 00087 * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00088 * the routine will use iterative refinement to try to get a small 00089 * error and error bounds. Refinement calculates the residual to at 00090 * least twice the working precision. 00091 * 00092 * 6. If equilibration was used, the matrix X is premultiplied by 00093 * diag(R) so that it solves the original system before 00094 * equilibration. 00095 * 00096 * Arguments 00097 * ========= 00098 * 00099 * Some optional parameters are bundled in the PARAMS array. These 00100 * settings determine how refinement is performed, but often the 00101 * defaults are acceptable. If the defaults are acceptable, users 00102 * can pass NPARAMS = 0 which prevents the source code from accessing 00103 * the PARAMS argument. 00104 * 00105 * FACT (input) CHARACTER*1 00106 * Specifies whether or not the factored form of the matrix A is 00107 * supplied on entry, and if not, whether the matrix A should be 00108 * equilibrated before it is factored. 00109 * = 'F': On entry, AF and IPIV contain the factored form of A. 00110 * If EQUED is not 'N', the matrix A has been 00111 * equilibrated with scaling factors given by S. 00112 * A, AF, and IPIV are not modified. 00113 * = 'N': The matrix A will be copied to AF and factored. 00114 * = 'E': The matrix A will be equilibrated if necessary, then 00115 * copied to AF and factored. 00116 * 00117 * UPLO (input) CHARACTER*1 00118 * = 'U': Upper triangle of A is stored; 00119 * = 'L': Lower triangle of A is stored. 00120 * 00121 * N (input) INTEGER 00122 * The number of linear equations, i.e., the order of the 00123 * matrix A. N >= 0. 00124 * 00125 * NRHS (input) INTEGER 00126 * The number of right hand sides, i.e., the number of columns 00127 * of the matrices B and X. NRHS >= 0. 00128 * 00129 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00130 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00131 * upper triangular part of A contains the upper triangular 00132 * part of the matrix A, and the strictly lower triangular 00133 * part of A is not referenced. If UPLO = 'L', the leading 00134 * N-by-N lower triangular part of A contains the lower 00135 * triangular part of the matrix A, and the strictly upper 00136 * triangular part of A is not referenced. 00137 * 00138 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00139 * diag(S)*A*diag(S). 00140 * 00141 * LDA (input) INTEGER 00142 * The leading dimension of the array A. LDA >= max(1,N). 00143 * 00144 * AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) 00145 * If FACT = 'F', then AF is an input argument and on entry 00146 * contains the block diagonal matrix D and the multipliers 00147 * used to obtain the factor U or L from the factorization A = 00148 * U*D*U**T or A = L*D*L**T as computed by DSYTRF. 00149 * 00150 * If FACT = 'N', then AF is an output argument and on exit 00151 * returns the block diagonal matrix D and the multipliers 00152 * used to obtain the factor U or L from the factorization A = 00153 * U*D*U**T or A = L*D*L**T. 00154 * 00155 * LDAF (input) INTEGER 00156 * The leading dimension of the array AF. LDAF >= max(1,N). 00157 * 00158 * IPIV (input or output) INTEGER array, dimension (N) 00159 * If FACT = 'F', then IPIV is an input argument and on entry 00160 * contains details of the interchanges and the block 00161 * structure of D, as determined by DSYTRF. If IPIV(k) > 0, 00162 * then rows and columns k and IPIV(k) were interchanged and 00163 * D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and 00164 * IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and 00165 * -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 00166 * diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, 00167 * then rows and columns k+1 and -IPIV(k) were interchanged 00168 * and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00169 * 00170 * If FACT = 'N', then IPIV is an output argument and on exit 00171 * contains details of the interchanges and the block 00172 * structure of D, as determined by DSYTRF. 00173 * 00174 * EQUED (input or output) CHARACTER*1 00175 * Specifies the form of equilibration that was done. 00176 * = 'N': No equilibration (always true if FACT = 'N'). 00177 * = 'Y': Both row and column equilibration, i.e., A has been 00178 * replaced by diag(S) * A * diag(S). 00179 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00180 * output argument. 00181 * 00182 * S (input or output) DOUBLE PRECISION array, dimension (N) 00183 * The scale factors for A. If EQUED = 'Y', A is multiplied on 00184 * the left and right by diag(S). S is an input argument if FACT = 00185 * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00186 * = 'Y', each element of S must be positive. If S is output, each 00187 * element of S is a power of the radix. If S is input, each element 00188 * of S should be a power of the radix to ensure a reliable solution 00189 * and error estimates. Scaling by powers of the radix does not cause 00190 * rounding errors unless the result underflows or overflows. 00191 * Rounding errors during scaling lead to refining with a matrix that 00192 * is not equivalent to the input matrix, producing error estimates 00193 * that may not be reliable. 00194 * 00195 * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) 00196 * On entry, the N-by-NRHS right hand side matrix B. 00197 * On exit, 00198 * if EQUED = 'N', B is not modified; 00199 * if EQUED = 'Y', B is overwritten by diag(S)*B; 00200 * 00201 * LDB (input) INTEGER 00202 * The leading dimension of the array B. LDB >= max(1,N). 00203 * 00204 * X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) 00205 * If INFO = 0, the N-by-NRHS solution matrix X to the original 00206 * system of equations. Note that A and B are modified on exit if 00207 * EQUED .ne. 'N', and the solution to the equilibrated system is 00208 * inv(diag(S))*X. 00209 * 00210 * LDX (input) INTEGER 00211 * The leading dimension of the array X. LDX >= max(1,N). 00212 * 00213 * RCOND (output) DOUBLE PRECISION 00214 * Reciprocal scaled condition number. This is an estimate of the 00215 * reciprocal Skeel condition number of the matrix A after 00216 * equilibration (if done). If this is less than the machine 00217 * precision (in particular, if it is zero), the matrix is singular 00218 * to working precision. Note that the error may still be small even 00219 * if this number is very small and the matrix appears ill- 00220 * conditioned. 00221 * 00222 * RPVGRW (output) DOUBLE PRECISION 00223 * Reciprocal pivot growth. On exit, this contains the reciprocal 00224 * pivot growth factor norm(A)/norm(U). The "max absolute element" 00225 * norm is used. If this is much less than 1, then the stability of 00226 * the LU factorization of the (equilibrated) matrix A could be poor. 00227 * This also means that the solution X, estimated condition numbers, 00228 * and error bounds could be unreliable. If factorization fails with 00229 * 0<INFO<=N, then this contains the reciprocal pivot growth factor 00230 * for the leading INFO columns of A. 00231 * 00232 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00233 * Componentwise relative backward error. This is the 00234 * componentwise relative backward error of each solution vector X(j) 00235 * (i.e., the smallest relative change in any element of A or B that 00236 * makes X(j) an exact solution). 00237 * 00238 * N_ERR_BNDS (input) INTEGER 00239 * Number of error bounds to return for each right hand side 00240 * and each type (normwise or componentwise). See ERR_BNDS_NORM and 00241 * ERR_BNDS_COMP below. 00242 * 00243 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00244 * For each right-hand side, this array contains information about 00245 * various error bounds and condition numbers corresponding to the 00246 * normwise relative error, which is defined as follows: 00247 * 00248 * Normwise relative error in the ith solution vector: 00249 * max_j (abs(XTRUE(j,i) - X(j,i))) 00250 * ------------------------------ 00251 * max_j abs(X(j,i)) 00252 * 00253 * The array is indexed by the type of error information as described 00254 * below. There currently are up to three pieces of information 00255 * returned. 00256 * 00257 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00258 * right-hand side. 00259 * 00260 * The second index in ERR_BNDS_NORM(:,err) contains the following 00261 * three fields: 00262 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00263 * reciprocal condition number is less than the threshold 00264 * sqrt(n) * dlamch('Epsilon'). 00265 * 00266 * err = 2 "Guaranteed" error bound: The estimated forward error, 00267 * almost certainly within a factor of 10 of the true error 00268 * so long as the next entry is greater than the threshold 00269 * sqrt(n) * dlamch('Epsilon'). This error bound should only 00270 * be trusted if the previous boolean is true. 00271 * 00272 * err = 3 Reciprocal condition number: Estimated normwise 00273 * reciprocal condition number. Compared with the threshold 00274 * sqrt(n) * dlamch('Epsilon') to determine if the error 00275 * estimate is "guaranteed". These reciprocal condition 00276 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00277 * appropriately scaled matrix Z. 00278 * Let Z = S*A, where S scales each row by a power of the 00279 * radix so all absolute row sums of Z are approximately 1. 00280 * 00281 * See Lapack Working Note 165 for further details and extra 00282 * cautions. 00283 * 00284 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00285 * For each right-hand side, this array contains information about 00286 * various error bounds and condition numbers corresponding to the 00287 * componentwise relative error, which is defined as follows: 00288 * 00289 * Componentwise relative error in the ith solution vector: 00290 * abs(XTRUE(j,i) - X(j,i)) 00291 * max_j ---------------------- 00292 * abs(X(j,i)) 00293 * 00294 * The array is indexed by the right-hand side i (on which the 00295 * componentwise relative error depends), and the type of error 00296 * information as described below. There currently are up to three 00297 * pieces of information returned for each right-hand side. If 00298 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00299 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00300 * the first (:,N_ERR_BNDS) entries are returned. 00301 * 00302 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00303 * right-hand side. 00304 * 00305 * The second index in ERR_BNDS_COMP(:,err) contains the following 00306 * three fields: 00307 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00308 * reciprocal condition number is less than the threshold 00309 * sqrt(n) * dlamch('Epsilon'). 00310 * 00311 * err = 2 "Guaranteed" error bound: The estimated forward error, 00312 * almost certainly within a factor of 10 of the true error 00313 * so long as the next entry is greater than the threshold 00314 * sqrt(n) * dlamch('Epsilon'). This error bound should only 00315 * be trusted if the previous boolean is true. 00316 * 00317 * err = 3 Reciprocal condition number: Estimated componentwise 00318 * reciprocal condition number. Compared with the threshold 00319 * sqrt(n) * dlamch('Epsilon') to determine if the error 00320 * estimate is "guaranteed". These reciprocal condition 00321 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00322 * appropriately scaled matrix Z. 00323 * Let Z = S*(A*diag(x)), where x is the solution for the 00324 * current right-hand side and S scales each row of 00325 * A*diag(x) by a power of the radix so all absolute row 00326 * sums of Z are approximately 1. 00327 * 00328 * See Lapack Working Note 165 for further details and extra 00329 * cautions. 00330 * 00331 * NPARAMS (input) INTEGER 00332 * Specifies the number of parameters set in PARAMS. If .LE. 0, the 00333 * PARAMS array is never referenced and default values are used. 00334 * 00335 * PARAMS (input / output) DOUBLE PRECISION array, dimension (NPARAMS) 00336 * Specifies algorithm parameters. If an entry is .LT. 0.0, then 00337 * that entry will be filled with default value used for that 00338 * parameter. Only positions up to NPARAMS are accessed; defaults 00339 * are used for higher-numbered parameters. 00340 * 00341 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00342 * refinement or not. 00343 * Default: 1.0D+0 00344 * = 0.0 : No refinement is performed, and no error bounds are 00345 * computed. 00346 * = 1.0 : Use the extra-precise refinement algorithm. 00347 * (other values are reserved for future use) 00348 * 00349 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00350 * computations allowed for refinement. 00351 * Default: 10 00352 * Aggressive: Set to 100 to permit convergence using approximate 00353 * factorizations or factorizations other than LU. If 00354 * the factorization uses a technique other than 00355 * Gaussian elimination, the guarantees in 00356 * err_bnds_norm and err_bnds_comp may no longer be 00357 * trustworthy. 00358 * 00359 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00360 * will attempt to find a solution with small componentwise 00361 * relative error in the double-precision algorithm. Positive 00362 * is true, 0.0 is false. 00363 * Default: 1.0 (attempt componentwise convergence) 00364 * 00365 * WORK (workspace) DOUBLE PRECISION array, dimension (4*N) 00366 * 00367 * IWORK (workspace) INTEGER array, dimension (N) 00368 * 00369 * INFO (output) INTEGER 00370 * = 0: Successful exit. The solution to every right-hand side is 00371 * guaranteed. 00372 * < 0: If INFO = -i, the i-th argument had an illegal value 00373 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00374 * has been completed, but the factor U is exactly singular, so 00375 * the solution and error bounds could not be computed. RCOND = 0 00376 * is returned. 00377 * = N+J: The solution corresponding to the Jth right-hand side is 00378 * not guaranteed. The solutions corresponding to other right- 00379 * hand sides K with K > J may not be guaranteed as well, but 00380 * only the first such right-hand side is reported. If a small 00381 * componentwise error is not requested (PARAMS(3) = 0.0) then 00382 * the Jth right-hand side is the first with a normwise error 00383 * bound that is not guaranteed (the smallest J such 00384 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00385 * the Jth right-hand side is the first with either a normwise or 00386 * componentwise error bound that is not guaranteed (the smallest 00387 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00388 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00389 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00390 * about all of the right-hand sides check ERR_BNDS_NORM or 00391 * ERR_BNDS_COMP. 00392 * 00393 * ================================================================== 00394 * 00395 * .. Parameters .. 00396 DOUBLE PRECISION ZERO, ONE 00397 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00398 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00399 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00400 INTEGER CMP_ERR_I, PIV_GROWTH_I 00401 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00402 $ BERR_I = 3 ) 00403 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00404 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00405 $ PIV_GROWTH_I = 9 ) 00406 * .. 00407 * .. Local Scalars .. 00408 LOGICAL EQUIL, NOFACT, RCEQU 00409 INTEGER INFEQU, J 00410 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 00411 * .. 00412 * .. External Functions .. 00413 EXTERNAL LSAME, DLAMCH, DLA_SYRPVGRW 00414 LOGICAL LSAME 00415 DOUBLE PRECISION DLAMCH, DLA_SYRPVGRW 00416 * .. 00417 * .. External Subroutines .. 00418 EXTERNAL DSYCON, DSYEQUB, DSYTRF, DSYTRS, 00419 $ DLACPY, DLAQSY, XERBLA, DLASCL2, DSYRFSX 00420 * .. 00421 * .. Intrinsic Functions .. 00422 INTRINSIC MAX, MIN 00423 * .. 00424 * .. Executable Statements .. 00425 * 00426 INFO = 0 00427 NOFACT = LSAME( FACT, 'N' ) 00428 EQUIL = LSAME( FACT, 'E' ) 00429 SMLNUM = DLAMCH( 'Safe minimum' ) 00430 BIGNUM = ONE / SMLNUM 00431 IF( NOFACT .OR. EQUIL ) THEN 00432 EQUED = 'N' 00433 RCEQU = .FALSE. 00434 ELSE 00435 RCEQU = LSAME( EQUED, 'Y' ) 00436 ENDIF 00437 * 00438 * Default is failure. If an input parameter is wrong or 00439 * factorization fails, make everything look horrible. Only the 00440 * pivot growth is set here, the rest is initialized in DSYRFSX. 00441 * 00442 RPVGRW = ZERO 00443 * 00444 * Test the input parameters. PARAMS is not tested until DSYRFSX. 00445 * 00446 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00447 $ LSAME( FACT, 'F' ) ) THEN 00448 INFO = -1 00449 ELSE IF( .NOT.LSAME(UPLO, 'U') .AND. 00450 $ .NOT.LSAME(UPLO, 'L') ) THEN 00451 INFO = -2 00452 ELSE IF( N.LT.0 ) THEN 00453 INFO = -3 00454 ELSE IF( NRHS.LT.0 ) THEN 00455 INFO = -4 00456 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00457 INFO = -6 00458 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00459 INFO = -8 00460 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00461 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00462 INFO = -9 00463 ELSE 00464 IF ( RCEQU ) THEN 00465 SMIN = BIGNUM 00466 SMAX = ZERO 00467 DO 10 J = 1, N 00468 SMIN = MIN( SMIN, S( J ) ) 00469 SMAX = MAX( SMAX, S( J ) ) 00470 10 CONTINUE 00471 IF( SMIN.LE.ZERO ) THEN 00472 INFO = -10 00473 ELSE IF( N.GT.0 ) THEN 00474 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00475 ELSE 00476 SCOND = ONE 00477 END IF 00478 END IF 00479 IF( INFO.EQ.0 ) THEN 00480 IF( LDB.LT.MAX( 1, N ) ) THEN 00481 INFO = -12 00482 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00483 INFO = -14 00484 END IF 00485 END IF 00486 END IF 00487 * 00488 IF( INFO.NE.0 ) THEN 00489 CALL XERBLA( 'DSYSVXX', -INFO ) 00490 RETURN 00491 END IF 00492 * 00493 IF( EQUIL ) THEN 00494 * 00495 * Compute row and column scalings to equilibrate the matrix A. 00496 * 00497 CALL DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU ) 00498 IF( INFEQU.EQ.0 ) THEN 00499 * 00500 * Equilibrate the matrix. 00501 * 00502 CALL DLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00503 RCEQU = LSAME( EQUED, 'Y' ) 00504 END IF 00505 END IF 00506 * 00507 * Scale the right-hand side. 00508 * 00509 IF( RCEQU ) CALL DLASCL2( N, NRHS, S, B, LDB ) 00510 * 00511 IF( NOFACT .OR. EQUIL ) THEN 00512 * 00513 * Compute the LDL^T or UDU^T factorization of A. 00514 * 00515 CALL DLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00516 CALL DSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO ) 00517 * 00518 * Return if INFO is non-zero. 00519 * 00520 IF( INFO.GT.0 ) THEN 00521 * 00522 * Pivot in column INFO is exactly 0 00523 * Compute the reciprocal pivot growth factor of the 00524 * leading rank-deficient INFO columns of A. 00525 * 00526 IF ( N.GT.0 ) 00527 $ RPVGRW = DLA_SYRPVGRW(UPLO, N, INFO, A, LDA, AF, 00528 $ LDAF, IPIV, WORK ) 00529 RETURN 00530 END IF 00531 END IF 00532 * 00533 * Compute the reciprocal pivot growth factor RPVGRW. 00534 * 00535 IF ( N.GT.0 ) 00536 $ RPVGRW = DLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, 00537 $ IPIV, WORK ) 00538 * 00539 * Compute the solution matrix X. 00540 * 00541 CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00542 CALL DSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 00543 * 00544 * Use iterative refinement to improve the computed solution and 00545 * compute error bounds and backward error estimates for it. 00546 * 00547 CALL DSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, 00548 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00549 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) 00550 * 00551 * Scale solutions. 00552 * 00553 IF ( RCEQU ) THEN 00554 CALL DLASCL2 ( N, NRHS, S, X, LDX ) 00555 END IF 00556 * 00557 RETURN 00558 * 00559 * End of DSYSVXX 00560 * 00561 END
1.7.2