00001 SUBROUTINE CSYSVXX( 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, RWORK, 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 IPIV( * ) 00024 COMPLEX 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, * ), RWORK( * ) 00029 * .. 00030 * 00031 * Purpose 00032 * ======= 00033 * 00034 * CSYSVXX uses the diagonal pivoting factorization to compute the 00035 * solution to a complex system of linear equations A * X = B, where 00036 * A is an N-by-N symmetric matrix and X and B are N-by-NRHS 00037 * matrices. 00038 * 00039 * If requested, both normwise and maximum componentwise error bounds 00040 * are returned. CSYSVXX 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 * CSYSVXX 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 * CSYSVXX 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 CSYSVXX 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 LU decomposition is used to factor 00069 * the matrix A (after equilibration if FACT = 'E') as 00070 * 00071 * A = U * D * U**T, if UPLO = 'U', or 00072 * A = L * D * L**T, if UPLO = 'L', 00073 * 00074 * where U (or L) is a product of permutation and unit upper (lower) 00075 * triangular matrices, and D is symmetric and block diagonal with 00076 * 1-by-1 and 2-by-2 diagonal blocks. 00077 * 00078 * 3. If some D(i,i)=0, so that D is exactly singular, then the 00079 * routine returns with INFO = i. Otherwise, the factored form of A 00080 * is used to estimate the condition number of the matrix A (see 00081 * argument RCOND). If the reciprocal of the condition number is 00082 * less than machine precision, the routine still goes on to solve 00083 * for X and compute error bounds as described below. 00084 * 00085 * 4. The system of equations is solved for X using the factored form 00086 * of A. 00087 * 00088 * 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), 00089 * the routine will use iterative refinement to try to get a small 00090 * error and error bounds. Refinement calculates the residual to at 00091 * least twice the working precision. 00092 * 00093 * 6. If equilibration was used, the matrix X is premultiplied by 00094 * diag(R) so that it solves the original system before 00095 * equilibration. 00096 * 00097 * Arguments 00098 * ========= 00099 * 00100 * Some optional parameters are bundled in the PARAMS array. These 00101 * settings determine how refinement is performed, but often the 00102 * defaults are acceptable. If the defaults are acceptable, users 00103 * can pass NPARAMS = 0 which prevents the source code from accessing 00104 * the PARAMS argument. 00105 * 00106 * FACT (input) CHARACTER*1 00107 * Specifies whether or not the factored form of the matrix A is 00108 * supplied on entry, and if not, whether the matrix A should be 00109 * equilibrated before it is factored. 00110 * = 'F': On entry, AF and IPIV contain the factored form of A. 00111 * If EQUED is not 'N', the matrix A has been 00112 * equilibrated with scaling factors given by S. 00113 * A, AF, and IPIV are not modified. 00114 * = 'N': The matrix A will be copied to AF and factored. 00115 * = 'E': The matrix A will be equilibrated if necessary, then 00116 * copied to AF and factored. 00117 * 00118 * UPLO (input) CHARACTER*1 00119 * = 'U': Upper triangle of A is stored; 00120 * = 'L': Lower triangle of A is stored. 00121 * 00122 * N (input) INTEGER 00123 * The number of linear equations, i.e., the order of the 00124 * matrix A. N >= 0. 00125 * 00126 * NRHS (input) INTEGER 00127 * The number of right hand sides, i.e., the number of columns 00128 * of the matrices B and X. NRHS >= 0. 00129 * 00130 * A (input/output) COMPLEX array, dimension (LDA,N) 00131 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00132 * upper triangular part of A contains the upper triangular 00133 * part of the matrix A, and the strictly lower triangular 00134 * part of A is not referenced. If UPLO = 'L', the leading 00135 * N-by-N lower triangular part of A contains the lower 00136 * triangular part of the matrix A, and the strictly upper 00137 * triangular part of A is not referenced. 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) COMPLEX array, dimension (LDAF,N) 00146 * If FACT = 'F', then AF is an input argument and on entry 00147 * contains the block diagonal matrix D and the multipliers 00148 * used to obtain the factor U or L from the factorization A = 00149 * U*D*U**T or A = L*D*L**T as computed by SSYTRF. 00150 * 00151 * If FACT = 'N', then AF is an output argument and on exit 00152 * returns the block diagonal matrix D and the multipliers 00153 * used to obtain the factor U or L from the factorization A = 00154 * U*D*U**T or A = L*D*L**T. 00155 * 00156 * LDAF (input) INTEGER 00157 * The leading dimension of the array AF. LDAF >= max(1,N). 00158 * 00159 * IPIV (input or output) INTEGER array, dimension (N) 00160 * If FACT = 'F', then IPIV is an input argument and on entry 00161 * contains details of the interchanges and the block 00162 * structure of D, as determined by SSYTRF. If IPIV(k) > 0, 00163 * then rows and columns k and IPIV(k) were interchanged and 00164 * D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and 00165 * IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and 00166 * -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 00167 * diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, 00168 * then rows and columns k+1 and -IPIV(k) were interchanged 00169 * and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00170 * 00171 * If FACT = 'N', then IPIV is an output argument and on exit 00172 * contains details of the interchanges and the block 00173 * structure of D, as determined by SSYTRF. 00174 * 00175 * EQUED (input or output) CHARACTER*1 00176 * Specifies the form of equilibration that was done. 00177 * = 'N': No equilibration (always true if FACT = 'N'). 00178 * = 'Y': Both row and column equilibration, i.e., A has been 00179 * replaced by diag(S) * A * diag(S). 00180 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00181 * output argument. 00182 * 00183 * S (input or output) REAL array, dimension (N) 00184 * The scale factors for A. If EQUED = 'Y', A is multiplied on 00185 * the left and right by diag(S). S is an input argument if FACT = 00186 * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00187 * = 'Y', each element of S must be positive. If S is output, each 00188 * element of S is a power of the radix. If S is input, each element 00189 * of S should be a power of the radix to ensure a reliable solution 00190 * and error estimates. Scaling by powers of the radix does not cause 00191 * rounding errors unless the result underflows or overflows. 00192 * Rounding errors during scaling lead to refining with a matrix that 00193 * is not equivalent to the input matrix, producing error estimates 00194 * that may not be reliable. 00195 * 00196 * B (input/output) COMPLEX array, dimension (LDB,NRHS) 00197 * On entry, the N-by-NRHS right hand side matrix B. 00198 * On exit, 00199 * if EQUED = 'N', B is not modified; 00200 * if EQUED = 'Y', B is overwritten by diag(S)*B; 00201 * 00202 * LDB (input) INTEGER 00203 * The leading dimension of the array B. LDB >= max(1,N). 00204 * 00205 * X (output) COMPLEX array, dimension (LDX,NRHS) 00206 * If INFO = 0, the N-by-NRHS solution matrix X to the original 00207 * system of equations. Note that A and B are modified on exit if 00208 * EQUED .ne. 'N', and the solution to the equilibrated system is 00209 * inv(diag(S))*X. 00210 * 00211 * LDX (input) INTEGER 00212 * The leading dimension of the array X. LDX >= max(1,N). 00213 * 00214 * RCOND (output) REAL 00215 * Reciprocal scaled condition number. This is an estimate of the 00216 * reciprocal Skeel condition number of the matrix A after 00217 * equilibration (if done). If this is less than the machine 00218 * precision (in particular, if it is zero), the matrix is singular 00219 * to working precision. Note that the error may still be small even 00220 * if this number is very small and the matrix appears ill- 00221 * conditioned. 00222 * 00223 * RPVGRW (output) REAL 00224 * Reciprocal pivot growth. On exit, this contains the reciprocal 00225 * pivot growth factor norm(A)/norm(U). The "max absolute element" 00226 * norm is used. If this is much less than 1, then the stability of 00227 * the LU factorization of the (equilibrated) matrix A could be poor. 00228 * This also means that the solution X, estimated condition numbers, 00229 * and error bounds could be unreliable. If factorization fails with 00230 * 0<INFO<=N, then this contains the reciprocal pivot growth factor 00231 * for the leading INFO columns of A. 00232 * 00233 * BERR (output) REAL array, dimension (NRHS) 00234 * Componentwise relative backward error. This is the 00235 * componentwise relative backward error of each solution vector X(j) 00236 * (i.e., the smallest relative change in any element of A or B that 00237 * makes X(j) an exact solution). 00238 * 00239 * N_ERR_BNDS (input) INTEGER 00240 * Number of error bounds to return for each right hand side 00241 * and each type (normwise or componentwise). See ERR_BNDS_NORM and 00242 * ERR_BNDS_COMP below. 00243 * 00244 * ERR_BNDS_NORM (output) REAL array, dimension (NRHS, N_ERR_BNDS) 00245 * For each right-hand side, this array contains information about 00246 * various error bounds and condition numbers corresponding to the 00247 * normwise relative error, which is defined as follows: 00248 * 00249 * Normwise relative error in the ith solution vector: 00250 * max_j (abs(XTRUE(j,i) - X(j,i))) 00251 * ------------------------------ 00252 * max_j abs(X(j,i)) 00253 * 00254 * The array is indexed by the type of error information as described 00255 * below. There currently are up to three pieces of information 00256 * returned. 00257 * 00258 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00259 * right-hand side. 00260 * 00261 * The second index in ERR_BNDS_NORM(:,err) contains the following 00262 * three fields: 00263 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00264 * reciprocal condition number is less than the threshold 00265 * sqrt(n) * slamch('Epsilon'). 00266 * 00267 * err = 2 "Guaranteed" error bound: The estimated forward error, 00268 * almost certainly within a factor of 10 of the true error 00269 * so long as the next entry is greater than the threshold 00270 * sqrt(n) * slamch('Epsilon'). This error bound should only 00271 * be trusted if the previous boolean is true. 00272 * 00273 * err = 3 Reciprocal condition number: Estimated normwise 00274 * reciprocal condition number. Compared with the threshold 00275 * sqrt(n) * slamch('Epsilon') to determine if the error 00276 * estimate is "guaranteed". These reciprocal condition 00277 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00278 * appropriately scaled matrix Z. 00279 * Let Z = S*A, where S scales each row by a power of the 00280 * radix so all absolute row sums of Z are approximately 1. 00281 * 00282 * See Lapack Working Note 165 for further details and extra 00283 * cautions. 00284 * 00285 * ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS) 00286 * For each right-hand side, this array contains information about 00287 * various error bounds and condition numbers corresponding to the 00288 * componentwise relative error, which is defined as follows: 00289 * 00290 * Componentwise relative error in the ith solution vector: 00291 * abs(XTRUE(j,i) - X(j,i)) 00292 * max_j ---------------------- 00293 * abs(X(j,i)) 00294 * 00295 * The array is indexed by the right-hand side i (on which the 00296 * componentwise relative error depends), and the type of error 00297 * information as described below. There currently are up to three 00298 * pieces of information returned for each right-hand side. If 00299 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00300 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00301 * the first (:,N_ERR_BNDS) entries are returned. 00302 * 00303 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00304 * right-hand side. 00305 * 00306 * The second index in ERR_BNDS_COMP(:,err) contains the following 00307 * three fields: 00308 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00309 * reciprocal condition number is less than the threshold 00310 * sqrt(n) * slamch('Epsilon'). 00311 * 00312 * err = 2 "Guaranteed" error bound: The estimated forward error, 00313 * almost certainly within a factor of 10 of the true error 00314 * so long as the next entry is greater than the threshold 00315 * sqrt(n) * slamch('Epsilon'). This error bound should only 00316 * be trusted if the previous boolean is true. 00317 * 00318 * err = 3 Reciprocal condition number: Estimated componentwise 00319 * reciprocal condition number. Compared with the threshold 00320 * sqrt(n) * slamch('Epsilon') to determine if the error 00321 * estimate is "guaranteed". These reciprocal condition 00322 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00323 * appropriately scaled matrix Z. 00324 * Let Z = S*(A*diag(x)), where x is the solution for the 00325 * current right-hand side and S scales each row of 00326 * A*diag(x) by a power of the radix so all absolute row 00327 * sums of Z are approximately 1. 00328 * 00329 * See Lapack Working Note 165 for further details and extra 00330 * cautions. 00331 * 00332 * NPARAMS (input) INTEGER 00333 * Specifies the number of parameters set in PARAMS. If .LE. 0, the 00334 * PARAMS array is never referenced and default values are used. 00335 * 00336 * PARAMS (input / output) REAL array, dimension NPARAMS 00337 * Specifies algorithm parameters. If an entry is .LT. 0.0, then 00338 * that entry will be filled with default value used for that 00339 * parameter. Only positions up to NPARAMS are accessed; defaults 00340 * are used for higher-numbered parameters. 00341 * 00342 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00343 * refinement or not. 00344 * Default: 1.0 00345 * = 0.0 : No refinement is performed, and no error bounds are 00346 * computed. 00347 * = 1.0 : Use the double-precision refinement algorithm, 00348 * possibly with doubled-single computations if the 00349 * compilation environment does not support DOUBLE 00350 * PRECISION. 00351 * (other values are reserved for future use) 00352 * 00353 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00354 * computations allowed for refinement. 00355 * Default: 10 00356 * Aggressive: Set to 100 to permit convergence using approximate 00357 * factorizations or factorizations other than LU. If 00358 * the factorization uses a technique other than 00359 * Gaussian elimination, the guarantees in 00360 * err_bnds_norm and err_bnds_comp may no longer be 00361 * trustworthy. 00362 * 00363 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00364 * will attempt to find a solution with small componentwise 00365 * relative error in the double-precision algorithm. Positive 00366 * is true, 0.0 is false. 00367 * Default: 1.0 (attempt componentwise convergence) 00368 * 00369 * WORK (workspace) COMPLEX array, dimension (2*N) 00370 * 00371 * RWORK (workspace) REAL array, dimension (2*N) 00372 * 00373 * INFO (output) INTEGER 00374 * = 0: Successful exit. The solution to every right-hand side is 00375 * guaranteed. 00376 * < 0: If INFO = -i, the i-th argument had an illegal value 00377 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00378 * has been completed, but the factor U is exactly singular, so 00379 * the solution and error bounds could not be computed. RCOND = 0 00380 * is returned. 00381 * = N+J: The solution corresponding to the Jth right-hand side is 00382 * not guaranteed. The solutions corresponding to other right- 00383 * hand sides K with K > J may not be guaranteed as well, but 00384 * only the first such right-hand side is reported. If a small 00385 * componentwise error is not requested (PARAMS(3) = 0.0) then 00386 * the Jth right-hand side is the first with a normwise error 00387 * bound that is not guaranteed (the smallest J such 00388 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00389 * the Jth right-hand side is the first with either a normwise or 00390 * componentwise error bound that is not guaranteed (the smallest 00391 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00392 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00393 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00394 * about all of the right-hand sides check ERR_BNDS_NORM or 00395 * ERR_BNDS_COMP. 00396 * 00397 * ================================================================== 00398 * 00399 * .. Parameters .. 00400 REAL ZERO, ONE 00401 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00402 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00403 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00404 INTEGER CMP_ERR_I, PIV_GROWTH_I 00405 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00406 $ BERR_I = 3 ) 00407 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00408 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00409 $ PIV_GROWTH_I = 9 ) 00410 * .. 00411 * .. Local Scalars .. 00412 LOGICAL EQUIL, NOFACT, RCEQU 00413 INTEGER INFEQU, J 00414 REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 00415 * .. 00416 * .. External Functions .. 00417 EXTERNAL LSAME, SLAMCH, CLA_SYRPVGRW 00418 LOGICAL LSAME 00419 REAL SLAMCH, CLA_SYRPVGRW 00420 * .. 00421 * .. External Subroutines .. 00422 EXTERNAL CSYCON, CSYEQUB, CSYTRF, CSYTRS, CLACPY, 00423 $ CLAQSY, XERBLA, CLASCL2, CSYRFSX 00424 * .. 00425 * .. Intrinsic Functions .. 00426 INTRINSIC MAX, MIN 00427 * .. 00428 * .. Executable Statements .. 00429 * 00430 INFO = 0 00431 NOFACT = LSAME( FACT, 'N' ) 00432 EQUIL = LSAME( FACT, 'E' ) 00433 SMLNUM = SLAMCH( 'Safe minimum' ) 00434 BIGNUM = ONE / SMLNUM 00435 IF( NOFACT .OR. EQUIL ) THEN 00436 EQUED = 'N' 00437 RCEQU = .FALSE. 00438 ELSE 00439 RCEQU = LSAME( EQUED, 'Y' ) 00440 ENDIF 00441 * 00442 * Default is failure. If an input parameter is wrong or 00443 * factorization fails, make everything look horrible. Only the 00444 * pivot growth is set here, the rest is initialized in CSYRFSX. 00445 * 00446 RPVGRW = ZERO 00447 * 00448 * Test the input parameters. PARAMS is not tested until CSYRFSX. 00449 * 00450 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00451 $ LSAME( FACT, 'F' ) ) THEN 00452 INFO = -1 00453 ELSE IF( .NOT.LSAME(UPLO, 'U') .AND. 00454 $ .NOT.LSAME(UPLO, 'L') ) THEN 00455 INFO = -2 00456 ELSE IF( N.LT.0 ) THEN 00457 INFO = -3 00458 ELSE IF( NRHS.LT.0 ) THEN 00459 INFO = -4 00460 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00461 INFO = -6 00462 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00463 INFO = -8 00464 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00465 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00466 INFO = -9 00467 ELSE 00468 IF ( RCEQU ) THEN 00469 SMIN = BIGNUM 00470 SMAX = ZERO 00471 DO 10 J = 1, N 00472 SMIN = MIN( SMIN, S( J ) ) 00473 SMAX = MAX( SMAX, S( J ) ) 00474 10 CONTINUE 00475 IF( SMIN.LE.ZERO ) THEN 00476 INFO = -10 00477 ELSE IF( N.GT.0 ) THEN 00478 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00479 ELSE 00480 SCOND = ONE 00481 END IF 00482 END IF 00483 IF( INFO.EQ.0 ) THEN 00484 IF( LDB.LT.MAX( 1, N ) ) THEN 00485 INFO = -12 00486 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00487 INFO = -14 00488 END IF 00489 END IF 00490 END IF 00491 * 00492 IF( INFO.NE.0 ) THEN 00493 CALL XERBLA( 'CSYSVXX', -INFO ) 00494 RETURN 00495 END IF 00496 * 00497 IF( EQUIL ) THEN 00498 * 00499 * Compute row and column scalings to equilibrate the matrix A. 00500 * 00501 CALL CSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU ) 00502 IF( INFEQU.EQ.0 ) THEN 00503 * 00504 * Equilibrate the matrix. 00505 * 00506 CALL CLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00507 RCEQU = LSAME( EQUED, 'Y' ) 00508 END IF 00509 00510 END IF 00511 * 00512 * Scale the right hand-side. 00513 * 00514 IF( RCEQU ) CALL CLASCL2( N, NRHS, S, B, LDB ) 00515 * 00516 IF( NOFACT .OR. EQUIL ) THEN 00517 * 00518 * Compute the LDL^T or UDU^T factorization of A. 00519 * 00520 CALL CLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00521 CALL CSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO ) 00522 * 00523 * Return if INFO is non-zero. 00524 * 00525 IF( INFO.GT.0 ) THEN 00526 * 00527 * Pivot in column INFO is exactly 0 00528 * Compute the reciprocal pivot growth factor of the 00529 * leading rank-deficient INFO columns of A. 00530 * 00531 IF ( N.GT.0 ) 00532 $ RPVGRW = CLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, 00533 $ LDAF, IPIV, RWORK ) 00534 RETURN 00535 END IF 00536 END IF 00537 * 00538 * Compute the reciprocal pivot growth factor RPVGRW. 00539 * 00540 IF ( N.GT.0 ) 00541 $ RPVGRW = CLA_SYRPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, 00542 $ IPIV, RWORK ) 00543 * 00544 * Compute the solution matrix X. 00545 * 00546 CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00547 CALL CSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 00548 * 00549 * Use iterative refinement to improve the computed solution and 00550 * compute error bounds and backward error estimates for it. 00551 * 00552 CALL CSYRFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, 00553 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00554 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 00555 * 00556 * Scale solutions. 00557 * 00558 IF ( RCEQU ) THEN 00559 CALL CLASCL2 (N, NRHS, S, X, LDX ) 00560 END IF 00561 * 00562 RETURN 00563 * 00564 * End of CSYSVXX 00565 * 00566 END
1.7.2