00001 SUBROUTINE ZHESVXX( 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 DOUBLE PRECISION RCOND, RPVGRW 00021 * .. 00022 * .. Array Arguments .. 00023 INTEGER IPIV( * ) 00024 COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ), 00025 $ WORK( * ), X( LDX, * ) 00026 DOUBLE PRECISION S( * ), PARAMS( * ), BERR( * ), RWORK( * ), 00027 $ ERR_BNDS_NORM( NRHS, * ), 00028 $ ERR_BNDS_COMP( NRHS, * ) 00029 * .. 00030 * 00031 * Purpose 00032 * ======= 00033 * 00034 * ZHESVXX uses the diagonal pivoting factorization to compute the 00035 * solution to a complex*16 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. ZHESVXX 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 * ZHESVXX 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 * ZHESVXX 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 ZHESVXX would itself produce. 00053 * 00054 * Description 00055 * =========== 00056 * 00057 * The following steps are performed: 00058 * 00059 * 1. If FACT = 'E', double precision 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 * N (input) INTEGER 00119 * The number of linear equations, i.e., the order of the 00120 * matrix A. N >= 0. 00121 * 00122 * NRHS (input) INTEGER 00123 * The number of right hand sides, i.e., the number of columns 00124 * of the matrices B and X. NRHS >= 0. 00125 * 00126 * A (input/output) COMPLEX*16 array, dimension (LDA,N) 00127 * The symmetric matrix A. If UPLO = 'U', the leading N-by-N 00128 * upper triangular part of A contains the upper triangular 00129 * part of the matrix A, and the strictly lower triangular 00130 * part of A is not referenced. If UPLO = 'L', the leading 00131 * N-by-N lower triangular part of A contains the lower 00132 * triangular part of the matrix A, and the strictly upper 00133 * triangular part of A is not referenced. 00134 * 00135 * On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by 00136 * diag(S)*A*diag(S). 00137 * 00138 * LDA (input) INTEGER 00139 * The leading dimension of the array A. LDA >= max(1,N). 00140 * 00141 * AF (input or output) COMPLEX*16 array, dimension (LDAF,N) 00142 * If FACT = 'F', then AF is an input argument and on entry 00143 * contains the block diagonal matrix D and the multipliers 00144 * used to obtain the factor U or L from the factorization A = 00145 * U*D*U**T or A = L*D*L**T as computed by DSYTRF. 00146 * 00147 * If FACT = 'N', then AF is an output argument and on exit 00148 * returns the block diagonal matrix D and the multipliers 00149 * used to obtain the factor U or L from the factorization A = 00150 * U*D*U**T or A = L*D*L**T. 00151 * 00152 * LDAF (input) INTEGER 00153 * The leading dimension of the array AF. LDAF >= max(1,N). 00154 * 00155 * IPIV (input or output) INTEGER array, dimension (N) 00156 * If FACT = 'F', then IPIV is an input argument and on entry 00157 * contains details of the interchanges and the block 00158 * structure of D, as determined by ZHETRF. If IPIV(k) > 0, 00159 * then rows and columns k and IPIV(k) were interchanged and 00160 * D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and 00161 * IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and 00162 * -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 00163 * diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, 00164 * then rows and columns k+1 and -IPIV(k) were interchanged 00165 * and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. 00166 * 00167 * If FACT = 'N', then IPIV is an output argument and on exit 00168 * contains details of the interchanges and the block 00169 * structure of D, as determined by ZHETRF. 00170 * 00171 * EQUED (input or output) CHARACTER*1 00172 * Specifies the form of equilibration that was done. 00173 * = 'N': No equilibration (always true if FACT = 'N'). 00174 * = 'Y': Both row and column equilibration, i.e., A has been 00175 * replaced by diag(S) * A * diag(S). 00176 * EQUED is an input argument if FACT = 'F'; otherwise, it is an 00177 * output argument. 00178 * 00179 * S (input or output) DOUBLE PRECISION array, dimension (N) 00180 * The scale factors for A. If EQUED = 'Y', A is multiplied on 00181 * the left and right by diag(S). S is an input argument if FACT = 00182 * 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED 00183 * = 'Y', each element of S must be positive. If S is output, each 00184 * element of S is a power of the radix. If S is input, each element 00185 * of S should be a power of the radix to ensure a reliable solution 00186 * and error estimates. Scaling by powers of the radix does not cause 00187 * rounding errors unless the result underflows or overflows. 00188 * Rounding errors during scaling lead to refining with a matrix that 00189 * is not equivalent to the input matrix, producing error estimates 00190 * that may not be reliable. 00191 * 00192 * B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) 00193 * On entry, the N-by-NRHS right hand side matrix B. 00194 * On exit, 00195 * if EQUED = 'N', B is not modified; 00196 * if EQUED = 'Y', B is overwritten by diag(S)*B; 00197 * 00198 * LDB (input) INTEGER 00199 * The leading dimension of the array B. LDB >= max(1,N). 00200 * 00201 * X (output) COMPLEX*16 array, dimension (LDX,NRHS) 00202 * If INFO = 0, the N-by-NRHS solution matrix X to the original 00203 * system of equations. Note that A and B are modified on exit if 00204 * EQUED .ne. 'N', and the solution to the equilibrated system is 00205 * inv(diag(S))*X. 00206 * 00207 * LDX (input) INTEGER 00208 * The leading dimension of the array X. LDX >= max(1,N). 00209 * 00210 * RCOND (output) DOUBLE PRECISION 00211 * Reciprocal scaled condition number. This is an estimate of the 00212 * reciprocal Skeel condition number of the matrix A after 00213 * equilibration (if done). If this is less than the machine 00214 * precision (in particular, if it is zero), the matrix is singular 00215 * to working precision. Note that the error may still be small even 00216 * if this number is very small and the matrix appears ill- 00217 * conditioned. 00218 * 00219 * RPVGRW (output) DOUBLE PRECISION 00220 * Reciprocal pivot growth. On exit, this contains the reciprocal 00221 * pivot growth factor norm(A)/norm(U). The "max absolute element" 00222 * norm is used. If this is much less than 1, then the stability of 00223 * the LU factorization of the (equilibrated) matrix A could be poor. 00224 * This also means that the solution X, estimated condition numbers, 00225 * and error bounds could be unreliable. If factorization fails with 00226 * 0<INFO<=N, then this contains the reciprocal pivot growth factor 00227 * for the leading INFO columns of A. 00228 * 00229 * BERR (output) DOUBLE PRECISION array, dimension (NRHS) 00230 * Componentwise relative backward error. This is the 00231 * componentwise relative backward error of each solution vector X(j) 00232 * (i.e., the smallest relative change in any element of A or B that 00233 * makes X(j) an exact solution). 00234 * 00235 * N_ERR_BNDS (input) INTEGER 00236 * Number of error bounds to return for each right hand side 00237 * and each type (normwise or componentwise). See ERR_BNDS_NORM and 00238 * ERR_BNDS_COMP below. 00239 * 00240 * ERR_BNDS_NORM (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00241 * For each right-hand side, this array contains information about 00242 * various error bounds and condition numbers corresponding to the 00243 * normwise relative error, which is defined as follows: 00244 * 00245 * Normwise relative error in the ith solution vector: 00246 * max_j (abs(XTRUE(j,i) - X(j,i))) 00247 * ------------------------------ 00248 * max_j abs(X(j,i)) 00249 * 00250 * The array is indexed by the type of error information as described 00251 * below. There currently are up to three pieces of information 00252 * returned. 00253 * 00254 * The first index in ERR_BNDS_NORM(i,:) corresponds to the ith 00255 * right-hand side. 00256 * 00257 * The second index in ERR_BNDS_NORM(:,err) contains the following 00258 * three fields: 00259 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00260 * reciprocal condition number is less than the threshold 00261 * sqrt(n) * dlamch('Epsilon'). 00262 * 00263 * err = 2 "Guaranteed" error bound: The estimated forward error, 00264 * almost certainly within a factor of 10 of the true error 00265 * so long as the next entry is greater than the threshold 00266 * sqrt(n) * dlamch('Epsilon'). This error bound should only 00267 * be trusted if the previous boolean is true. 00268 * 00269 * err = 3 Reciprocal condition number: Estimated normwise 00270 * reciprocal condition number. Compared with the threshold 00271 * sqrt(n) * dlamch('Epsilon') to determine if the error 00272 * estimate is "guaranteed". These reciprocal condition 00273 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00274 * appropriately scaled matrix Z. 00275 * Let Z = S*A, where S scales each row by a power of the 00276 * radix so all absolute row sums of Z are approximately 1. 00277 * 00278 * See Lapack Working Note 165 for further details and extra 00279 * cautions. 00280 * 00281 * ERR_BNDS_COMP (output) DOUBLE PRECISION array, dimension (NRHS, N_ERR_BNDS) 00282 * For each right-hand side, this array contains information about 00283 * various error bounds and condition numbers corresponding to the 00284 * componentwise relative error, which is defined as follows: 00285 * 00286 * Componentwise relative error in the ith solution vector: 00287 * abs(XTRUE(j,i) - X(j,i)) 00288 * max_j ---------------------- 00289 * abs(X(j,i)) 00290 * 00291 * The array is indexed by the right-hand side i (on which the 00292 * componentwise relative error depends), and the type of error 00293 * information as described below. There currently are up to three 00294 * pieces of information returned for each right-hand side. If 00295 * componentwise accuracy is not requested (PARAMS(3) = 0.0), then 00296 * ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most 00297 * the first (:,N_ERR_BNDS) entries are returned. 00298 * 00299 * The first index in ERR_BNDS_COMP(i,:) corresponds to the ith 00300 * right-hand side. 00301 * 00302 * The second index in ERR_BNDS_COMP(:,err) contains the following 00303 * three fields: 00304 * err = 1 "Trust/don't trust" boolean. Trust the answer if the 00305 * reciprocal condition number is less than the threshold 00306 * sqrt(n) * dlamch('Epsilon'). 00307 * 00308 * err = 2 "Guaranteed" error bound: The estimated forward error, 00309 * almost certainly within a factor of 10 of the true error 00310 * so long as the next entry is greater than the threshold 00311 * sqrt(n) * dlamch('Epsilon'). This error bound should only 00312 * be trusted if the previous boolean is true. 00313 * 00314 * err = 3 Reciprocal condition number: Estimated componentwise 00315 * reciprocal condition number. Compared with the threshold 00316 * sqrt(n) * dlamch('Epsilon') to determine if the error 00317 * estimate is "guaranteed". These reciprocal condition 00318 * numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some 00319 * appropriately scaled matrix Z. 00320 * Let Z = S*(A*diag(x)), where x is the solution for the 00321 * current right-hand side and S scales each row of 00322 * A*diag(x) by a power of the radix so all absolute row 00323 * sums of Z are approximately 1. 00324 * 00325 * See Lapack Working Note 165 for further details and extra 00326 * cautions. 00327 * 00328 * NPARAMS (input) INTEGER 00329 * Specifies the number of parameters set in PARAMS. If .LE. 0, the 00330 * PARAMS array is never referenced and default values are used. 00331 * 00332 * PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS 00333 * Specifies algorithm parameters. If an entry is .LT. 0.0, then 00334 * that entry will be filled with default value used for that 00335 * parameter. Only positions up to NPARAMS are accessed; defaults 00336 * are used for higher-numbered parameters. 00337 * 00338 * PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative 00339 * refinement or not. 00340 * Default: 1.0D+0 00341 * = 0.0 : No refinement is performed, and no error bounds are 00342 * computed. 00343 * = 1.0 : Use the extra-precise refinement algorithm. 00344 * (other values are reserved for future use) 00345 * 00346 * PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual 00347 * computations allowed for refinement. 00348 * Default: 10 00349 * Aggressive: Set to 100 to permit convergence using approximate 00350 * factorizations or factorizations other than LU. If 00351 * the factorization uses a technique other than 00352 * Gaussian elimination, the guarantees in 00353 * err_bnds_norm and err_bnds_comp may no longer be 00354 * trustworthy. 00355 * 00356 * PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code 00357 * will attempt to find a solution with small componentwise 00358 * relative error in the double-precision algorithm. Positive 00359 * is true, 0.0 is false. 00360 * Default: 1.0 (attempt componentwise convergence) 00361 * 00362 * WORK (workspace) COMPLEX*16 array, dimension (2*N) 00363 * 00364 * RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) 00365 * 00366 * INFO (output) INTEGER 00367 * = 0: Successful exit. The solution to every right-hand side is 00368 * guaranteed. 00369 * < 0: If INFO = -i, the i-th argument had an illegal value 00370 * > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization 00371 * has been completed, but the factor U is exactly singular, so 00372 * the solution and error bounds could not be computed. RCOND = 0 00373 * is returned. 00374 * = N+J: The solution corresponding to the Jth right-hand side is 00375 * not guaranteed. The solutions corresponding to other right- 00376 * hand sides K with K > J may not be guaranteed as well, but 00377 * only the first such right-hand side is reported. If a small 00378 * componentwise error is not requested (PARAMS(3) = 0.0) then 00379 * the Jth right-hand side is the first with a normwise error 00380 * bound that is not guaranteed (the smallest J such 00381 * that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) 00382 * the Jth right-hand side is the first with either a normwise or 00383 * componentwise error bound that is not guaranteed (the smallest 00384 * J such that either ERR_BNDS_NORM(J,1) = 0.0 or 00385 * ERR_BNDS_COMP(J,1) = 0.0). See the definition of 00386 * ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information 00387 * about all of the right-hand sides check ERR_BNDS_NORM or 00388 * ERR_BNDS_COMP. 00389 * 00390 * ================================================================== 00391 * 00392 * .. Parameters .. 00393 DOUBLE PRECISION ZERO, ONE 00394 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00395 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I 00396 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I 00397 INTEGER CMP_ERR_I, PIV_GROWTH_I 00398 PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2, 00399 $ BERR_I = 3 ) 00400 PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 ) 00401 PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8, 00402 $ PIV_GROWTH_I = 9 ) 00403 * .. 00404 * .. Local Scalars .. 00405 LOGICAL EQUIL, NOFACT, RCEQU 00406 INTEGER INFEQU, J 00407 DOUBLE PRECISION AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM 00408 * .. 00409 * .. External Functions .. 00410 EXTERNAL LSAME, DLAMCH, ZLA_HERPVGRW 00411 LOGICAL LSAME 00412 DOUBLE PRECISION DLAMCH, ZLA_HERPVGRW 00413 * .. 00414 * .. External Subroutines .. 00415 EXTERNAL ZHECON, ZHEEQUB, ZHETRF, ZHETRS, ZLACPY, 00416 $ ZLAQHE, XERBLA, ZLASCL2, ZHERFSX 00417 * .. 00418 * .. Intrinsic Functions .. 00419 INTRINSIC MAX, MIN 00420 * .. 00421 * .. Executable Statements .. 00422 * 00423 INFO = 0 00424 NOFACT = LSAME( FACT, 'N' ) 00425 EQUIL = LSAME( FACT, 'E' ) 00426 SMLNUM = DLAMCH( 'Safe minimum' ) 00427 BIGNUM = ONE / SMLNUM 00428 IF( NOFACT .OR. EQUIL ) THEN 00429 EQUED = 'N' 00430 RCEQU = .FALSE. 00431 ELSE 00432 RCEQU = LSAME( EQUED, 'Y' ) 00433 ENDIF 00434 * 00435 * Default is failure. If an input parameter is wrong or 00436 * factorization fails, make everything look horrible. Only the 00437 * pivot growth is set here, the rest is initialized in ZHERFSX. 00438 * 00439 RPVGRW = ZERO 00440 * 00441 * Test the input parameters. PARAMS is not tested until ZHERFSX. 00442 * 00443 IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT. 00444 $ LSAME( FACT, 'F' ) ) THEN 00445 INFO = -1 00446 ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. 00447 $ .NOT.LSAME( UPLO, 'L' ) ) THEN 00448 INFO = -2 00449 ELSE IF( N.LT.0 ) THEN 00450 INFO = -3 00451 ELSE IF( NRHS.LT.0 ) THEN 00452 INFO = -4 00453 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00454 INFO = -6 00455 ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN 00456 INFO = -8 00457 ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT. 00458 $ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN 00459 INFO = -9 00460 ELSE 00461 IF ( RCEQU ) THEN 00462 SMIN = BIGNUM 00463 SMAX = ZERO 00464 DO 10 J = 1, N 00465 SMIN = MIN( SMIN, S( J ) ) 00466 SMAX = MAX( SMAX, S( J ) ) 00467 10 CONTINUE 00468 IF( SMIN.LE.ZERO ) THEN 00469 INFO = -10 00470 ELSE IF( N.GT.0 ) THEN 00471 SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM ) 00472 ELSE 00473 SCOND = ONE 00474 END IF 00475 END IF 00476 IF( INFO.EQ.0 ) THEN 00477 IF( LDB.LT.MAX( 1, N ) ) THEN 00478 INFO = -12 00479 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN 00480 INFO = -14 00481 END IF 00482 END IF 00483 END IF 00484 * 00485 IF( INFO.NE.0 ) THEN 00486 CALL XERBLA( 'ZHESVXX', -INFO ) 00487 RETURN 00488 END IF 00489 * 00490 IF( EQUIL ) THEN 00491 * 00492 * Compute row and column scalings to equilibrate the matrix A. 00493 * 00494 CALL ZHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU ) 00495 IF( INFEQU.EQ.0 ) THEN 00496 * 00497 * Equilibrate the matrix. 00498 * 00499 CALL ZLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) 00500 RCEQU = LSAME( EQUED, 'Y' ) 00501 END IF 00502 END IF 00503 * 00504 * Scale the right-hand side. 00505 * 00506 IF( RCEQU ) CALL ZLASCL2( N, NRHS, S, B, LDB ) 00507 * 00508 IF( NOFACT .OR. EQUIL ) THEN 00509 * 00510 * Compute the LDL^T or UDU^T factorization of A. 00511 * 00512 CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF ) 00513 CALL ZHETRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO ) 00514 * 00515 * Return if INFO is non-zero. 00516 * 00517 IF( INFO.GT.0 ) THEN 00518 * 00519 * Pivot in column INFO is exactly 0 00520 * Compute the reciprocal pivot growth factor of the 00521 * leading rank-deficient INFO columns of A. 00522 * 00523 IF( N.GT.0 ) 00524 $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, 00525 $ IPIV, RWORK ) 00526 RETURN 00527 END IF 00528 END IF 00529 * 00530 * Compute the reciprocal pivot growth factor RPVGRW. 00531 * 00532 IF( N.GT.0 ) 00533 $ RPVGRW = ZLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV, 00534 $ RWORK ) 00535 * 00536 * Compute the solution matrix X. 00537 * 00538 CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) 00539 CALL ZHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO ) 00540 * 00541 * Use iterative refinement to improve the computed solution and 00542 * compute error bounds and backward error estimates for it. 00543 * 00544 CALL ZHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV, 00545 $ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, 00546 $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO ) 00547 * 00548 * Scale solutions. 00549 * 00550 IF ( RCEQU ) THEN 00551 CALL ZLASCL2 ( N, NRHS, S, X, LDX ) 00552 END IF 00553 * 00554 RETURN 00555 * 00556 * End of ZHESVXX 00557 * 00558 END
1.7.2