00001 /* zposvx.f -- translated by f2c (version 20061008). 00002 You must link the resulting object file with libf2c: 00003 on Microsoft Windows system, link with libf2c.lib; 00004 on Linux or Unix systems, link with .../path/to/libf2c.a -lm 00005 or, if you install libf2c.a in a standard place, with -lf2c -lm 00006 -- in that order, at the end of the command line, as in 00007 cc *.o -lf2c -lm 00008 Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., 00009 00010 http://www.netlib.org/f2c/libf2c.zip 00011 */ 00012 00013 #include "f2c.h" 00014 #include "blaswrap.h" 00015 00016 /* Subroutine */ int zposvx_(char *fact, char *uplo, integer *n, integer * 00017 nrhs, doublecomplex *a, integer *lda, doublecomplex *af, integer * 00018 ldaf, char *equed, doublereal *s, doublecomplex *b, integer *ldb, 00019 doublecomplex *x, integer *ldx, doublereal *rcond, doublereal *ferr, 00020 doublereal *berr, doublecomplex *work, doublereal *rwork, integer * 00021 info) 00022 { 00023 /* System generated locals */ 00024 integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, 00025 x_offset, i__1, i__2, i__3, i__4, i__5; 00026 doublereal d__1, d__2; 00027 doublecomplex z__1; 00028 00029 /* Local variables */ 00030 integer i__, j; 00031 doublereal amax, smin, smax; 00032 extern logical lsame_(char *, char *); 00033 doublereal scond, anorm; 00034 logical equil, rcequ; 00035 extern doublereal dlamch_(char *); 00036 logical nofact; 00037 extern /* Subroutine */ int xerbla_(char *, integer *); 00038 doublereal bignum; 00039 extern doublereal zlanhe_(char *, char *, integer *, doublecomplex *, 00040 integer *, doublereal *); 00041 extern /* Subroutine */ int zlaqhe_(char *, integer *, doublecomplex *, 00042 integer *, doublereal *, doublereal *, doublereal *, char *); 00043 integer infequ; 00044 extern /* Subroutine */ int zlacpy_(char *, integer *, integer *, 00045 doublecomplex *, integer *, doublecomplex *, integer *), 00046 zpocon_(char *, integer *, doublecomplex *, integer *, doublereal 00047 *, doublereal *, doublecomplex *, doublereal *, integer *) 00048 ; 00049 doublereal smlnum; 00050 extern /* Subroutine */ int zpoequ_(integer *, doublecomplex *, integer *, 00051 doublereal *, doublereal *, doublereal *, integer *), zporfs_( 00052 char *, integer *, integer *, doublecomplex *, integer *, 00053 doublecomplex *, integer *, doublecomplex *, integer *, 00054 doublecomplex *, integer *, doublereal *, doublereal *, 00055 doublecomplex *, doublereal *, integer *), zpotrf_(char *, 00056 integer *, doublecomplex *, integer *, integer *), 00057 zpotrs_(char *, integer *, integer *, doublecomplex *, integer *, 00058 doublecomplex *, integer *, integer *); 00059 00060 00061 /* -- LAPACK driver routine (version 3.2) -- */ 00062 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00063 /* November 2006 */ 00064 00065 /* .. Scalar Arguments .. */ 00066 /* .. */ 00067 /* .. Array Arguments .. */ 00068 /* .. */ 00069 00070 /* Purpose */ 00071 /* ======= */ 00072 00073 /* ZPOSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to */ 00074 /* compute the solution to a complex system of linear equations */ 00075 /* A * X = B, */ 00076 /* where A is an N-by-N Hermitian positive definite matrix and X and B */ 00077 /* are N-by-NRHS matrices. */ 00078 00079 /* Error bounds on the solution and a condition estimate are also */ 00080 /* provided. */ 00081 00082 /* Description */ 00083 /* =========== */ 00084 00085 /* The following steps are performed: */ 00086 00087 /* 1. If FACT = 'E', real scaling factors are computed to equilibrate */ 00088 /* the system: */ 00089 /* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */ 00090 /* Whether or not the system will be equilibrated depends on the */ 00091 /* scaling of the matrix A, but if equilibration is used, A is */ 00092 /* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */ 00093 00094 /* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ 00095 /* factor the matrix A (after equilibration if FACT = 'E') as */ 00096 /* A = U**H* U, if UPLO = 'U', or */ 00097 /* A = L * L**H, if UPLO = 'L', */ 00098 /* where U is an upper triangular matrix and L is a lower triangular */ 00099 /* matrix. */ 00100 00101 /* 3. If the leading i-by-i principal minor is not positive definite, */ 00102 /* then the routine returns with INFO = i. Otherwise, the factored */ 00103 /* form of A is used to estimate the condition number of the matrix */ 00104 /* A. If the reciprocal of the condition number is less than machine */ 00105 /* precision, INFO = N+1 is returned as a warning, but the routine */ 00106 /* still goes on to solve for X and compute error bounds as */ 00107 /* described below. */ 00108 00109 /* 4. The system of equations is solved for X using the factored form */ 00110 /* of A. */ 00111 00112 /* 5. Iterative refinement is applied to improve the computed solution */ 00113 /* matrix and calculate error bounds and backward error estimates */ 00114 /* for it. */ 00115 00116 /* 6. If equilibration was used, the matrix X is premultiplied by */ 00117 /* diag(S) so that it solves the original system before */ 00118 /* equilibration. */ 00119 00120 /* Arguments */ 00121 /* ========= */ 00122 00123 /* FACT (input) CHARACTER*1 */ 00124 /* Specifies whether or not the factored form of the matrix A is */ 00125 /* supplied on entry, and if not, whether the matrix A should be */ 00126 /* equilibrated before it is factored. */ 00127 /* = 'F': On entry, AF contains the factored form of A. */ 00128 /* If EQUED = 'Y', the matrix A has been equilibrated */ 00129 /* with scaling factors given by S. A and AF will not */ 00130 /* be modified. */ 00131 /* = 'N': The matrix A will be copied to AF and factored. */ 00132 /* = 'E': The matrix A will be equilibrated if necessary, then */ 00133 /* copied to AF and factored. */ 00134 00135 /* UPLO (input) CHARACTER*1 */ 00136 /* = 'U': Upper triangle of A is stored; */ 00137 /* = 'L': Lower triangle of A is stored. */ 00138 00139 /* N (input) INTEGER */ 00140 /* The number of linear equations, i.e., the order of the */ 00141 /* matrix A. N >= 0. */ 00142 00143 /* NRHS (input) INTEGER */ 00144 /* The number of right hand sides, i.e., the number of columns */ 00145 /* of the matrices B and X. NRHS >= 0. */ 00146 00147 /* A (input/output) COMPLEX*16 array, dimension (LDA,N) */ 00148 /* On entry, the Hermitian matrix A, except if FACT = 'F' and */ 00149 /* EQUED = 'Y', then A must contain the equilibrated matrix */ 00150 /* diag(S)*A*diag(S). If UPLO = 'U', the leading */ 00151 /* N-by-N upper triangular part of A contains the upper */ 00152 /* triangular part of the matrix A, and the strictly lower */ 00153 /* triangular part of A is not referenced. If UPLO = 'L', the */ 00154 /* leading N-by-N lower triangular part of A contains the lower */ 00155 /* triangular part of the matrix A, and the strictly upper */ 00156 /* triangular part of A is not referenced. A is not modified if */ 00157 /* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ 00158 00159 /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ 00160 /* diag(S)*A*diag(S). */ 00161 00162 /* LDA (input) INTEGER */ 00163 /* The leading dimension of the array A. LDA >= max(1,N). */ 00164 00165 /* AF (input or output) COMPLEX*16 array, dimension (LDAF,N) */ 00166 /* If FACT = 'F', then AF is an input argument and on entry */ 00167 /* contains the triangular factor U or L from the Cholesky */ 00168 /* factorization A = U**H*U or A = L*L**H, in the same storage */ 00169 /* format as A. If EQUED .ne. 'N', then AF is the factored form */ 00170 /* of the equilibrated matrix diag(S)*A*diag(S). */ 00171 00172 /* If FACT = 'N', then AF is an output argument and on exit */ 00173 /* returns the triangular factor U or L from the Cholesky */ 00174 /* factorization A = U**H*U or A = L*L**H of the original */ 00175 /* matrix A. */ 00176 00177 /* If FACT = 'E', then AF is an output argument and on exit */ 00178 /* returns the triangular factor U or L from the Cholesky */ 00179 /* factorization A = U**H*U or A = L*L**H of the equilibrated */ 00180 /* matrix A (see the description of A for the form of the */ 00181 /* equilibrated matrix). */ 00182 00183 /* LDAF (input) INTEGER */ 00184 /* The leading dimension of the array AF. LDAF >= max(1,N). */ 00185 00186 /* EQUED (input or output) CHARACTER*1 */ 00187 /* Specifies the form of equilibration that was done. */ 00188 /* = 'N': No equilibration (always true if FACT = 'N'). */ 00189 /* = 'Y': Equilibration was done, i.e., A has been replaced by */ 00190 /* diag(S) * A * diag(S). */ 00191 /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ 00192 /* output argument. */ 00193 00194 /* S (input or output) DOUBLE PRECISION array, dimension (N) */ 00195 /* The scale factors for A; not accessed if EQUED = 'N'. S is */ 00196 /* an input argument if FACT = 'F'; otherwise, S is an output */ 00197 /* argument. If FACT = 'F' and EQUED = 'Y', each element of S */ 00198 /* must be positive. */ 00199 00200 /* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS) */ 00201 /* On entry, the N-by-NRHS righthand side matrix B. */ 00202 /* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */ 00203 /* B is overwritten by diag(S) * B. */ 00204 00205 /* LDB (input) INTEGER */ 00206 /* The leading dimension of the array B. LDB >= max(1,N). */ 00207 00208 /* X (output) COMPLEX*16 array, dimension (LDX,NRHS) */ 00209 /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ 00210 /* the original system of equations. Note that if EQUED = 'Y', */ 00211 /* A and B are modified on exit, and the solution to the */ 00212 /* equilibrated system is inv(diag(S))*X. */ 00213 00214 /* LDX (input) INTEGER */ 00215 /* The leading dimension of the array X. LDX >= max(1,N). */ 00216 00217 /* RCOND (output) DOUBLE PRECISION */ 00218 /* The estimate of the reciprocal condition number of the matrix */ 00219 /* A after equilibration (if done). If RCOND is less than the */ 00220 /* machine precision (in particular, if RCOND = 0), the matrix */ 00221 /* is singular to working precision. This condition is */ 00222 /* indicated by a return code of INFO > 0. */ 00223 00224 /* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */ 00225 /* The estimated forward error bound for each solution vector */ 00226 /* X(j) (the j-th column of the solution matrix X). */ 00227 /* If XTRUE is the true solution corresponding to X(j), FERR(j) */ 00228 /* is an estimated upper bound for the magnitude of the largest */ 00229 /* element in (X(j) - XTRUE) divided by the magnitude of the */ 00230 /* largest element in X(j). The estimate is as reliable as */ 00231 /* the estimate for RCOND, and is almost always a slight */ 00232 /* overestimate of the true error. */ 00233 00234 /* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */ 00235 /* The componentwise relative backward error of each solution */ 00236 /* vector X(j) (i.e., the smallest relative change in */ 00237 /* any element of A or B that makes X(j) an exact solution). */ 00238 00239 /* WORK (workspace) COMPLEX*16 array, dimension (2*N) */ 00240 00241 /* RWORK (workspace) DOUBLE PRECISION array, dimension (N) */ 00242 00243 /* INFO (output) INTEGER */ 00244 /* = 0: successful exit */ 00245 /* < 0: if INFO = -i, the i-th argument had an illegal value */ 00246 /* > 0: if INFO = i, and i is */ 00247 /* <= N: the leading minor of order i of A is */ 00248 /* not positive definite, so the factorization */ 00249 /* could not be completed, and the solution has not */ 00250 /* been computed. RCOND = 0 is returned. */ 00251 /* = N+1: U is nonsingular, but RCOND is less than machine */ 00252 /* precision, meaning that the matrix is singular */ 00253 /* to working precision. Nevertheless, the */ 00254 /* solution and error bounds are computed because */ 00255 /* there are a number of situations where the */ 00256 /* computed solution can be more accurate than the */ 00257 /* value of RCOND would suggest. */ 00258 00259 /* ===================================================================== */ 00260 00261 /* .. Parameters .. */ 00262 /* .. */ 00263 /* .. Local Scalars .. */ 00264 /* .. */ 00265 /* .. External Functions .. */ 00266 /* .. */ 00267 /* .. External Subroutines .. */ 00268 /* .. */ 00269 /* .. Intrinsic Functions .. */ 00270 /* .. */ 00271 /* .. Executable Statements .. */ 00272 00273 /* Parameter adjustments */ 00274 a_dim1 = *lda; 00275 a_offset = 1 + a_dim1; 00276 a -= a_offset; 00277 af_dim1 = *ldaf; 00278 af_offset = 1 + af_dim1; 00279 af -= af_offset; 00280 --s; 00281 b_dim1 = *ldb; 00282 b_offset = 1 + b_dim1; 00283 b -= b_offset; 00284 x_dim1 = *ldx; 00285 x_offset = 1 + x_dim1; 00286 x -= x_offset; 00287 --ferr; 00288 --berr; 00289 --work; 00290 --rwork; 00291 00292 /* Function Body */ 00293 *info = 0; 00294 nofact = lsame_(fact, "N"); 00295 equil = lsame_(fact, "E"); 00296 if (nofact || equil) { 00297 *(unsigned char *)equed = 'N'; 00298 rcequ = FALSE_; 00299 } else { 00300 rcequ = lsame_(equed, "Y"); 00301 smlnum = dlamch_("Safe minimum"); 00302 bignum = 1. / smlnum; 00303 } 00304 00305 /* Test the input parameters. */ 00306 00307 if (! nofact && ! equil && ! lsame_(fact, "F")) { 00308 *info = -1; 00309 } else if (! lsame_(uplo, "U") && ! lsame_(uplo, 00310 "L")) { 00311 *info = -2; 00312 } else if (*n < 0) { 00313 *info = -3; 00314 } else if (*nrhs < 0) { 00315 *info = -4; 00316 } else if (*lda < max(1,*n)) { 00317 *info = -6; 00318 } else if (*ldaf < max(1,*n)) { 00319 *info = -8; 00320 } else if (lsame_(fact, "F") && ! (rcequ || lsame_( 00321 equed, "N"))) { 00322 *info = -9; 00323 } else { 00324 if (rcequ) { 00325 smin = bignum; 00326 smax = 0.; 00327 i__1 = *n; 00328 for (j = 1; j <= i__1; ++j) { 00329 /* Computing MIN */ 00330 d__1 = smin, d__2 = s[j]; 00331 smin = min(d__1,d__2); 00332 /* Computing MAX */ 00333 d__1 = smax, d__2 = s[j]; 00334 smax = max(d__1,d__2); 00335 /* L10: */ 00336 } 00337 if (smin <= 0.) { 00338 *info = -10; 00339 } else if (*n > 0) { 00340 scond = max(smin,smlnum) / min(smax,bignum); 00341 } else { 00342 scond = 1.; 00343 } 00344 } 00345 if (*info == 0) { 00346 if (*ldb < max(1,*n)) { 00347 *info = -12; 00348 } else if (*ldx < max(1,*n)) { 00349 *info = -14; 00350 } 00351 } 00352 } 00353 00354 if (*info != 0) { 00355 i__1 = -(*info); 00356 xerbla_("ZPOSVX", &i__1); 00357 return 0; 00358 } 00359 00360 if (equil) { 00361 00362 /* Compute row and column scalings to equilibrate the matrix A. */ 00363 00364 zpoequ_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ); 00365 if (infequ == 0) { 00366 00367 /* Equilibrate the matrix. */ 00368 00369 zlaqhe_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); 00370 rcequ = lsame_(equed, "Y"); 00371 } 00372 } 00373 00374 /* Scale the right hand side. */ 00375 00376 if (rcequ) { 00377 i__1 = *nrhs; 00378 for (j = 1; j <= i__1; ++j) { 00379 i__2 = *n; 00380 for (i__ = 1; i__ <= i__2; ++i__) { 00381 i__3 = i__ + j * b_dim1; 00382 i__4 = i__; 00383 i__5 = i__ + j * b_dim1; 00384 z__1.r = s[i__4] * b[i__5].r, z__1.i = s[i__4] * b[i__5].i; 00385 b[i__3].r = z__1.r, b[i__3].i = z__1.i; 00386 /* L20: */ 00387 } 00388 /* L30: */ 00389 } 00390 } 00391 00392 if (nofact || equil) { 00393 00394 /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ 00395 00396 zlacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); 00397 zpotrf_(uplo, n, &af[af_offset], ldaf, info); 00398 00399 /* Return if INFO is non-zero. */ 00400 00401 if (*info > 0) { 00402 *rcond = 0.; 00403 return 0; 00404 } 00405 } 00406 00407 /* Compute the norm of the matrix A. */ 00408 00409 anorm = zlanhe_("1", uplo, n, &a[a_offset], lda, &rwork[1]); 00410 00411 /* Compute the reciprocal of the condition number of A. */ 00412 00413 zpocon_(uplo, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &rwork[1], 00414 info); 00415 00416 /* Compute the solution matrix X. */ 00417 00418 zlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); 00419 zpotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info); 00420 00421 /* Use iterative refinement to improve the computed solution and */ 00422 /* compute error bounds and backward error estimates for it. */ 00423 00424 zporfs_(uplo, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &b[ 00425 b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1], & 00426 rwork[1], info); 00427 00428 /* Transform the solution matrix X to a solution of the original */ 00429 /* system. */ 00430 00431 if (rcequ) { 00432 i__1 = *nrhs; 00433 for (j = 1; j <= i__1; ++j) { 00434 i__2 = *n; 00435 for (i__ = 1; i__ <= i__2; ++i__) { 00436 i__3 = i__ + j * x_dim1; 00437 i__4 = i__; 00438 i__5 = i__ + j * x_dim1; 00439 z__1.r = s[i__4] * x[i__5].r, z__1.i = s[i__4] * x[i__5].i; 00440 x[i__3].r = z__1.r, x[i__3].i = z__1.i; 00441 /* L40: */ 00442 } 00443 /* L50: */ 00444 } 00445 i__1 = *nrhs; 00446 for (j = 1; j <= i__1; ++j) { 00447 ferr[j] /= scond; 00448 /* L60: */ 00449 } 00450 } 00451 00452 /* Set INFO = N+1 if the matrix is singular to working precision. */ 00453 00454 if (*rcond < dlamch_("Epsilon")) { 00455 *info = *n + 1; 00456 } 00457 00458 return 0; 00459 00460 /* End of ZPOSVX */ 00461 00462 } /* zposvx_ */