00001 /* sposvx.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 sposvx_(char *fact, char *uplo, integer *n, integer * 00017 nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed, 00018 real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond, 00019 real *ferr, real *berr, real *work, integer *iwork, integer *info) 00020 { 00021 /* System generated locals */ 00022 integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, 00023 x_offset, i__1, i__2; 00024 real r__1, r__2; 00025 00026 /* Local variables */ 00027 integer i__, j; 00028 real amax, smin, smax; 00029 extern logical lsame_(char *, char *); 00030 real scond, anorm; 00031 logical equil, rcequ; 00032 extern doublereal slamch_(char *); 00033 logical nofact; 00034 extern /* Subroutine */ int xerbla_(char *, integer *); 00035 real bignum; 00036 integer infequ; 00037 extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, 00038 integer *, real *, integer *), spocon_(char *, integer *, 00039 real *, integer *, real *, real *, real *, integer *, integer *); 00040 extern doublereal slansy_(char *, char *, integer *, real *, integer *, 00041 real *); 00042 real smlnum; 00043 extern /* Subroutine */ int slaqsy_(char *, integer *, real *, integer *, 00044 real *, real *, real *, char *), spoequ_(integer * 00045 , real *, integer *, real *, real *, real *, integer *), sporfs_( 00046 char *, integer *, integer *, real *, integer *, real *, integer * 00047 , real *, integer *, real *, integer *, real *, real *, real *, 00048 integer *, integer *), spotrf_(char *, integer *, real *, 00049 integer *, integer *), spotrs_(char *, integer *, integer 00050 *, real *, integer *, real *, integer *, integer *); 00051 00052 00053 /* -- LAPACK driver routine (version 3.2) -- */ 00054 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00055 /* November 2006 */ 00056 00057 /* .. Scalar Arguments .. */ 00058 /* .. */ 00059 /* .. Array Arguments .. */ 00060 /* .. */ 00061 00062 /* Purpose */ 00063 /* ======= */ 00064 00065 /* SPOSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to */ 00066 /* compute the solution to a real system of linear equations */ 00067 /* A * X = B, */ 00068 /* where A is an N-by-N symmetric positive definite matrix and X and B */ 00069 /* are N-by-NRHS matrices. */ 00070 00071 /* Error bounds on the solution and a condition estimate are also */ 00072 /* provided. */ 00073 00074 /* Description */ 00075 /* =========== */ 00076 00077 /* The following steps are performed: */ 00078 00079 /* 1. If FACT = 'E', real scaling factors are computed to equilibrate */ 00080 /* the system: */ 00081 /* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */ 00082 /* Whether or not the system will be equilibrated depends on the */ 00083 /* scaling of the matrix A, but if equilibration is used, A is */ 00084 /* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */ 00085 00086 /* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ 00087 /* factor the matrix A (after equilibration if FACT = 'E') as */ 00088 /* A = U**T* U, if UPLO = 'U', or */ 00089 /* A = L * L**T, if UPLO = 'L', */ 00090 /* where U is an upper triangular matrix and L is a lower triangular */ 00091 /* matrix. */ 00092 00093 /* 3. If the leading i-by-i principal minor is not positive definite, */ 00094 /* then the routine returns with INFO = i. Otherwise, the factored */ 00095 /* form of A is used to estimate the condition number of the matrix */ 00096 /* A. If the reciprocal of the condition number is less than machine */ 00097 /* precision, INFO = N+1 is returned as a warning, but the routine */ 00098 /* still goes on to solve for X and compute error bounds as */ 00099 /* described below. */ 00100 00101 /* 4. The system of equations is solved for X using the factored form */ 00102 /* of A. */ 00103 00104 /* 5. Iterative refinement is applied to improve the computed solution */ 00105 /* matrix and calculate error bounds and backward error estimates */ 00106 /* for it. */ 00107 00108 /* 6. If equilibration was used, the matrix X is premultiplied by */ 00109 /* diag(S) so that it solves the original system before */ 00110 /* equilibration. */ 00111 00112 /* Arguments */ 00113 /* ========= */ 00114 00115 /* FACT (input) CHARACTER*1 */ 00116 /* Specifies whether or not the factored form of the matrix A is */ 00117 /* supplied on entry, and if not, whether the matrix A should be */ 00118 /* equilibrated before it is factored. */ 00119 /* = 'F': On entry, AF contains the factored form of A. */ 00120 /* If EQUED = 'Y', the matrix A has been equilibrated */ 00121 /* with scaling factors given by S. A and AF will not */ 00122 /* be modified. */ 00123 /* = 'N': The matrix A will be copied to AF and factored. */ 00124 /* = 'E': The matrix A will be equilibrated if necessary, then */ 00125 /* copied to AF and factored. */ 00126 00127 /* UPLO (input) CHARACTER*1 */ 00128 /* = 'U': Upper triangle of A is stored; */ 00129 /* = 'L': Lower triangle of A is stored. */ 00130 00131 /* N (input) INTEGER */ 00132 /* The number of linear equations, i.e., the order of the */ 00133 /* matrix A. N >= 0. */ 00134 00135 /* NRHS (input) INTEGER */ 00136 /* The number of right hand sides, i.e., the number of columns */ 00137 /* of the matrices B and X. NRHS >= 0. */ 00138 00139 /* A (input/output) REAL array, dimension (LDA,N) */ 00140 /* On entry, the symmetric matrix A, except if FACT = 'F' and */ 00141 /* EQUED = 'Y', then A must contain the equilibrated matrix */ 00142 /* diag(S)*A*diag(S). If UPLO = 'U', the leading */ 00143 /* N-by-N upper triangular part of A contains the upper */ 00144 /* triangular part of the matrix A, and the strictly lower */ 00145 /* triangular part of A is not referenced. If UPLO = 'L', the */ 00146 /* leading N-by-N lower triangular part of A contains the lower */ 00147 /* triangular part of the matrix A, and the strictly upper */ 00148 /* triangular part of A is not referenced. A is not modified if */ 00149 /* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ 00150 00151 /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ 00152 /* diag(S)*A*diag(S). */ 00153 00154 /* LDA (input) INTEGER */ 00155 /* The leading dimension of the array A. LDA >= max(1,N). */ 00156 00157 /* AF (input or output) REAL array, dimension (LDAF,N) */ 00158 /* If FACT = 'F', then AF is an input argument and on entry */ 00159 /* contains the triangular factor U or L from the Cholesky */ 00160 /* factorization A = U**T*U or A = L*L**T, in the same storage */ 00161 /* format as A. If EQUED .ne. 'N', then AF is the factored form */ 00162 /* of the equilibrated matrix diag(S)*A*diag(S). */ 00163 00164 /* If FACT = 'N', then AF is an output argument and on exit */ 00165 /* returns the triangular factor U or L from the Cholesky */ 00166 /* factorization A = U**T*U or A = L*L**T of the original */ 00167 /* matrix A. */ 00168 00169 /* If FACT = 'E', then AF is an output argument and on exit */ 00170 /* returns the triangular factor U or L from the Cholesky */ 00171 /* factorization A = U**T*U or A = L*L**T of the equilibrated */ 00172 /* matrix A (see the description of A for the form of the */ 00173 /* equilibrated matrix). */ 00174 00175 /* LDAF (input) INTEGER */ 00176 /* The leading dimension of the array AF. LDAF >= max(1,N). */ 00177 00178 /* EQUED (input or output) CHARACTER*1 */ 00179 /* Specifies the form of equilibration that was done. */ 00180 /* = 'N': No equilibration (always true if FACT = 'N'). */ 00181 /* = 'Y': Equilibration was done, i.e., A has been replaced by */ 00182 /* diag(S) * A * diag(S). */ 00183 /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ 00184 /* output argument. */ 00185 00186 /* S (input or output) REAL array, dimension (N) */ 00187 /* The scale factors for A; not accessed if EQUED = 'N'. S is */ 00188 /* an input argument if FACT = 'F'; otherwise, S is an output */ 00189 /* argument. If FACT = 'F' and EQUED = 'Y', each element of S */ 00190 /* must be positive. */ 00191 00192 /* B (input/output) REAL array, dimension (LDB,NRHS) */ 00193 /* On entry, the N-by-NRHS right hand side matrix B. */ 00194 /* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */ 00195 /* B is overwritten by diag(S) * B. */ 00196 00197 /* LDB (input) INTEGER */ 00198 /* The leading dimension of the array B. LDB >= max(1,N). */ 00199 00200 /* X (output) REAL array, dimension (LDX,NRHS) */ 00201 /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ 00202 /* the original system of equations. Note that if EQUED = 'Y', */ 00203 /* A and B are modified on exit, and the solution to the */ 00204 /* equilibrated system is inv(diag(S))*X. */ 00205 00206 /* LDX (input) INTEGER */ 00207 /* The leading dimension of the array X. LDX >= max(1,N). */ 00208 00209 /* RCOND (output) REAL */ 00210 /* The estimate of the reciprocal condition number of the matrix */ 00211 /* A after equilibration (if done). If RCOND is less than the */ 00212 /* machine precision (in particular, if RCOND = 0), the matrix */ 00213 /* is singular to working precision. This condition is */ 00214 /* indicated by a return code of INFO > 0. */ 00215 00216 /* FERR (output) REAL array, dimension (NRHS) */ 00217 /* The estimated forward error bound for each solution vector */ 00218 /* X(j) (the j-th column of the solution matrix X). */ 00219 /* If XTRUE is the true solution corresponding to X(j), FERR(j) */ 00220 /* is an estimated upper bound for the magnitude of the largest */ 00221 /* element in (X(j) - XTRUE) divided by the magnitude of the */ 00222 /* largest element in X(j). The estimate is as reliable as */ 00223 /* the estimate for RCOND, and is almost always a slight */ 00224 /* overestimate of the true error. */ 00225 00226 /* BERR (output) REAL array, dimension (NRHS) */ 00227 /* The componentwise relative backward error of each solution */ 00228 /* vector X(j) (i.e., the smallest relative change in */ 00229 /* any element of A or B that makes X(j) an exact solution). */ 00230 00231 /* WORK (workspace) REAL array, dimension (3*N) */ 00232 00233 /* IWORK (workspace) INTEGER array, dimension (N) */ 00234 00235 /* INFO (output) INTEGER */ 00236 /* = 0: successful exit */ 00237 /* < 0: if INFO = -i, the i-th argument had an illegal value */ 00238 /* > 0: if INFO = i, and i is */ 00239 /* <= N: the leading minor of order i of A is */ 00240 /* not positive definite, so the factorization */ 00241 /* could not be completed, and the solution has not */ 00242 /* been computed. RCOND = 0 is returned. */ 00243 /* = N+1: U is nonsingular, but RCOND is less than machine */ 00244 /* precision, meaning that the matrix is singular */ 00245 /* to working precision. Nevertheless, the */ 00246 /* solution and error bounds are computed because */ 00247 /* there are a number of situations where the */ 00248 /* computed solution can be more accurate than the */ 00249 /* value of RCOND would suggest. */ 00250 00251 /* ===================================================================== */ 00252 00253 /* .. Parameters .. */ 00254 /* .. */ 00255 /* .. Local Scalars .. */ 00256 /* .. */ 00257 /* .. External Functions .. */ 00258 /* .. */ 00259 /* .. External Subroutines .. */ 00260 /* .. */ 00261 /* .. Intrinsic Functions .. */ 00262 /* .. */ 00263 /* .. Executable Statements .. */ 00264 00265 /* Parameter adjustments */ 00266 a_dim1 = *lda; 00267 a_offset = 1 + a_dim1; 00268 a -= a_offset; 00269 af_dim1 = *ldaf; 00270 af_offset = 1 + af_dim1; 00271 af -= af_offset; 00272 --s; 00273 b_dim1 = *ldb; 00274 b_offset = 1 + b_dim1; 00275 b -= b_offset; 00276 x_dim1 = *ldx; 00277 x_offset = 1 + x_dim1; 00278 x -= x_offset; 00279 --ferr; 00280 --berr; 00281 --work; 00282 --iwork; 00283 00284 /* Function Body */ 00285 *info = 0; 00286 nofact = lsame_(fact, "N"); 00287 equil = lsame_(fact, "E"); 00288 if (nofact || equil) { 00289 *(unsigned char *)equed = 'N'; 00290 rcequ = FALSE_; 00291 } else { 00292 rcequ = lsame_(equed, "Y"); 00293 smlnum = slamch_("Safe minimum"); 00294 bignum = 1.f / smlnum; 00295 } 00296 00297 /* Test the input parameters. */ 00298 00299 if (! nofact && ! equil && ! lsame_(fact, "F")) { 00300 *info = -1; 00301 } else if (! lsame_(uplo, "U") && ! lsame_(uplo, 00302 "L")) { 00303 *info = -2; 00304 } else if (*n < 0) { 00305 *info = -3; 00306 } else if (*nrhs < 0) { 00307 *info = -4; 00308 } else if (*lda < max(1,*n)) { 00309 *info = -6; 00310 } else if (*ldaf < max(1,*n)) { 00311 *info = -8; 00312 } else if (lsame_(fact, "F") && ! (rcequ || lsame_( 00313 equed, "N"))) { 00314 *info = -9; 00315 } else { 00316 if (rcequ) { 00317 smin = bignum; 00318 smax = 0.f; 00319 i__1 = *n; 00320 for (j = 1; j <= i__1; ++j) { 00321 /* Computing MIN */ 00322 r__1 = smin, r__2 = s[j]; 00323 smin = dmin(r__1,r__2); 00324 /* Computing MAX */ 00325 r__1 = smax, r__2 = s[j]; 00326 smax = dmax(r__1,r__2); 00327 /* L10: */ 00328 } 00329 if (smin <= 0.f) { 00330 *info = -10; 00331 } else if (*n > 0) { 00332 scond = dmax(smin,smlnum) / dmin(smax,bignum); 00333 } else { 00334 scond = 1.f; 00335 } 00336 } 00337 if (*info == 0) { 00338 if (*ldb < max(1,*n)) { 00339 *info = -12; 00340 } else if (*ldx < max(1,*n)) { 00341 *info = -14; 00342 } 00343 } 00344 } 00345 00346 if (*info != 0) { 00347 i__1 = -(*info); 00348 xerbla_("SPOSVX", &i__1); 00349 return 0; 00350 } 00351 00352 if (equil) { 00353 00354 /* Compute row and column scalings to equilibrate the matrix A. */ 00355 00356 spoequ_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ); 00357 if (infequ == 0) { 00358 00359 /* Equilibrate the matrix. */ 00360 00361 slaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); 00362 rcequ = lsame_(equed, "Y"); 00363 } 00364 } 00365 00366 /* Scale the right hand side. */ 00367 00368 if (rcequ) { 00369 i__1 = *nrhs; 00370 for (j = 1; j <= i__1; ++j) { 00371 i__2 = *n; 00372 for (i__ = 1; i__ <= i__2; ++i__) { 00373 b[i__ + j * b_dim1] = s[i__] * b[i__ + j * b_dim1]; 00374 /* L20: */ 00375 } 00376 /* L30: */ 00377 } 00378 } 00379 00380 if (nofact || equil) { 00381 00382 /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ 00383 00384 slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); 00385 spotrf_(uplo, n, &af[af_offset], ldaf, info); 00386 00387 /* Return if INFO is non-zero. */ 00388 00389 if (*info > 0) { 00390 *rcond = 0.f; 00391 return 0; 00392 } 00393 } 00394 00395 /* Compute the norm of the matrix A. */ 00396 00397 anorm = slansy_("1", uplo, n, &a[a_offset], lda, &work[1]); 00398 00399 /* Compute the reciprocal of the condition number of A. */ 00400 00401 spocon_(uplo, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &iwork[1], 00402 info); 00403 00404 /* Compute the solution matrix X. */ 00405 00406 slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); 00407 spotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info); 00408 00409 /* Use iterative refinement to improve the computed solution and */ 00410 /* compute error bounds and backward error estimates for it. */ 00411 00412 sporfs_(uplo, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &b[ 00413 b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1], & 00414 iwork[1], info); 00415 00416 /* Transform the solution matrix X to a solution of the original */ 00417 /* system. */ 00418 00419 if (rcequ) { 00420 i__1 = *nrhs; 00421 for (j = 1; j <= i__1; ++j) { 00422 i__2 = *n; 00423 for (i__ = 1; i__ <= i__2; ++i__) { 00424 x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1]; 00425 /* L40: */ 00426 } 00427 /* L50: */ 00428 } 00429 i__1 = *nrhs; 00430 for (j = 1; j <= i__1; ++j) { 00431 ferr[j] /= scond; 00432 /* L60: */ 00433 } 00434 } 00435 00436 /* Set INFO = N+1 if the matrix is singular to working precision. */ 00437 00438 if (*rcond < slamch_("Epsilon")) { 00439 *info = *n + 1; 00440 } 00441 00442 return 0; 00443 00444 /* End of SPOSVX */ 00445 00446 } /* sposvx_ */