00001 /* sppsvx.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 /* Table of constant values */ 00017 00018 static integer c__1 = 1; 00019 00020 /* Subroutine */ int sppsvx_(char *fact, char *uplo, integer *n, integer * 00021 nrhs, real *ap, real *afp, char *equed, real *s, real *b, integer * 00022 ldb, real *x, integer *ldx, real *rcond, real *ferr, real *berr, real 00023 *work, integer *iwork, integer *info) 00024 { 00025 /* System generated locals */ 00026 integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2; 00027 real r__1, r__2; 00028 00029 /* Local variables */ 00030 integer i__, j; 00031 real amax, smin, smax; 00032 extern logical lsame_(char *, char *); 00033 real scond, anorm; 00034 logical equil, rcequ; 00035 extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 00036 integer *); 00037 extern doublereal slamch_(char *); 00038 logical nofact; 00039 extern /* Subroutine */ int xerbla_(char *, integer *); 00040 real bignum; 00041 integer infequ; 00042 extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, 00043 integer *, real *, integer *); 00044 extern doublereal slansp_(char *, char *, integer *, real *, real *); 00045 extern /* Subroutine */ int sppcon_(char *, integer *, real *, real *, 00046 real *, real *, integer *, integer *), slaqsp_(char *, 00047 integer *, real *, real *, real *, real *, char *) 00048 ; 00049 real smlnum; 00050 extern /* Subroutine */ int sppequ_(char *, integer *, real *, real *, 00051 real *, real *, integer *), spprfs_(char *, integer *, 00052 integer *, real *, real *, real *, integer *, real *, integer *, 00053 real *, real *, real *, integer *, integer *), spptrf_( 00054 char *, integer *, real *, integer *), spptrs_(char *, 00055 integer *, integer *, real *, real *, integer *, integer *); 00056 00057 00058 /* -- LAPACK driver routine (version 3.2) -- */ 00059 /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ 00060 /* November 2006 */ 00061 00062 /* .. Scalar Arguments .. */ 00063 /* .. */ 00064 /* .. Array Arguments .. */ 00065 /* .. */ 00066 00067 /* Purpose */ 00068 /* ======= */ 00069 00070 /* SPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to */ 00071 /* compute the solution to a real system of linear equations */ 00072 /* A * X = B, */ 00073 /* where A is an N-by-N symmetric positive definite matrix stored in */ 00074 /* packed format and X and B are N-by-NRHS matrices. */ 00075 00076 /* Error bounds on the solution and a condition estimate are also */ 00077 /* provided. */ 00078 00079 /* Description */ 00080 /* =========== */ 00081 00082 /* The following steps are performed: */ 00083 00084 /* 1. If FACT = 'E', real scaling factors are computed to equilibrate */ 00085 /* the system: */ 00086 /* diag(S) * A * diag(S) * inv(diag(S)) * X = diag(S) * B */ 00087 /* Whether or not the system will be equilibrated depends on the */ 00088 /* scaling of the matrix A, but if equilibration is used, A is */ 00089 /* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */ 00090 00091 /* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ 00092 /* factor the matrix A (after equilibration if FACT = 'E') as */ 00093 /* A = U**T* U, if UPLO = 'U', or */ 00094 /* A = L * L**T, if UPLO = 'L', */ 00095 /* where U is an upper triangular matrix and L is a lower triangular */ 00096 /* matrix. */ 00097 00098 /* 3. If the leading i-by-i principal minor is not positive definite, */ 00099 /* then the routine returns with INFO = i. Otherwise, the factored */ 00100 /* form of A is used to estimate the condition number of the matrix */ 00101 /* A. If the reciprocal of the condition number is less than machine */ 00102 /* precision, INFO = N+1 is returned as a warning, but the routine */ 00103 /* still goes on to solve for X and compute error bounds as */ 00104 /* described below. */ 00105 00106 /* 4. The system of equations is solved for X using the factored form */ 00107 /* of A. */ 00108 00109 /* 5. Iterative refinement is applied to improve the computed solution */ 00110 /* matrix and calculate error bounds and backward error estimates */ 00111 /* for it. */ 00112 00113 /* 6. If equilibration was used, the matrix X is premultiplied by */ 00114 /* diag(S) so that it solves the original system before */ 00115 /* equilibration. */ 00116 00117 /* Arguments */ 00118 /* ========= */ 00119 00120 /* FACT (input) CHARACTER*1 */ 00121 /* Specifies whether or not the factored form of the matrix A is */ 00122 /* supplied on entry, and if not, whether the matrix A should be */ 00123 /* equilibrated before it is factored. */ 00124 /* = 'F': On entry, AFP contains the factored form of A. */ 00125 /* If EQUED = 'Y', the matrix A has been equilibrated */ 00126 /* with scaling factors given by S. AP and AFP will not */ 00127 /* be modified. */ 00128 /* = 'N': The matrix A will be copied to AFP and factored. */ 00129 /* = 'E': The matrix A will be equilibrated if necessary, then */ 00130 /* copied to AFP and factored. */ 00131 00132 /* UPLO (input) CHARACTER*1 */ 00133 /* = 'U': Upper triangle of A is stored; */ 00134 /* = 'L': Lower triangle of A is stored. */ 00135 00136 /* N (input) INTEGER */ 00137 /* The number of linear equations, i.e., the order of the */ 00138 /* matrix A. N >= 0. */ 00139 00140 /* NRHS (input) INTEGER */ 00141 /* The number of right hand sides, i.e., the number of columns */ 00142 /* of the matrices B and X. NRHS >= 0. */ 00143 00144 /* AP (input/output) REAL array, dimension (N*(N+1)/2) */ 00145 /* On entry, the upper or lower triangle of the symmetric matrix */ 00146 /* A, packed columnwise in a linear array, except if FACT = 'F' */ 00147 /* and EQUED = 'Y', then A must contain the equilibrated matrix */ 00148 /* diag(S)*A*diag(S). The j-th column of A is stored in the */ 00149 /* array AP as follows: */ 00150 /* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */ 00151 /* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */ 00152 /* See below for further details. A is not modified if */ 00153 /* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ 00154 00155 /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ 00156 /* diag(S)*A*diag(S). */ 00157 00158 /* AFP (input or output) REAL array, dimension */ 00159 /* (N*(N+1)/2) */ 00160 /* If FACT = 'F', then AFP is an input argument and on entry */ 00161 /* contains the triangular factor U or L from the Cholesky */ 00162 /* factorization A = U'*U or A = L*L', in the same storage */ 00163 /* format as A. If EQUED .ne. 'N', then AFP is the factored */ 00164 /* form of the equilibrated matrix A. */ 00165 00166 /* If FACT = 'N', then AFP is an output argument and on exit */ 00167 /* returns the triangular factor U or L from the Cholesky */ 00168 /* factorization A = U'*U or A = L*L' of the original matrix A. */ 00169 00170 /* If FACT = 'E', then AFP is an output argument and on exit */ 00171 /* returns the triangular factor U or L from the Cholesky */ 00172 /* factorization A = U'*U or A = L*L' of the equilibrated */ 00173 /* matrix A (see the description of AP for the form of the */ 00174 /* equilibrated matrix). */ 00175 00176 /* EQUED (input or output) CHARACTER*1 */ 00177 /* Specifies the form of equilibration that was done. */ 00178 /* = 'N': No equilibration (always true if FACT = 'N'). */ 00179 /* = 'Y': Equilibration was done, i.e., A has been replaced by */ 00180 /* diag(S) * A * diag(S). */ 00181 /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ 00182 /* output argument. */ 00183 00184 /* S (input or output) REAL array, dimension (N) */ 00185 /* The scale factors for A; not accessed if EQUED = 'N'. S is */ 00186 /* an input argument if FACT = 'F'; otherwise, S is an output */ 00187 /* argument. If FACT = 'F' and EQUED = 'Y', each element of S */ 00188 /* must be positive. */ 00189 00190 /* B (input/output) REAL array, dimension (LDB,NRHS) */ 00191 /* On entry, the N-by-NRHS right hand side matrix B. */ 00192 /* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */ 00193 /* B is overwritten by diag(S) * B. */ 00194 00195 /* LDB (input) INTEGER */ 00196 /* The leading dimension of the array B. LDB >= max(1,N). */ 00197 00198 /* X (output) REAL array, dimension (LDX,NRHS) */ 00199 /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ 00200 /* the original system of equations. Note that if EQUED = 'Y', */ 00201 /* A and B are modified on exit, and the solution to the */ 00202 /* equilibrated system is inv(diag(S))*X. */ 00203 00204 /* LDX (input) INTEGER */ 00205 /* The leading dimension of the array X. LDX >= max(1,N). */ 00206 00207 /* RCOND (output) REAL */ 00208 /* The estimate of the reciprocal condition number of the matrix */ 00209 /* A after equilibration (if done). If RCOND is less than the */ 00210 /* machine precision (in particular, if RCOND = 0), the matrix */ 00211 /* is singular to working precision. This condition is */ 00212 /* indicated by a return code of INFO > 0. */ 00213 00214 /* FERR (output) REAL array, dimension (NRHS) */ 00215 /* The estimated forward error bound for each solution vector */ 00216 /* X(j) (the j-th column of the solution matrix X). */ 00217 /* If XTRUE is the true solution corresponding to X(j), FERR(j) */ 00218 /* is an estimated upper bound for the magnitude of the largest */ 00219 /* element in (X(j) - XTRUE) divided by the magnitude of the */ 00220 /* largest element in X(j). The estimate is as reliable as */ 00221 /* the estimate for RCOND, and is almost always a slight */ 00222 /* overestimate of the true error. */ 00223 00224 /* BERR (output) REAL array, dimension (NRHS) */ 00225 /* The componentwise relative backward error of each solution */ 00226 /* vector X(j) (i.e., the smallest relative change in */ 00227 /* any element of A or B that makes X(j) an exact solution). */ 00228 00229 /* WORK (workspace) REAL array, dimension (3*N) */ 00230 00231 /* IWORK (workspace) INTEGER array, dimension (N) */ 00232 00233 /* INFO (output) INTEGER */ 00234 /* = 0: successful exit */ 00235 /* < 0: if INFO = -i, the i-th argument had an illegal value */ 00236 /* > 0: if INFO = i, and i is */ 00237 /* <= N: the leading minor of order i of A is */ 00238 /* not positive definite, so the factorization */ 00239 /* could not be completed, and the solution has not */ 00240 /* been computed. RCOND = 0 is returned. */ 00241 /* = N+1: U is nonsingular, but RCOND is less than machine */ 00242 /* precision, meaning that the matrix is singular */ 00243 /* to working precision. Nevertheless, the */ 00244 /* solution and error bounds are computed because */ 00245 /* there are a number of situations where the */ 00246 /* computed solution can be more accurate than the */ 00247 /* value of RCOND would suggest. */ 00248 00249 /* Further Details */ 00250 /* =============== */ 00251 00252 /* The packed storage scheme is illustrated by the following example */ 00253 /* when N = 4, UPLO = 'U': */ 00254 00255 /* Two-dimensional storage of the symmetric matrix A: */ 00256 00257 /* a11 a12 a13 a14 */ 00258 /* a22 a23 a24 */ 00259 /* a33 a34 (aij = conjg(aji)) */ 00260 /* a44 */ 00261 00262 /* Packed storage of the upper triangle of A: */ 00263 00264 /* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] */ 00265 00266 /* ===================================================================== */ 00267 00268 /* .. Parameters .. */ 00269 /* .. */ 00270 /* .. Local Scalars .. */ 00271 /* .. */ 00272 /* .. External Functions .. */ 00273 /* .. */ 00274 /* .. External Subroutines .. */ 00275 /* .. */ 00276 /* .. Intrinsic Functions .. */ 00277 /* .. */ 00278 /* .. Executable Statements .. */ 00279 00280 /* Parameter adjustments */ 00281 --ap; 00282 --afp; 00283 --s; 00284 b_dim1 = *ldb; 00285 b_offset = 1 + b_dim1; 00286 b -= b_offset; 00287 x_dim1 = *ldx; 00288 x_offset = 1 + x_dim1; 00289 x -= x_offset; 00290 --ferr; 00291 --berr; 00292 --work; 00293 --iwork; 00294 00295 /* Function Body */ 00296 *info = 0; 00297 nofact = lsame_(fact, "N"); 00298 equil = lsame_(fact, "E"); 00299 if (nofact || equil) { 00300 *(unsigned char *)equed = 'N'; 00301 rcequ = FALSE_; 00302 } else { 00303 rcequ = lsame_(equed, "Y"); 00304 smlnum = slamch_("Safe minimum"); 00305 bignum = 1.f / smlnum; 00306 } 00307 00308 /* Test the input parameters. */ 00309 00310 if (! nofact && ! equil && ! lsame_(fact, "F")) { 00311 *info = -1; 00312 } else if (! lsame_(uplo, "U") && ! lsame_(uplo, 00313 "L")) { 00314 *info = -2; 00315 } else if (*n < 0) { 00316 *info = -3; 00317 } else if (*nrhs < 0) { 00318 *info = -4; 00319 } else if (lsame_(fact, "F") && ! (rcequ || lsame_( 00320 equed, "N"))) { 00321 *info = -7; 00322 } else { 00323 if (rcequ) { 00324 smin = bignum; 00325 smax = 0.f; 00326 i__1 = *n; 00327 for (j = 1; j <= i__1; ++j) { 00328 /* Computing MIN */ 00329 r__1 = smin, r__2 = s[j]; 00330 smin = dmin(r__1,r__2); 00331 /* Computing MAX */ 00332 r__1 = smax, r__2 = s[j]; 00333 smax = dmax(r__1,r__2); 00334 /* L10: */ 00335 } 00336 if (smin <= 0.f) { 00337 *info = -8; 00338 } else if (*n > 0) { 00339 scond = dmax(smin,smlnum) / dmin(smax,bignum); 00340 } else { 00341 scond = 1.f; 00342 } 00343 } 00344 if (*info == 0) { 00345 if (*ldb < max(1,*n)) { 00346 *info = -10; 00347 } else if (*ldx < max(1,*n)) { 00348 *info = -12; 00349 } 00350 } 00351 } 00352 00353 if (*info != 0) { 00354 i__1 = -(*info); 00355 xerbla_("SPPSVX", &i__1); 00356 return 0; 00357 } 00358 00359 if (equil) { 00360 00361 /* Compute row and column scalings to equilibrate the matrix A. */ 00362 00363 sppequ_(uplo, n, &ap[1], &s[1], &scond, &amax, &infequ); 00364 if (infequ == 0) { 00365 00366 /* Equilibrate the matrix. */ 00367 00368 slaqsp_(uplo, n, &ap[1], &s[1], &scond, &amax, equed); 00369 rcequ = lsame_(equed, "Y"); 00370 } 00371 } 00372 00373 /* Scale the right-hand side. */ 00374 00375 if (rcequ) { 00376 i__1 = *nrhs; 00377 for (j = 1; j <= i__1; ++j) { 00378 i__2 = *n; 00379 for (i__ = 1; i__ <= i__2; ++i__) { 00380 b[i__ + j * b_dim1] = s[i__] * b[i__ + j * b_dim1]; 00381 /* L20: */ 00382 } 00383 /* L30: */ 00384 } 00385 } 00386 00387 if (nofact || equil) { 00388 00389 /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ 00390 00391 i__1 = *n * (*n + 1) / 2; 00392 scopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1); 00393 spptrf_(uplo, n, &afp[1], info); 00394 00395 /* Return if INFO is non-zero. */ 00396 00397 if (*info > 0) { 00398 *rcond = 0.f; 00399 return 0; 00400 } 00401 } 00402 00403 /* Compute the norm of the matrix A. */ 00404 00405 anorm = slansp_("I", uplo, n, &ap[1], &work[1]); 00406 00407 /* Compute the reciprocal of the condition number of A. */ 00408 00409 sppcon_(uplo, n, &afp[1], &anorm, rcond, &work[1], &iwork[1], info); 00410 00411 /* Compute the solution matrix X. */ 00412 00413 slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); 00414 spptrs_(uplo, n, nrhs, &afp[1], &x[x_offset], ldx, info); 00415 00416 /* Use iterative refinement to improve the computed solution and */ 00417 /* compute error bounds and backward error estimates for it. */ 00418 00419 spprfs_(uplo, n, nrhs, &ap[1], &afp[1], &b[b_offset], ldb, &x[x_offset], 00420 ldx, &ferr[1], &berr[1], &work[1], &iwork[1], info); 00421 00422 /* Transform the solution matrix X to a solution of the original */ 00423 /* system. */ 00424 00425 if (rcequ) { 00426 i__1 = *nrhs; 00427 for (j = 1; j <= i__1; ++j) { 00428 i__2 = *n; 00429 for (i__ = 1; i__ <= i__2; ++i__) { 00430 x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1]; 00431 /* L40: */ 00432 } 00433 /* L50: */ 00434 } 00435 i__1 = *nrhs; 00436 for (j = 1; j <= i__1; ++j) { 00437 ferr[j] /= scond; 00438 /* L60: */ 00439 } 00440 } 00441 00442 /* Set INFO = N+1 if the matrix is singular to working precision. */ 00443 00444 if (*rcond < slamch_("Epsilon")) { 00445 *info = *n + 1; 00446 } 00447 00448 return 0; 00449 00450 /* End of SPPSVX */ 00451 00452 } /* sppsvx_ */