00001 /* dppsvx.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 dppsvx_(char *fact, char *uplo, integer *n, integer * 00021 nrhs, doublereal *ap, doublereal *afp, char *equed, doublereal *s, 00022 doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal * 00023 rcond, doublereal *ferr, doublereal *berr, doublereal *work, integer * 00024 iwork, integer *info) 00025 { 00026 /* System generated locals */ 00027 integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2; 00028 doublereal d__1, d__2; 00029 00030 /* Local variables */ 00031 integer i__, j; 00032 doublereal amax, smin, smax; 00033 extern logical lsame_(char *, char *); 00034 doublereal scond, anorm; 00035 extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 00036 doublereal *, integer *); 00037 logical equil, rcequ; 00038 extern doublereal dlamch_(char *); 00039 logical nofact; 00040 extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 00041 doublereal *, integer *, doublereal *, integer *), 00042 xerbla_(char *, integer *); 00043 doublereal bignum; 00044 extern doublereal dlansp_(char *, char *, integer *, doublereal *, 00045 doublereal *); 00046 extern /* Subroutine */ int dppcon_(char *, integer *, doublereal *, 00047 doublereal *, doublereal *, doublereal *, integer *, integer *), dlaqsp_(char *, integer *, doublereal *, doublereal *, 00048 doublereal *, doublereal *, char *); 00049 integer infequ; 00050 extern /* Subroutine */ int dppequ_(char *, integer *, doublereal *, 00051 doublereal *, doublereal *, doublereal *, integer *), 00052 dpprfs_(char *, integer *, integer *, doublereal *, doublereal *, 00053 doublereal *, integer *, doublereal *, integer *, doublereal *, 00054 doublereal *, doublereal *, integer *, integer *), 00055 dpptrf_(char *, integer *, doublereal *, integer *); 00056 doublereal smlnum; 00057 extern /* Subroutine */ int dpptrs_(char *, integer *, integer *, 00058 doublereal *, doublereal *, 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 /* DPPSVX uses the Cholesky factorization A = U**T*U or A = L*L**T to */ 00074 /* compute the solution to a real system of linear equations */ 00075 /* A * X = B, */ 00076 /* where A is an N-by-N symmetric positive definite matrix stored in */ 00077 /* packed format and X and B 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**T* U, if UPLO = 'U', or */ 00097 /* A = L * L**T, 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, AFP contains the factored form of A. */ 00128 /* If EQUED = 'Y', the matrix A has been equilibrated */ 00129 /* with scaling factors given by S. AP and AFP will not */ 00130 /* be modified. */ 00131 /* = 'N': The matrix A will be copied to AFP and factored. */ 00132 /* = 'E': The matrix A will be equilibrated if necessary, then */ 00133 /* copied to AFP 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 /* AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) */ 00148 /* On entry, the upper or lower triangle of the symmetric matrix */ 00149 /* A, packed columnwise in a linear array, except if FACT = 'F' */ 00150 /* and EQUED = 'Y', then A must contain the equilibrated matrix */ 00151 /* diag(S)*A*diag(S). The j-th column of A is stored in the */ 00152 /* array AP as follows: */ 00153 /* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; */ 00154 /* if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. */ 00155 /* See below for further details. A is not modified if */ 00156 /* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. */ 00157 00158 /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ 00159 /* diag(S)*A*diag(S). */ 00160 00161 /* AFP (input or output) DOUBLE PRECISION array, dimension */ 00162 /* (N*(N+1)/2) */ 00163 /* If FACT = 'F', then AFP is an input argument and on entry */ 00164 /* contains the triangular factor U or L from the Cholesky */ 00165 /* factorization A = U'*U or A = L*L', in the same storage */ 00166 /* format as A. If EQUED .ne. 'N', then AFP is the factored */ 00167 /* form of the equilibrated matrix A. */ 00168 00169 /* If FACT = 'N', then AFP is an output argument and on exit */ 00170 /* returns the triangular factor U or L from the Cholesky */ 00171 /* factorization A = U'*U or A = L*L' of the original matrix A. */ 00172 00173 /* If FACT = 'E', then AFP is an output argument and on exit */ 00174 /* returns the triangular factor U or L from the Cholesky */ 00175 /* factorization A = U'*U or A = L*L' of the equilibrated */ 00176 /* matrix A (see the description of AP for the form of the */ 00177 /* equilibrated matrix). */ 00178 00179 /* EQUED (input or output) CHARACTER*1 */ 00180 /* Specifies the form of equilibration that was done. */ 00181 /* = 'N': No equilibration (always true if FACT = 'N'). */ 00182 /* = 'Y': Equilibration was done, i.e., A has been replaced by */ 00183 /* diag(S) * A * diag(S). */ 00184 /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ 00185 /* output argument. */ 00186 00187 /* S (input or output) DOUBLE PRECISION array, dimension (N) */ 00188 /* The scale factors for A; not accessed if EQUED = 'N'. S is */ 00189 /* an input argument if FACT = 'F'; otherwise, S is an output */ 00190 /* argument. If FACT = 'F' and EQUED = 'Y', each element of S */ 00191 /* must be positive. */ 00192 00193 /* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */ 00194 /* On entry, the N-by-NRHS right hand side matrix B. */ 00195 /* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y', */ 00196 /* 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) DOUBLE PRECISION array, dimension (LDX,NRHS) */ 00202 /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ 00203 /* the original system of equations. Note that if EQUED = 'Y', */ 00204 /* A and B are modified on exit, and the solution to the */ 00205 /* equilibrated system is 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 /* The estimate of the reciprocal condition number of the matrix */ 00212 /* A after equilibration (if done). If RCOND is less than the */ 00213 /* machine precision (in particular, if RCOND = 0), the matrix */ 00214 /* is singular to working precision. This condition is */ 00215 /* indicated by a return code of INFO > 0. */ 00216 00217 /* FERR (output) DOUBLE PRECISION array, dimension (NRHS) */ 00218 /* The estimated forward error bound for each solution vector */ 00219 /* X(j) (the j-th column of the solution matrix X). */ 00220 /* If XTRUE is the true solution corresponding to X(j), FERR(j) */ 00221 /* is an estimated upper bound for the magnitude of the largest */ 00222 /* element in (X(j) - XTRUE) divided by the magnitude of the */ 00223 /* largest element in X(j). The estimate is as reliable as */ 00224 /* the estimate for RCOND, and is almost always a slight */ 00225 /* overestimate of the true error. */ 00226 00227 /* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */ 00228 /* The componentwise relative backward error of each solution */ 00229 /* vector X(j) (i.e., the smallest relative change in */ 00230 /* any element of A or B that makes X(j) an exact solution). */ 00231 00232 /* WORK (workspace) DOUBLE PRECISION array, dimension (3*N) */ 00233 00234 /* IWORK (workspace) INTEGER array, dimension (N) */ 00235 00236 /* INFO (output) INTEGER */ 00237 /* = 0: successful exit */ 00238 /* < 0: if INFO = -i, the i-th argument had an illegal value */ 00239 /* > 0: if INFO = i, and i is */ 00240 /* <= N: the leading minor of order i of A is */ 00241 /* not positive definite, so the factorization */ 00242 /* could not be completed, and the solution has not */ 00243 /* been computed. RCOND = 0 is returned. */ 00244 /* = N+1: U is nonsingular, but RCOND is less than machine */ 00245 /* precision, meaning that the matrix is singular */ 00246 /* to working precision. Nevertheless, the */ 00247 /* solution and error bounds are computed because */ 00248 /* there are a number of situations where the */ 00249 /* computed solution can be more accurate than the */ 00250 /* value of RCOND would suggest. */ 00251 00252 /* Further Details */ 00253 /* =============== */ 00254 00255 /* The packed storage scheme is illustrated by the following example */ 00256 /* when N = 4, UPLO = 'U': */ 00257 00258 /* Two-dimensional storage of the symmetric matrix A: */ 00259 00260 /* a11 a12 a13 a14 */ 00261 /* a22 a23 a24 */ 00262 /* a33 a34 (aij = conjg(aji)) */ 00263 /* a44 */ 00264 00265 /* Packed storage of the upper triangle of A: */ 00266 00267 /* AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ] */ 00268 00269 /* ===================================================================== */ 00270 00271 /* .. Parameters .. */ 00272 /* .. */ 00273 /* .. Local Scalars .. */ 00274 /* .. */ 00275 /* .. External Functions .. */ 00276 /* .. */ 00277 /* .. External Subroutines .. */ 00278 /* .. */ 00279 /* .. Intrinsic Functions .. */ 00280 /* .. */ 00281 /* .. Executable Statements .. */ 00282 00283 /* Parameter adjustments */ 00284 --ap; 00285 --afp; 00286 --s; 00287 b_dim1 = *ldb; 00288 b_offset = 1 + b_dim1; 00289 b -= b_offset; 00290 x_dim1 = *ldx; 00291 x_offset = 1 + x_dim1; 00292 x -= x_offset; 00293 --ferr; 00294 --berr; 00295 --work; 00296 --iwork; 00297 00298 /* Function Body */ 00299 *info = 0; 00300 nofact = lsame_(fact, "N"); 00301 equil = lsame_(fact, "E"); 00302 if (nofact || equil) { 00303 *(unsigned char *)equed = 'N'; 00304 rcequ = FALSE_; 00305 } else { 00306 rcequ = lsame_(equed, "Y"); 00307 smlnum = dlamch_("Safe minimum"); 00308 bignum = 1. / smlnum; 00309 } 00310 00311 /* Test the input parameters. */ 00312 00313 if (! nofact && ! equil && ! lsame_(fact, "F")) { 00314 *info = -1; 00315 } else if (! lsame_(uplo, "U") && ! lsame_(uplo, 00316 "L")) { 00317 *info = -2; 00318 } else if (*n < 0) { 00319 *info = -3; 00320 } else if (*nrhs < 0) { 00321 *info = -4; 00322 } else if (lsame_(fact, "F") && ! (rcequ || lsame_( 00323 equed, "N"))) { 00324 *info = -7; 00325 } else { 00326 if (rcequ) { 00327 smin = bignum; 00328 smax = 0.; 00329 i__1 = *n; 00330 for (j = 1; j <= i__1; ++j) { 00331 /* Computing MIN */ 00332 d__1 = smin, d__2 = s[j]; 00333 smin = min(d__1,d__2); 00334 /* Computing MAX */ 00335 d__1 = smax, d__2 = s[j]; 00336 smax = max(d__1,d__2); 00337 /* L10: */ 00338 } 00339 if (smin <= 0.) { 00340 *info = -8; 00341 } else if (*n > 0) { 00342 scond = max(smin,smlnum) / min(smax,bignum); 00343 } else { 00344 scond = 1.; 00345 } 00346 } 00347 if (*info == 0) { 00348 if (*ldb < max(1,*n)) { 00349 *info = -10; 00350 } else if (*ldx < max(1,*n)) { 00351 *info = -12; 00352 } 00353 } 00354 } 00355 00356 if (*info != 0) { 00357 i__1 = -(*info); 00358 xerbla_("DPPSVX", &i__1); 00359 return 0; 00360 } 00361 00362 if (equil) { 00363 00364 /* Compute row and column scalings to equilibrate the matrix A. */ 00365 00366 dppequ_(uplo, n, &ap[1], &s[1], &scond, &amax, &infequ); 00367 if (infequ == 0) { 00368 00369 /* Equilibrate the matrix. */ 00370 00371 dlaqsp_(uplo, n, &ap[1], &s[1], &scond, &amax, equed); 00372 rcequ = lsame_(equed, "Y"); 00373 } 00374 } 00375 00376 /* Scale the right-hand side. */ 00377 00378 if (rcequ) { 00379 i__1 = *nrhs; 00380 for (j = 1; j <= i__1; ++j) { 00381 i__2 = *n; 00382 for (i__ = 1; i__ <= i__2; ++i__) { 00383 b[i__ + j * b_dim1] = s[i__] * b[i__ + j * b_dim1]; 00384 /* L20: */ 00385 } 00386 /* L30: */ 00387 } 00388 } 00389 00390 if (nofact || equil) { 00391 00392 /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ 00393 00394 i__1 = *n * (*n + 1) / 2; 00395 dcopy_(&i__1, &ap[1], &c__1, &afp[1], &c__1); 00396 dpptrf_(uplo, n, &afp[1], info); 00397 00398 /* Return if INFO is non-zero. */ 00399 00400 if (*info > 0) { 00401 *rcond = 0.; 00402 return 0; 00403 } 00404 } 00405 00406 /* Compute the norm of the matrix A. */ 00407 00408 anorm = dlansp_("I", uplo, n, &ap[1], &work[1]); 00409 00410 /* Compute the reciprocal of the condition number of A. */ 00411 00412 dppcon_(uplo, n, &afp[1], &anorm, rcond, &work[1], &iwork[1], info); 00413 00414 /* Compute the solution matrix X. */ 00415 00416 dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); 00417 dpptrs_(uplo, n, nrhs, &afp[1], &x[x_offset], ldx, info); 00418 00419 /* Use iterative refinement to improve the computed solution and */ 00420 /* compute error bounds and backward error estimates for it. */ 00421 00422 dpprfs_(uplo, n, nrhs, &ap[1], &afp[1], &b[b_offset], ldb, &x[x_offset], 00423 ldx, &ferr[1], &berr[1], &work[1], &iwork[1], info); 00424 00425 /* Transform the solution matrix X to a solution of the original */ 00426 /* system. */ 00427 00428 if (rcequ) { 00429 i__1 = *nrhs; 00430 for (j = 1; j <= i__1; ++j) { 00431 i__2 = *n; 00432 for (i__ = 1; i__ <= i__2; ++i__) { 00433 x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1]; 00434 /* L40: */ 00435 } 00436 /* L50: */ 00437 } 00438 i__1 = *nrhs; 00439 for (j = 1; j <= i__1; ++j) { 00440 ferr[j] /= scond; 00441 /* L60: */ 00442 } 00443 } 00444 00445 /* Set INFO = N+1 if the matrix is singular to working precision. */ 00446 00447 if (*rcond < dlamch_("Epsilon")) { 00448 *info = *n + 1; 00449 } 00450 00451 return 0; 00452 00453 /* End of DPPSVX */ 00454 00455 } /* dppsvx_ */