--- /dev/null
+#include <stdlib.h>
+#include <string.h>
+#include <math.h>
+#include "kmath.h"
+
+/**************************************
+ *** Pseudo-random number generator ***
+ **************************************/
+
+/*
+ 64-bit Mersenne Twister pseudorandom number generator. Adapted from:
+
+ http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/VERSIONS/C-LANG/mt19937-64.c
+
+ which was written by Takuji Nishimura and Makoto Matsumoto and released
+ under the 3-clause BSD license.
+*/
+
+#define KR_NN 312
+#define KR_MM 156
+#define KR_UM 0xFFFFFFFF80000000ULL /* Most significant 33 bits */
+#define KR_LM 0x7FFFFFFFULL /* Least significant 31 bits */
+
+struct _krand_t {
+ int mti;
+ krint64_t mt[KR_NN];
+};
+
+static void kr_srand0(krint64_t seed, krand_t *kr)
+{
+ kr->mt[0] = seed;
+ for (kr->mti = 1; kr->mti < KR_NN; ++kr->mti)
+ kr->mt[kr->mti] = 6364136223846793005ULL * (kr->mt[kr->mti - 1] ^ (kr->mt[kr->mti - 1] >> 62)) + kr->mti;
+}
+
+krand_t *kr_srand(krint64_t seed)
+{
+ krand_t *kr;
+ kr = malloc(sizeof(krand_t));
+ kr_srand0(seed, kr);
+ return kr;
+}
+
+krint64_t kr_rand(krand_t *kr)
+{
+ krint64_t x;
+ static const krint64_t mag01[2] = { 0, 0xB5026F5AA96619E9ULL };
+ if (kr->mti >= KR_NN) {
+ int i;
+ if (kr->mti == KR_NN + 1) kr_srand0(5489ULL, kr);
+ for (i = 0; i < KR_NN - KR_MM; ++i) {
+ x = (kr->mt[i] & KR_UM) | (kr->mt[i+1] & KR_LM);
+ kr->mt[i] = kr->mt[i + KR_MM] ^ (x>>1) ^ mag01[(int)(x&1)];
+ }
+ for (; i < KR_NN - 1; ++i) {
+ x = (kr->mt[i] & KR_UM) | (kr->mt[i+1] & KR_LM);
+ kr->mt[i] = kr->mt[i + (KR_MM - KR_NN)] ^ (x>>1) ^ mag01[(int)(x&1)];
+ }
+ x = (kr->mt[KR_NN - 1] & KR_UM) | (kr->mt[0] & KR_LM);
+ kr->mt[KR_NN - 1] = kr->mt[KR_MM - 1] ^ (x>>1) ^ mag01[(int)(x&1)];
+ kr->mti = 0;
+ }
+ x = kr->mt[kr->mti++];
+ x ^= (x >> 29) & 0x5555555555555555ULL;
+ x ^= (x << 17) & 0x71D67FFFEDA60000ULL;
+ x ^= (x << 37) & 0xFFF7EEE000000000ULL;
+ x ^= (x >> 43);
+ return x;
+}
+
+#ifdef _KR_MAIN
+int main(int argc, char *argv[])
+{
+ long i, N = 200000000;
+ krand_t *kr;
+ if (argc > 1) N = atol(argv[1]);
+ kr = kr_srand(11);
+ for (i = 0; i < N; ++i) kr_rand(kr);
+// for (i = 0; i < N; ++i) lrand48();
+ free(kr);
+ return 0;
+}
+#endif
+
+/******************************
+ *** Non-linear programming ***
+ ******************************/
+
+/* Hooke-Jeeves algorithm for nonlinear minimization
+
+ Based on the pseudocodes by Bell and Pike (CACM 9(9):684-685), and
+ the revision by Tomlin and Smith (CACM 12(11):637-638). Both of the
+ papers are comments on Kaupe's Algorithm 178 "Direct Search" (ACM
+ 6(6):313-314). The original algorithm was designed by Hooke and
+ Jeeves (ACM 8:212-229). This program is further revised according to
+ Johnson's implementation at Netlib (opt/hooke.c).
+
+ Hooke-Jeeves algorithm is very simple and it works quite well on a
+ few examples. However, it might fail to converge due to its heuristic
+ nature. A possible improvement, as is suggested by Johnson, may be to
+ choose a small r at the beginning to quickly approach to the minimum
+ and a large r at later step to hit the minimum.
+ */
+
+static double __kmin_hj_aux(kmin_f func, int n, double *x1, void *data, double fx1, double *dx, int *n_calls)
+{
+ int k, j = *n_calls;
+ double ftmp;
+ for (k = 0; k != n; ++k) {
+ x1[k] += dx[k];
+ ftmp = func(n, x1, data); ++j;
+ if (ftmp < fx1) fx1 = ftmp;
+ else { /* search the opposite direction */
+ dx[k] = 0.0 - dx[k];
+ x1[k] += dx[k] + dx[k];
+ ftmp = func(n, x1, data); ++j;
+ if (ftmp < fx1) fx1 = ftmp;
+ else x1[k] -= dx[k]; /* back to the original x[k] */
+ }
+ }
+ *n_calls = j;
+ return fx1; /* here: fx1=f(n,x1) */
+}
+
+double kmin_hj(kmin_f func, int n, double *x, void *data, double r, double eps, int max_calls)
+{
+ double fx, fx1, *x1, *dx, radius;
+ int k, n_calls = 0;
+ x1 = (double*)calloc(n, sizeof(double));
+ dx = (double*)calloc(n, sizeof(double));
+ for (k = 0; k != n; ++k) { /* initial directions, based on MGJ */
+ dx[k] = fabs(x[k]) * r;
+ if (dx[k] == 0) dx[k] = r;
+ }
+ radius = r;
+ fx1 = fx = func(n, x, data); ++n_calls;
+ for (;;) {
+ memcpy(x1, x, n * sizeof(double)); /* x1 = x */
+ fx1 = __kmin_hj_aux(func, n, x1, data, fx, dx, &n_calls);
+ while (fx1 < fx) {
+ for (k = 0; k != n; ++k) {
+ double t = x[k];
+ dx[k] = x1[k] > x[k]? fabs(dx[k]) : 0.0 - fabs(dx[k]);
+ x[k] = x1[k];
+ x1[k] = x1[k] + x1[k] - t;
+ }
+ fx = fx1;
+ if (n_calls >= max_calls) break;
+ fx1 = func(n, x1, data); ++n_calls;
+ fx1 = __kmin_hj_aux(func, n, x1, data, fx1, dx, &n_calls);
+ if (fx1 >= fx) break;
+ for (k = 0; k != n; ++k)
+ if (fabs(x1[k] - x[k]) > .5 * fabs(dx[k])) break;
+ if (k == n) break;
+ }
+ if (radius >= eps) {
+ if (n_calls >= max_calls) break;
+ radius *= r;
+ for (k = 0; k != n; ++k) dx[k] *= r;
+ } else break; /* converge */
+ }
+ free(x1); free(dx);
+ return fx1;
+}
+
+// I copied this function somewhere several years ago with some of my modifications, but I forgot the source.
+double kmin_brent(kmin1_f func, double a, double b, void *data, double tol, double *xmin)
+{
+ double bound, u, r, q, fu, tmp, fa, fb, fc, c;
+ const double gold1 = 1.6180339887;
+ const double gold2 = 0.3819660113;
+ const double tiny = 1e-20;
+ const int max_iter = 100;
+
+ double e, d, w, v, mid, tol1, tol2, p, eold, fv, fw;
+ int iter;
+
+ fa = func(a, data); fb = func(b, data);
+ if (fb > fa) { // swap, such that f(a) > f(b)
+ tmp = a; a = b; b = tmp;
+ tmp = fa; fa = fb; fb = tmp;
+ }
+ c = b + gold1 * (b - a), fc = func(c, data); // golden section extrapolation
+ while (fb > fc) {
+ bound = b + 100.0 * (c - b); // the farthest point where we want to go
+ r = (b - a) * (fb - fc);
+ q = (b - c) * (fb - fa);
+ if (fabs(q - r) < tiny) { // avoid 0 denominator
+ tmp = q > r? tiny : 0.0 - tiny;
+ } else tmp = q - r;
+ u = b - ((b - c) * q - (b - a) * r) / (2.0 * tmp); // u is the parabolic extrapolation point
+ if ((b > u && u > c) || (b < u && u < c)) { // u lies between b and c
+ fu = func(u, data);
+ if (fu < fc) { // (b,u,c) bracket the minimum
+ a = b; b = u; fa = fb; fb = fu;
+ break;
+ } else if (fu > fb) { // (a,b,u) bracket the minimum
+ c = u; fc = fu;
+ break;
+ }
+ u = c + gold1 * (c - b); fu = func(u, data); // golden section extrapolation
+ } else if ((c > u && u > bound) || (c < u && u < bound)) { // u lies between c and bound
+ fu = func(u, data);
+ if (fu < fc) { // fb > fc > fu
+ b = c; c = u; u = c + gold1 * (c - b);
+ fb = fc; fc = fu; fu = func(u, data);
+ } else { // (b,c,u) bracket the minimum
+ a = b; b = c; c = u;
+ fa = fb; fb = fc; fc = fu;
+ break;
+ }
+ } else if ((u > bound && bound > c) || (u < bound && bound < c)) { // u goes beyond the bound
+ u = bound; fu = func(u, data);
+ } else { // u goes the other way around, use golden section extrapolation
+ u = c + gold1 * (c - b); fu = func(u, data);
+ }
+ a = b; b = c; c = u;
+ fa = fb; fb = fc; fc = fu;
+ }
+ if (a > c) u = a, a = c, c = u; // swap
+
+ // now, a<b<c, fa>fb and fb<fc, move on to Brent's algorithm
+ e = d = 0.0;
+ w = v = b; fv = fw = fb;
+ for (iter = 0; iter != max_iter; ++iter) {
+ mid = 0.5 * (a + c);
+ tol2 = 2.0 * (tol1 = tol * fabs(b) + tiny);
+ if (fabs(b - mid) <= (tol2 - 0.5 * (c - a))) {
+ *xmin = b; return fb; // found
+ }
+ if (fabs(e) > tol1) {
+ // related to parabolic interpolation
+ r = (b - w) * (fb - fv);
+ q = (b - v) * (fb - fw);
+ p = (b - v) * q - (b - w) * r;
+ q = 2.0 * (q - r);
+ if (q > 0.0) p = 0.0 - p;
+ else q = 0.0 - q;
+ eold = e; e = d;
+ if (fabs(p) >= fabs(0.5 * q * eold) || p <= q * (a - b) || p >= q * (c - b)) {
+ d = gold2 * (e = (b >= mid ? a - b : c - b));
+ } else {
+ d = p / q; u = b + d; // actual parabolic interpolation happens here
+ if (u - a < tol2 || c - u < tol2)
+ d = (mid > b)? tol1 : 0.0 - tol1;
+ }
+ } else d = gold2 * (e = (b >= mid ? a - b : c - b)); // golden section interpolation
+ u = fabs(d) >= tol1 ? b + d : b + (d > 0.0? tol1 : -tol1);
+ fu = func(u, data);
+ if (fu <= fb) { // u is the minimum point so far
+ if (u >= b) a = b;
+ else c = b;
+ v = w; w = b; b = u; fv = fw; fw = fb; fb = fu;
+ } else { // adjust (a,c) and (u,v,w)
+ if (u < b) a = u;
+ else c = u;
+ if (fu <= fw || w == b) {
+ v = w; w = u;
+ fv = fw; fw = fu;
+ } else if (fu <= fv || v == b || v == w) {
+ v = u; fv = fu;
+ }
+ }
+ }
+ *xmin = b;
+ return fb;
+}
+
+/*************************
+ *** Special functions ***
+ *************************/
+
+/* Log gamma function
+ * \log{\Gamma(z)}
+ * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
+ */
+double kf_lgamma(double z)
+{
+ double x = 0;
+ x += 0.1659470187408462e-06 / (z+7);
+ x += 0.9934937113930748e-05 / (z+6);
+ x -= 0.1385710331296526 / (z+5);
+ x += 12.50734324009056 / (z+4);
+ x -= 176.6150291498386 / (z+3);
+ x += 771.3234287757674 / (z+2);
+ x -= 1259.139216722289 / (z+1);
+ x += 676.5203681218835 / z;
+ x += 0.9999999999995183;
+ return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
+}
+
+/* complementary error function
+ * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
+ * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
+ */
+double kf_erfc(double x)
+{
+ const double p0 = 220.2068679123761;
+ const double p1 = 221.2135961699311;
+ const double p2 = 112.0792914978709;
+ const double p3 = 33.912866078383;
+ const double p4 = 6.37396220353165;
+ const double p5 = .7003830644436881;
+ const double p6 = .03526249659989109;
+ const double q0 = 440.4137358247522;
+ const double q1 = 793.8265125199484;
+ const double q2 = 637.3336333788311;
+ const double q3 = 296.5642487796737;
+ const double q4 = 86.78073220294608;
+ const double q5 = 16.06417757920695;
+ const double q6 = 1.755667163182642;
+ const double q7 = .08838834764831844;
+ double expntl, z, p;
+ z = fabs(x) * M_SQRT2;
+ if (z > 37.) return x > 0.? 0. : 2.;
+ expntl = exp(z * z * - .5);
+ if (z < 10. / M_SQRT2) // for small z
+ p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
+ / (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
+ else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
+ return x > 0.? 2. * p : 2. * (1. - p);
+}
+
+/* The following computes regularized incomplete gamma functions.
+ * Formulas are taken from Wiki, with additional input from Numerical
+ * Recipes in C (for modified Lentz's algorithm) and AS245
+ * (http://lib.stat.cmu.edu/apstat/245).
+ *
+ * A good online calculator is available at:
+ *
+ * http://www.danielsoper.com/statcalc/calc23.aspx
+ *
+ * It calculates upper incomplete gamma function, which equals
+ * kf_gammaq(s,z)*tgamma(s).
+ */
+
+#define KF_GAMMA_EPS 1e-14
+#define KF_TINY 1e-290
+
+// regularized lower incomplete gamma function, by series expansion
+static double _kf_gammap(double s, double z)
+{
+ double sum, x;
+ int k;
+ for (k = 1, sum = x = 1.; k < 100; ++k) {
+ sum += (x *= z / (s + k));
+ if (x / sum < KF_GAMMA_EPS) break;
+ }
+ return exp(s * log(z) - z - kf_lgamma(s + 1.) + log(sum));
+}
+// regularized upper incomplete gamma function, by continued fraction
+static double _kf_gammaq(double s, double z)
+{
+ int j;
+ double C, D, f;
+ f = 1. + z - s; C = f; D = 0.;
+ // Modified Lentz's algorithm for computing continued fraction
+ // See Numerical Recipes in C, 2nd edition, section 5.2
+ for (j = 1; j < 100; ++j) {
+ double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
+ D = b + a * D;
+ if (D < KF_TINY) D = KF_TINY;
+ C = b + a / C;
+ if (C < KF_TINY) C = KF_TINY;
+ D = 1. / D;
+ d = C * D;
+ f *= d;
+ if (fabs(d - 1.) < KF_GAMMA_EPS) break;
+ }
+ return exp(s * log(z) - z - kf_lgamma(s) - log(f));
+}
+
+double kf_gammap(double s, double z)
+{
+ return z <= 1. || z < s? _kf_gammap(s, z) : 1. - _kf_gammaq(s, z);
+}
+
+double kf_gammaq(double s, double z)
+{
+ return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
+}
+
+/* Regularized incomplete beta function. The method is taken from
+ * Numerical Recipe in C, 2nd edition, section 6.4. The following web
+ * page calculates the incomplete beta function, which equals
+ * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
+ *
+ * http://www.danielsoper.com/statcalc/calc36.aspx
+ */
+static double kf_betai_aux(double a, double b, double x)
+{
+ double C, D, f;
+ int j;
+ if (x == 0.) return 0.;
+ if (x == 1.) return 1.;
+ f = 1.; C = f; D = 0.;
+ // Modified Lentz's algorithm for computing continued fraction
+ for (j = 1; j < 200; ++j) {
+ double aa, d;
+ int m = j>>1;
+ aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
+ : m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
+ D = 1. + aa * D;
+ if (D < KF_TINY) D = KF_TINY;
+ C = 1. + aa / C;
+ if (C < KF_TINY) C = KF_TINY;
+ D = 1. / D;
+ d = C * D;
+ f *= d;
+ if (fabs(d - 1.) < KF_GAMMA_EPS) break;
+ }
+ return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
+}
+double kf_betai(double a, double b, double x)
+{
+ return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
+}
+
+#ifdef KF_MAIN
+#include <stdio.h>
+int main(int argc, char *argv[])
+{
+ double x = 5.5, y = 3;
+ double a, b;
+ printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
+ printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
+ a = 2; b = 2; x = 0.5;
+ printf("incomplete-beta(%lg,%lg,%lg): %lg\n", a, b, x, kf_betai(a, b, x) / exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b)));
+ return 0;
+}
+#endif
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