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#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, n_iset;
double n_gset;
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 = (krand_t*)calloc(1, 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;
}
double kr_normal(krand_t *kr)
{
if (kr->n_iset == 0) {
double fac, rsq, v1, v2;
do {
v1 = 2.0 * kr_drand(kr) - 1.0;
v2 = 2.0 * kr_drand(kr) - 1.0;
rsq = v1 * v1 + v2 * v2;
} while (rsq >= 1.0 || rsq == 0.0);
fac = sqrt(-2.0 * log(rsq) / rsq);
kr->n_gset = v1 * fac;
kr->n_iset = 1;
return v2 * fac;
} else {
kr->n_iset = 0;
return kr->n_gset;
}
}
#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;
}
static inline float SIGN(float a, float b)
{
return b >= 0 ? (a >= 0 ? a : -a) : (a >= 0 ? -a : a);
}
double krf_brent(double x1, double x2, double tol, double (*func)(double, void*), void *data, int *err)
{
const int max_iter = 100;
const double eps = 3e-8f;
int i;
double a = x1, b = x2, c = x2, d, e, min1, min2;
double fa, fb, fc, p, q, r, s, tol1, xm;
*err = 0;
fa = func(a, data), fb = func(b, data);
if ((fa > 0.0f && fb > 0.0f) || (fa < 0.0f && fb < 0.0f)) {
*err = -1;
return 0.0f;
}
fc = fb;
for (i = 0; i < max_iter; ++i) {
if ((fb > 0.0f && fc > 0.0f) || (fb < 0.0f && fc < 0.0f)) {
c = a;
fc = fa;
e = d = b - a;
}
if (fabs(fc) < fabs(fb)) {
a = b, b = c, c = a;
fa = fb, fb = fc, fc = fa;
}
tol1 = 2.0f * eps * fabs(b) + 0.5f * tol;
xm = 0.5f * (c - b);
if (fabs(xm) <= tol1 || fb == 0.0f)
return b;
if (fabs(e) >= tol1 && fabs(fa) > fabs(fb)) {
s = fb / fa;
if (a == c) {
p = 2.0f * xm * s;
q = 1.0f - s;
} else {
q = fa / fc;
r = fb / fc;
p = s * (2.0f * xm * q * (q - r) - (b - a) * (r - 1.0f));
q = (q - 1.0f) * (r - 1.0f) * (s - 1.0f);
}
if (p > 0.0f) q = -q;
p = fabs(p);
min1 = 3.0f * xm * q - fabs(tol1 * q);
min2 = fabs(e * q);
if (2.0f * p < (min1 < min2 ? min1 : min2)) {
e = d;
d = p / q;
} else {
d = xm;
e = d;
}
} else {
d = xm;
e = d;
}
a = b;
fa = fb;
if (fabs(d) > tol1) b += d;
else b += SIGN(tol1, xm);
fb = func(b, data);
}
*err = -2;
return 0.0;
}
/*************************
*** 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);
}
/******************
*** Statistics ***
******************/
double km_ks_dist(int na, const double a[], int nb, const double b[]) // a[] and b[] MUST BE sorted
{
int ia = 0, ib = 0;
double fa = 0, fb = 0, sup = 0, na1 = 1. / na, nb1 = 1. / nb;
while (ia < na || ib < nb) {
if (ia == na) fb += nb1, ++ib;
else if (ib == nb) fa += na1, ++ia;
else if (a[ia] < b[ib]) fa += na1, ++ia;
else if (a[ia] > b[ib]) fb += nb1, ++ib;
else fa += na1, fb += nb1, ++ia, ++ib;
if (sup < fabs(fa - fb)) sup = fabs(fa - fb);
}
return sup;
}
#ifdef KF_MAIN
#include <stdio.h>
#include "ksort.h"
KSORT_INIT_GENERIC(double)
int main(int argc, char *argv[])
{
double x = 5.5, y = 3;
double a, b;
double xx[] = {0.22, -0.87, -2.39, -1.79, 0.37, -1.54, 1.28, -0.31, -0.74, 1.72, 0.38, -0.17, -0.62, -1.10, 0.30, 0.15, 2.30, 0.19, -0.50, -0.09};
double yy[] = {-5.13, -2.19, -2.43, -3.83, 0.50, -3.25, 4.32, 1.63, 5.18, -0.43, 7.11, 4.87, -3.10, -5.81, 3.76, 6.31, 2.58, 0.07, 5.76, 3.50};
ks_introsort(double, 20, xx); ks_introsort(double, 20, yy);
printf("K-S distance: %f\n", km_ks_dist(20, xx, 20, yy));
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