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author lsong10
date Fri, 26 Mar 2021 16:52:45 +0000
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1 #include <math.h>
2
3
4 /* Log gamma function
5 * \log{\Gamma(z)}
6 * AS245, 2nd algorithm, http://lib.stat.cmu.edu/apstat/245
7 */
8 double kf_lgamma(double z)
9 {
10 double x = 0;
11 x += 0.1659470187408462e-06 / (z+7);
12 x += 0.9934937113930748e-05 / (z+6);
13 x -= 0.1385710331296526 / (z+5);
14 x += 12.50734324009056 / (z+4);
15 x -= 176.6150291498386 / (z+3);
16 x += 771.3234287757674 / (z+2);
17 x -= 1259.139216722289 / (z+1);
18 x += 676.5203681218835 / z;
19 x += 0.9999999999995183;
20 return log(x) - 5.58106146679532777 - z + (z-0.5) * log(z+6.5);
21 }
22
23 /* complementary error function
24 * \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2} dt
25 * AS66, 2nd algorithm, http://lib.stat.cmu.edu/apstat/66
26 */
27 double kf_erfc(double x)
28 {
29 const double p0 = 220.2068679123761;
30 const double p1 = 221.2135961699311;
31 const double p2 = 112.0792914978709;
32 const double p3 = 33.912866078383;
33 const double p4 = 6.37396220353165;
34 const double p5 = .7003830644436881;
35 const double p6 = .03526249659989109;
36 const double q0 = 440.4137358247522;
37 const double q1 = 793.8265125199484;
38 const double q2 = 637.3336333788311;
39 const double q3 = 296.5642487796737;
40 const double q4 = 86.78073220294608;
41 const double q5 = 16.06417757920695;
42 const double q6 = 1.755667163182642;
43 const double q7 = .08838834764831844;
44 double expntl, z, p;
45 z = fabs(x) * M_SQRT2;
46 if (z > 37.) return x > 0.? 0. : 2.;
47 expntl = exp(z * z * - .5);
48 if (z < 10. / M_SQRT2) // for small z
49 p = expntl * ((((((p6 * z + p5) * z + p4) * z + p3) * z + p2) * z + p1) * z + p0)
50 / (((((((q7 * z + q6) * z + q5) * z + q4) * z + q3) * z + q2) * z + q1) * z + q0);
51 else p = expntl / 2.506628274631001 / (z + 1. / (z + 2. / (z + 3. / (z + 4. / (z + .65)))));
52 return x > 0.? 2. * p : 2. * (1. - p);
53 }
54
55 /* The following computes regularized incomplete gamma functions.
56 * Formulas are taken from Wiki, with additional input from Numerical
57 * Recipes in C (for modified Lentz's algorithm) and AS245
58 * (http://lib.stat.cmu.edu/apstat/245).
59 *
60 * A good online calculator is available at:
61 *
62 * http://www.danielsoper.com/statcalc/calc23.aspx
63 *
64 * It calculates upper incomplete gamma function, which equals
65 * kf_gammaq(s,z)*tgamma(s).
66 */
67
68 #define KF_GAMMA_EPS 1e-14
69 #define KF_TINY 1e-290
70
71 // regularized lower incomplete gamma function, by series expansion
72 static double _kf_gammap(double s, double z)
73 {
74 double sum, x;
75 int k;
76 for (k = 1, sum = x = 1.; k < 100; ++k) {
77 sum += (x *= z / (s + k));
78 if (x / sum < KF_GAMMA_EPS) break;
79 }
80 return exp(s * log(z) - z - kf_lgamma(s + 1.) + log(sum));
81 }
82 // regularized upper incomplete gamma function, by continued fraction
83 static double _kf_gammaq(double s, double z)
84 {
85 int j;
86 double C, D, f;
87 f = 1. + z - s; C = f; D = 0.;
88 // Modified Lentz's algorithm for computing continued fraction
89 // See Numerical Recipes in C, 2nd edition, section 5.2
90 for (j = 1; j < 100; ++j) {
91 double a = j * (s - j), b = (j<<1) + 1 + z - s, d;
92 D = b + a * D;
93 if (D < KF_TINY) D = KF_TINY;
94 C = b + a / C;
95 if (C < KF_TINY) C = KF_TINY;
96 D = 1. / D;
97 d = C * D;
98 f *= d;
99 if (fabs(d - 1.) < KF_GAMMA_EPS) break;
100 }
101 return exp(s * log(z) - z - kf_lgamma(s) - log(f));
102 }
103
104 double kf_gammap(double s, double z)
105 {
106 return z <= 1. || z < s? _kf_gammap(s, z) : 1. - _kf_gammaq(s, z);
107 }
108
109 double kf_gammaq(double s, double z)
110 {
111 return z <= 1. || z < s? 1. - _kf_gammap(s, z) : _kf_gammaq(s, z);
112 }
113
114 /* Regularized incomplete beta function. The method is taken from
115 * Numerical Recipe in C, 2nd edition, section 6.4. The following web
116 * page calculates the incomplete beta function, which equals
117 * kf_betai(a,b,x) * gamma(a) * gamma(b) / gamma(a+b):
118 *
119 * http://www.danielsoper.com/statcalc/calc36.aspx
120 */
121 static double kf_betai_aux(double a, double b, double x)
122 {
123 double C, D, f;
124 int j;
125 if (x == 0.) return 0.;
126 if (x == 1.) return 1.;
127 f = 1.; C = f; D = 0.;
128 // Modified Lentz's algorithm for computing continued fraction
129 for (j = 1; j < 200; ++j) {
130 double aa, d;
131 int m = j>>1;
132 aa = (j&1)? -(a + m) * (a + b + m) * x / ((a + 2*m) * (a + 2*m + 1))
133 : m * (b - m) * x / ((a + 2*m - 1) * (a + 2*m));
134 D = 1. + aa * D;
135 if (D < KF_TINY) D = KF_TINY;
136 C = 1. + aa / C;
137 if (C < KF_TINY) C = KF_TINY;
138 D = 1. / D;
139 d = C * D;
140 f *= d;
141 if (fabs(d - 1.) < KF_GAMMA_EPS) break;
142 }
143 return exp(kf_lgamma(a+b) - kf_lgamma(a) - kf_lgamma(b) + a * log(x) + b * log(1.-x)) / a / f;
144 }
145 double kf_betai(double a, double b, double x)
146 {
147 return x < (a + 1.) / (a + b + 2.)? kf_betai_aux(a, b, x) : 1. - kf_betai_aux(b, a, 1. - x);
148 }
149
150 #ifdef KF_MAIN
151 #include <stdio.h>
152 int main(int argc, char *argv[])
153 {
154 double x = 5.5, y = 3;
155 double a, b;
156 printf("erfc(%lg): %lg, %lg\n", x, erfc(x), kf_erfc(x));
157 printf("upper-gamma(%lg,%lg): %lg\n", x, y, kf_gammaq(y, x)*tgamma(y));
158 a = 2; b = 2; x = 0.5;
159 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)));
160 return 0;
161 }
162 #endif