| Geant4 Cross Reference |
1 /* igam.c
2 *
3 * Incomplete gamma integral
4 *
5 *
6 * DESCRIPTION:
7 *
8 * The function is defined by
9 *
10 * x
11 * -
12 * | | -t a-1
13 * igam(a,x) = | e t dt.
14 * | |
15 * -
16 * 0
17 *
18 *
19 * In this implementation both arguments must be positive.
20 * The integral is evaluated by either a power series or
21 * continued fraction expansion, depending on the relative
22 * values of a and x.
23 *
24 * ACCURACY:
25 *
26 * Relative error:
27 * arithmetic domain # trials peak rms
28 * IEEE 0,30 200000 3.6e-14 2.9e-15
29 * IEEE 0,100 300000 9.9e-14 1.5e-14
30 */
31 /* igamc()
32 *
33 * Complemented incomplete gamma integral
34 *
35 *
36 * DESCRIPTION:
37 *
38 * The function is defined by
39 *
40 *
41 * igamc(a,x) = 1 - igam(a,x)
42 *
43 * inf.
44 * -
45 * | | -t a-1
46 * = | e t dt.
47 * | |
48 * -
49 * x
50 *
51 *
52 * In this implementation both arguments must be positive.
53 * The integral is evaluated by either a power series or
54 * continued fraction expansion, depending on the relative
55 * values of a and x.
56 *
57 * ACCURACY:
58 *
59 * Tested at random a, x.
60 * a x Relative error:
61 * arithmetic domain domain # trials peak rms
62 * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15
63 * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15
64 */
65
66 /*
67 Cephes Math Library Release 2.8: June, 2000
68 Copyright 1985, 1987, 2000 by Stephen L. Moshier
69 */
70
71 #include "nf_specialFunctions.h"
72
73 #if defined __cplusplus
74 #include <cmath>
75 #include "G4Exp.hh"
76 #include "G4Log.hh"
77 namespace GIDI {
78 using namespace GIDI;
79 using namespace std;
80 #endif
81
82 static double big = 4.503599627370496e15;
83 static double biginv = 2.22044604925031308085e-16;
84
85 /*
86 ************************************************************
87 */
88 double nf_incompleteGammaFunctionComplementary( double a, double x, nfu_status *status ) {
89
90 double ans, ax, c, yc, r, t, y, z;
91 double pk, pkm1, pkm2, qk, qkm1, qkm2;
92
93 *status = nfu_badInput;
94 if( !isfinite( x ) ) return( x );
95 *status = nfu_Okay;
96
97 if( ( x <= 0 ) || ( a <= 0 ) ) return( 1.0 );
98 if( ( x < 1.0 ) || ( x < a ) ) return( nf_gammaFunction( a, status ) - nf_incompleteGammaFunction( a, x, status ) );
99
100 ax = G4Exp( a * G4Log( x ) - x );
101 if( ax == 0. ) return( 0.0 );
102
103 if( x < 10000. ) {
104 y = 1.0 - a; /* continued fraction */
105 z = x + y + 1.0;
106 c = 0.0;
107 pkm2 = 1.0;
108 qkm2 = x;
109 pkm1 = x + 1.0;
110 qkm1 = z * x;
111 ans = pkm1 / qkm1;
112
113 do {
114 c += 1.0;
115 y += 1.0;
116 z += 2.0;
117 yc = y * c;
118 pk = pkm1 * z - pkm2 * yc;
119 qk = qkm1 * z - qkm2 * yc;
120 if( qk != 0 ) {
121 r = pk / qk;
122 t = fabs( ( ans - r ) / r );
123 ans = r; }
124 else {
125 t = 1.0;
126 }
127 pkm2 = pkm1;
128 pkm1 = pk;
129 qkm2 = qkm1;
130 qkm1 = qk;
131 if( fabs( pk ) > big ) {
132 pkm2 *= biginv;
133 pkm1 *= biginv;
134 qkm2 *= biginv;
135 qkm1 *= biginv;
136 }
137 } while( t > DBL_EPSILON ); } // Loop checking, 11.06.2015, T. Koi
138 else { /* Asymptotic expansion. */
139 y = 1. / x;
140 r = a;
141 c = 1.;
142 ans = 1.;
143 do {
144 a -= 1.;
145 c *= a * y;
146 ans += c;
147 } while( fabs( c ) > 100 * ans * DBL_EPSILON ); // Loop checking, 11.06.2015, T. Koi
148 }
149
150 return( ans * ax );
151 }
152 /*
153 ************************************************************
154 */
155 double nf_incompleteGammaFunction( double a, double x, nfu_status *status ) {
156 /* left tail of incomplete gamma function:
157 *
158 * inf. k
159 * a -x - x
160 * x e > ----------
161 * - -
162 * k=0 | (a+k+1)
163 */
164 double ans, ax, c, r;
165
166 *status = nfu_badInput;
167 if( !isfinite( x ) ) return( x );
168 *status = nfu_Okay;
169
170 if( ( x <= 0 ) || ( a <= 0 ) ) return( 0.0 );
171 if( ( x > 1.0 ) && ( x > a ) ) return( nf_gammaFunction( a, status ) - nf_incompleteGammaFunctionComplementary( a, x, status ) );
172
173 ax = G4Exp( a * G4Log( x ) - x ); /* Compute x**a * exp(-x) */
174 if( ax == 0. ) return( 0.0 );
175
176 r = a; /* power series */
177 c = 1.0;
178 ans = 1.0;
179 do {
180 r += 1.0;
181 c *= x / r;
182 ans += c;
183 } while( c > ans * DBL_EPSILON ); // Loop checking, 11.06.2015, T. Koi
184
185 return( ans * ax / a );
186 }
187
188 #if defined __cplusplus
189 }
190 #endif
191