| Geant4 Cross Reference |
1 /* gamma.c
2 *
3 * Gamma function
4 *
5 * DESCRIPTION:
6 *
7 * Returns gamma function of the argument. The result is
8 * correctly signed, and the sign (+1 or -1) is also
9 * returned in a global (extern) variable named sgngam.
10 * This variable is also filled in by the logarithmic gamma
11 * function lgam().
12 *
13 * Arguments |x| <= 34 are reduced by recurrence and the function
14 * approximated by a rational function of degree 6/7 in the
15 * interval (2,3). Large arguments are handled by Stirling's
16 * formula. Large negative arguments are made positive using
17 * a reflection formula.
18 *
19 *
20 * ACCURACY:
21 *
22 * Relative error:
23 * arithmetic domain # trials peak rms
24 * IEEE -170,-33 20000 2.3e-15 3.3e-16
25 * IEEE -33, 33 20000 9.4e-16 2.2e-16
26 * IEEE 33, 171.6 20000 2.3e-15 3.2e-16
27 *
28 * Error for arguments outside the test range will be larger
29 * owing to error amplification by the exponential function.
30 *
31 */
32 /* lgam()
33 *
34 * Natural logarithm of gamma function
35 *
36 *
37 * DESCRIPTION:
38 *
39 * Returns the base e (2.718...) logarithm of the absolute
40 * value of the gamma function of the argument.
41 * The sign (+1 or -1) of the gamma function is returned in a
42 * global (extern) variable named sgngam.
43 *
44 * For arguments greater than 13, the logarithm of the gamma
45 * function is approximated by the logarithmic version of
46 * Stirling's formula using a polynomial approximation of
47 * degree 4. Arguments between -33 and +33 are reduced by
48 * recurrence to the interval [2,3] of a rational approximation.
49 * The cosecant reflection formula is employed for arguments
50 * less than -33.
51 *
52 * Arguments greater than MAXLGM return DBL_MAX and an error
53 * message. MAXLGM = 2.556348e305 for IEEE arithmetic.
54 *
55 *
56 * ACCURACY:
57 *
58 * arithmetic domain # trials peak rms
59 * IEEE 0, 3 28000 5.4e-16 1.1e-16
60 * IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
61 * The error criterion was relative when the function magnitude
62 * was greater than one but absolute when it was less than one.
63 *
64 * The following test used the relative error criterion, though
65 * at certain points the relative error could be much higher than
66 * indicated.
67 * IEEE -200, -4 10000 4.8e-16 1.3e-16
68 *
69 */
70 /* gamma.c */
71 /* gamma function */
72
73 /*
74 Cephes Math Library Release 2.8: June, 2000
75 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
76 */
77
78 #include "nf_specialFunctions.h"
79
80 #if defined __cplusplus
81 #include "G4Exp.hh"
82 #include "G4Log.hh"
83 #include "G4Pow.hh"
84 namespace GIDI {
85 using namespace GIDI;
86 using namespace std;
87 #endif
88
89 static double P[] = { 1.60119522476751861407E-4, 1.19135147006586384913E-3, 1.04213797561761569935E-2, 4.76367800457137231464E-2,
90 2.07448227648435975150E-1, 4.94214826801497100753E-1, 9.99999999999999996796E-1 };
91 static double Q[] = { -2.31581873324120129819E-5, 5.39605580493303397842E-4, -4.45641913851797240494E-3, 1.18139785222060435552E-2,
92 3.58236398605498653373E-2, -2.34591795718243348568E-1, 7.14304917030273074085E-2, 1.00000000000000000320E0 };
93 #define MAXGAM 171.624376956302725
94 static double LOGPI = 1.14472988584940017414;
95 static double SQTPI = 2.50662827463100050242E0;
96
97 /* Stirling's formula for the gamma function */
98 static double STIR[5] = { 7.873113957930936284e-4, -2.2954996161337812638e-4, -2.6813261780578123283e-3, 3.472222216054586673e-3, 8.3333333333348225713e-2 };
99 #define MAXSTIR 143.01608
100
101 static double stirf( double x, nfu_status *status );
102 static double lgam( double x, int *sgngam, nfu_status *status );
103 /*
104 ************************************************************
105 */
106 static double stirf( double x, nfu_status * /*status*/ ) {
107 /* Gamma function computed by Stirling's formula. The polynomial STIR is valid for 33 <= x <= 172. */
108
109 double y, w, v;
110
111 w = 1.0 / x;
112 w = 1.0 + w * nf_polevl( w, STIR, 4 );
113 y = G4Exp( x );
114 if( x > MAXSTIR ) { /* Avoid overflow in pow() */
115 v = G4Pow::GetInstance()->powA( x, 0.5 * x - 0.25 );
116 y = v * (v / y); }
117 else {
118 y = G4Pow::GetInstance()->powA( x, x - 0.5 ) / y;
119 }
120 y = SQTPI * y * w;
121 return( y );
122 }
123 /*
124 ************************************************************
125 */
126 double nf_gammaFunction( double x, nfu_status *status ) {
127
128 double p, q, z;
129 int i, sgngam = 1;
130
131 *status = nfu_badInput;
132 if( !isfinite( x ) ) return( x );
133 *status = nfu_Okay;
134
135 q = fabs( x );
136
137 if( q > 33.0 ) {
138 if( x < 0.0 ) {
139 p = floor( q );
140 if( p == q ) goto goverf;
141 i = (int) p;
142 if( ( i & 1 ) == 0 ) sgngam = -1;
143 z = q - p;
144 if( z > 0.5 ) {
145 p += 1.0;
146 z = q - p;
147 }
148 z = q * sin( M_PI * z );
149 if( z == 0.0 ) goto goverf;
150 z = M_PI / ( fabs( z ) * stirf( q, status ) );
151 }
152 else {
153 z = stirf( x, status );
154 }
155 return( sgngam * z );
156 }
157
158 z = 1.0;
159 while( x >= 3.0 ) {
160 x -= 1.0;
161 z *= x;
162 } // Loop checking, 11.06.2015, T. Koi
163
164 while( x < 0.0 ) {
165 if( x > -1.E-9 ) goto small;
166 z /= x;
167 x += 1.0;
168 } // Loop checking, 11.06.2015, T. Koi
169
170 while( x < 2.0 ) {
171 if( x < 1.e-9 ) goto small;
172 z /= x;
173 x += 1.0;
174 } // Loop checking, 11.06.2015, T. Koi
175
176 if( x == 2.0 ) return( z );
177
178 x -= 2.0;
179 p = nf_polevl( x, P, 6 );
180 q = nf_polevl( x, Q, 7 );
181 return( z * p / q );
182
183 small:
184 if( x == 0.0 ) goto goverf;
185 return( z / ( ( 1.0 + 0.5772156649015329 * x ) * x ) );
186
187 goverf:
188 return( sgngam * DBL_MAX );
189 }
190
191 /* A[]: Stirling's formula expansion of log gamma
192 * B[], C[]: log gamma function between 2 and 3
193 */
194 static double A[] = { 8.11614167470508450300E-4, -5.95061904284301438324E-4, 7.93650340457716943945E-4,
195 -2.77777777730099687205E-3, 8.33333333333331927722E-2 };
196 static double B[] = { -1.37825152569120859100E3, -3.88016315134637840924E4, -3.31612992738871184744E5,
197 -1.16237097492762307383E6, -1.72173700820839662146E6, -8.53555664245765465627E5 };
198 static double C[] = { -3.51815701436523470549E2, -1.70642106651881159223E4, -2.20528590553854454839E5,
199 -1.13933444367982507207E6, -2.53252307177582951285E6, -2.01889141433532773231E6 };
200 static double LS2PI = 0.91893853320467274178; /* log( sqrt( 2*pi ) ) */
201 #define MAXLGM 2.556348e305
202
203 /*
204 ************************************************************
205 */
206 double nf_logGammaFunction( double x, nfu_status *status ) {
207 /* Logarithm of gamma function */
208
209 int sgngam;
210
211 *status = nfu_badInput;
212 if( !isfinite( x ) ) return( x );
213 *status = nfu_Okay;
214 return( lgam( x, &sgngam, status ) );
215 }
216 /*
217 ************************************************************
218 */
219 static double lgam( double x, int *sgngam, nfu_status *status ) {
220
221 double p, q, u, w, z;
222 int i;
223
224 *sgngam = 1;
225
226 if( x < -34.0 ) {
227 q = -x;
228 w = lgam( q, sgngam, status ); /* note this modifies *sgngam! */
229 p = floor( q );
230 if( p == q ) goto lgsing;
231 i = (int) p;
232 if( ( i & 1 ) == 0 ) {
233 *sgngam = -1; }
234 else {
235 *sgngam = 1;
236 }
237 z = q - p;
238 if( z > 0.5 ) {
239 p += 1.0;
240 z = p - q;
241 }
242 z = q * sin( M_PI * z );
243 if( z == 0.0 ) goto lgsing;
244 z = LOGPI - G4Log( z ) - w;
245 return( z );
246 }
247
248 if( x < 13.0 ) {
249 z = 1.0;
250 p = 0.0;
251 u = x;
252 while( u >= 3.0 ) {
253 p -= 1.0;
254 u = x + p;
255 z *= u;
256 } // Loop checking, 11.06.2015, T. Koi
257 while( u < 2.0 ) {
258 if( u == 0.0 ) goto lgsing;
259 z /= u;
260 p += 1.0;
261 u = x + p;
262 } // Loop checking, 11.06.2015, T. Koi
263 if( z < 0.0 ) {
264 *sgngam = -1;
265 z = -z; }
266 else {
267 *sgngam = 1;
268 }
269 if( u == 2.0 ) return( G4Log( z ) );
270 p -= 2.0;
271 x = x + p;
272 p = x * nf_polevl( x, B, 5 ) / nf_p1evl( x, C, 6);
273 return( G4Log( z ) + p );
274 }
275
276 if( x > MAXLGM ) goto lgsing;
277 q = ( x - 0.5 ) * G4Log( x ) - x + LS2PI;
278 if( x > 1.0e8 ) return( q );
279
280 p = 1.0 / ( x * x );
281 if( x >= 1000.0 ) {
282 q += ( ( 7.9365079365079365079365e-4 * p - 2.7777777777777777777778e-3 ) * p + 0.0833333333333333333333 ) / x; }
283 else {
284 q += nf_polevl( p, A, 4 ) / x;
285 }
286 return( q );
287
288 lgsing:
289 return( *sgngam * DBL_MAX );
290 }
291
292 #if defined __cplusplus
293 }
294 #endif
295