[431c9e0] | 1 | /* igam.c |
---|
| 2 | * |
---|
| 3 | * Incomplete gamma integral |
---|
| 4 | * |
---|
| 5 | * |
---|
| 6 | * |
---|
| 7 | * SYNOPSIS: |
---|
| 8 | * |
---|
| 9 | * double a, x, y, igam(); |
---|
| 10 | * |
---|
| 11 | * y = igam( a, x ); |
---|
| 12 | * |
---|
| 13 | * DESCRIPTION: |
---|
| 14 | * |
---|
| 15 | * The function is defined by |
---|
| 16 | * |
---|
| 17 | * x |
---|
| 18 | * - |
---|
| 19 | * 1 | | -t a-1 |
---|
| 20 | * igam(a,x) = ----- | e t dt. |
---|
| 21 | * - | | |
---|
| 22 | * | (a) - |
---|
| 23 | * 0 |
---|
| 24 | * |
---|
| 25 | * |
---|
| 26 | * In this implementation both arguments must be positive. |
---|
| 27 | * The integral is evaluated by either a power series or |
---|
| 28 | * continued fraction expansion, depending on the relative |
---|
| 29 | * values of a and x. |
---|
| 30 | * |
---|
| 31 | * ACCURACY: |
---|
| 32 | * |
---|
| 33 | * Relative error: |
---|
| 34 | * arithmetic domain # trials peak rms |
---|
| 35 | * IEEE 0,30 200000 3.6e-14 2.9e-15 |
---|
| 36 | * IEEE 0,100 300000 9.9e-14 1.5e-14 |
---|
| 37 | */ |
---|
| 38 | /* igamc() |
---|
| 39 | * |
---|
| 40 | * Complemented incomplete gamma integral |
---|
| 41 | * |
---|
| 42 | * |
---|
| 43 | * |
---|
| 44 | * SYNOPSIS: |
---|
| 45 | * |
---|
| 46 | * double a, x, y, igamc(); |
---|
| 47 | * |
---|
| 48 | * y = igamc( a, x ); |
---|
| 49 | * |
---|
| 50 | * DESCRIPTION: |
---|
| 51 | * |
---|
| 52 | * The function is defined by |
---|
| 53 | * |
---|
| 54 | * |
---|
| 55 | * igamc(a,x) = 1 - igam(a,x) |
---|
| 56 | * |
---|
| 57 | * inf. |
---|
| 58 | * - |
---|
| 59 | * 1 | | -t a-1 |
---|
| 60 | * = ----- | e t dt. |
---|
| 61 | * - | | |
---|
| 62 | * | (a) - |
---|
| 63 | * x |
---|
| 64 | * |
---|
| 65 | * |
---|
| 66 | * In this implementation both arguments must be positive. |
---|
| 67 | * The integral is evaluated by either a power series or |
---|
| 68 | * continued fraction expansion, depending on the relative |
---|
| 69 | * values of a and x. |
---|
| 70 | * |
---|
| 71 | * ACCURACY: |
---|
| 72 | * |
---|
| 73 | * Tested at random a, x. |
---|
| 74 | * a x Relative error: |
---|
| 75 | * arithmetic domain domain # trials peak rms |
---|
| 76 | * IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 |
---|
| 77 | * IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 |
---|
| 78 | */ |
---|
| 79 | |
---|
| 80 | /* |
---|
| 81 | Cephes Math Library Release 2.8: June, 2000 |
---|
| 82 | Copyright 1985, 1987, 2000 by Stephen L. Moshier |
---|
| 83 | */ |
---|
| 84 | |
---|
| 85 | #include "mconf.h" |
---|
| 86 | #ifdef ANSIPROT |
---|
| 87 | extern double lgam ( double ); |
---|
| 88 | extern double exp ( double ); |
---|
| 89 | extern double log ( double ); |
---|
| 90 | extern double fabs ( double ); |
---|
| 91 | extern double igam ( double, double ); |
---|
| 92 | extern double igamc ( double, double ); |
---|
| 93 | #else |
---|
| 94 | double lgam(), exp(), log(), fabs(), igam(), igamc(); |
---|
| 95 | #endif |
---|
| 96 | |
---|
| 97 | extern double MACHEP, MAXLOG; |
---|
| 98 | static double big = 4.503599627370496e15; |
---|
| 99 | static double biginv = 2.22044604925031308085e-16; |
---|
| 100 | |
---|
| 101 | double igamc( a, x ) |
---|
| 102 | double a, x; |
---|
| 103 | { |
---|
| 104 | double ans, ax, c, yc, r, t, y, z; |
---|
| 105 | double pk, pkm1, pkm2, qk, qkm1, qkm2; |
---|
| 106 | |
---|
| 107 | if( (x <= 0) || ( a <= 0) ) |
---|
| 108 | return( 1.0 ); |
---|
| 109 | |
---|
| 110 | if( (x < 1.0) || (x < a) ) |
---|
| 111 | return( 1.0 - igam(a,x) ); |
---|
| 112 | |
---|
| 113 | ax = a * log(x) - x - lgam(a); |
---|
| 114 | if( ax < -MAXLOG ) |
---|
| 115 | { |
---|
| 116 | mtherr( "igamc", UNDERFLOW ); |
---|
| 117 | return( 0.0 ); |
---|
| 118 | } |
---|
| 119 | ax = exp(ax); |
---|
| 120 | |
---|
| 121 | /* continued fraction */ |
---|
| 122 | y = 1.0 - a; |
---|
| 123 | z = x + y + 1.0; |
---|
| 124 | c = 0.0; |
---|
| 125 | pkm2 = 1.0; |
---|
| 126 | qkm2 = x; |
---|
| 127 | pkm1 = x + 1.0; |
---|
| 128 | qkm1 = z * x; |
---|
| 129 | ans = pkm1/qkm1; |
---|
| 130 | |
---|
| 131 | do |
---|
| 132 | { |
---|
| 133 | c += 1.0; |
---|
| 134 | y += 1.0; |
---|
| 135 | z += 2.0; |
---|
| 136 | yc = y * c; |
---|
| 137 | pk = pkm1 * z - pkm2 * yc; |
---|
| 138 | qk = qkm1 * z - qkm2 * yc; |
---|
| 139 | if( qk != 0 ) |
---|
| 140 | { |
---|
| 141 | r = pk/qk; |
---|
| 142 | t = fabs( (ans - r)/r ); |
---|
| 143 | ans = r; |
---|
| 144 | } |
---|
| 145 | else |
---|
| 146 | t = 1.0; |
---|
| 147 | pkm2 = pkm1; |
---|
| 148 | pkm1 = pk; |
---|
| 149 | qkm2 = qkm1; |
---|
| 150 | qkm1 = qk; |
---|
| 151 | if( fabs(pk) > big ) |
---|
| 152 | { |
---|
| 153 | pkm2 *= biginv; |
---|
| 154 | pkm1 *= biginv; |
---|
| 155 | qkm2 *= biginv; |
---|
| 156 | qkm1 *= biginv; |
---|
| 157 | } |
---|
| 158 | } |
---|
| 159 | while( t > MACHEP ); |
---|
| 160 | |
---|
| 161 | return( ans * ax ); |
---|
| 162 | } |
---|
| 163 | |
---|
| 164 | |
---|
| 165 | |
---|
| 166 | /* left tail of incomplete gamma function: |
---|
| 167 | * |
---|
| 168 | * inf. k |
---|
| 169 | * a -x - x |
---|
| 170 | * x e > ---------- |
---|
| 171 | * - - |
---|
| 172 | * k=0 | (a+k+1) |
---|
| 173 | * |
---|
| 174 | */ |
---|
| 175 | |
---|
| 176 | double igam( a, x ) |
---|
| 177 | double a, x; |
---|
| 178 | { |
---|
| 179 | double ans, ax, c, r; |
---|
| 180 | |
---|
| 181 | if( (x <= 0) || ( a <= 0) ) |
---|
| 182 | return( 0.0 ); |
---|
| 183 | |
---|
| 184 | if( (x > 1.0) && (x > a ) ) |
---|
| 185 | return( 1.0 - igamc(a,x) ); |
---|
| 186 | |
---|
| 187 | /* Compute x**a * exp(-x) / gamma(a) */ |
---|
| 188 | ax = a * log(x) - x - lgam(a); |
---|
| 189 | if( ax < -MAXLOG ) |
---|
| 190 | { |
---|
| 191 | mtherr( "igam", UNDERFLOW ); |
---|
| 192 | return( 0.0 ); |
---|
| 193 | } |
---|
| 194 | ax = exp(ax); |
---|
| 195 | |
---|
| 196 | /* power series */ |
---|
| 197 | r = a; |
---|
| 198 | c = 1.0; |
---|
| 199 | ans = 1.0; |
---|
| 200 | |
---|
| 201 | do |
---|
| 202 | { |
---|
| 203 | r += 1.0; |
---|
| 204 | c *= x/r; |
---|
| 205 | ans += c; |
---|
| 206 | } |
---|
| 207 | while( c/ans > MACHEP ); |
---|
| 208 | |
---|
| 209 | return( ans * ax/a ); |
---|
| 210 | } |
---|