001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math.special; 018 019 import org.apache.commons.math.MathException; 020 import org.apache.commons.math.MaxIterationsExceededException; 021 import org.apache.commons.math.util.ContinuedFraction; 022 023 /** 024 * This is a utility class that provides computation methods related to the 025 * Gamma family of functions. 026 * 027 * @version $Revision: 780975 $ $Date: 2009-06-02 05:05:37 -0400 (Tue, 02 Jun 2009) $ 028 */ 029 public class Gamma { 030 031 /** 032 * <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a> 033 * @since 2.0 034 */ 035 public static final double GAMMA = 0.577215664901532860606512090082; 036 037 /** Maximum allowed numerical error. */ 038 private static final double DEFAULT_EPSILON = 10e-15; 039 040 /** Lanczos coefficients */ 041 private static final double[] lanczos = 042 { 043 0.99999999999999709182, 044 57.156235665862923517, 045 -59.597960355475491248, 046 14.136097974741747174, 047 -0.49191381609762019978, 048 .33994649984811888699e-4, 049 .46523628927048575665e-4, 050 -.98374475304879564677e-4, 051 .15808870322491248884e-3, 052 -.21026444172410488319e-3, 053 .21743961811521264320e-3, 054 -.16431810653676389022e-3, 055 .84418223983852743293e-4, 056 -.26190838401581408670e-4, 057 .36899182659531622704e-5, 058 }; 059 060 /** Avoid repeated computation of log of 2 PI in logGamma */ 061 private static final double HALF_LOG_2_PI = 0.5 * Math.log(2.0 * Math.PI); 062 063 064 /** 065 * Default constructor. Prohibit instantiation. 066 */ 067 private Gamma() { 068 super(); 069 } 070 071 /** 072 * Returns the natural logarithm of the gamma function Γ(x). 073 * 074 * The implementation of this method is based on: 075 * <ul> 076 * <li><a href="http://mathworld.wolfram.com/GammaFunction.html"> 077 * Gamma Function</a>, equation (28).</li> 078 * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html"> 079 * Lanczos Approximation</a>, equations (1) through (5).</li> 080 * <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on 081 * the computation of the convergent Lanczos complex Gamma approximation 082 * </a></li> 083 * </ul> 084 * 085 * @param x the value. 086 * @return log(Γ(x)) 087 */ 088 public static double logGamma(double x) { 089 double ret; 090 091 if (Double.isNaN(x) || (x <= 0.0)) { 092 ret = Double.NaN; 093 } else { 094 double g = 607.0 / 128.0; 095 096 double sum = 0.0; 097 for (int i = lanczos.length - 1; i > 0; --i) { 098 sum = sum + (lanczos[i] / (x + i)); 099 } 100 sum = sum + lanczos[0]; 101 102 double tmp = x + g + .5; 103 ret = ((x + .5) * Math.log(tmp)) - tmp + 104 HALF_LOG_2_PI + Math.log(sum / x); 105 } 106 107 return ret; 108 } 109 110 /** 111 * Returns the regularized gamma function P(a, x). 112 * 113 * @param a the a parameter. 114 * @param x the value. 115 * @return the regularized gamma function P(a, x) 116 * @throws MathException if the algorithm fails to converge. 117 */ 118 public static double regularizedGammaP(double a, double x) 119 throws MathException 120 { 121 return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); 122 } 123 124 125 /** 126 * Returns the regularized gamma function P(a, x). 127 * 128 * The implementation of this method is based on: 129 * <ul> 130 * <li> 131 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 132 * Regularized Gamma Function</a>, equation (1).</li> 133 * <li> 134 * <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html"> 135 * Incomplete Gamma Function</a>, equation (4).</li> 136 * <li> 137 * <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html"> 138 * Confluent Hypergeometric Function of the First Kind</a>, equation (1). 139 * </li> 140 * </ul> 141 * 142 * @param a the a parameter. 143 * @param x the value. 144 * @param epsilon When the absolute value of the nth item in the 145 * series is less than epsilon the approximation ceases 146 * to calculate further elements in the series. 147 * @param maxIterations Maximum number of "iterations" to complete. 148 * @return the regularized gamma function P(a, x) 149 * @throws MathException if the algorithm fails to converge. 150 */ 151 public static double regularizedGammaP(double a, 152 double x, 153 double epsilon, 154 int maxIterations) 155 throws MathException 156 { 157 double ret; 158 159 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) { 160 ret = Double.NaN; 161 } else if (x == 0.0) { 162 ret = 0.0; 163 } else if (a >= 1.0 && x > a) { 164 // use regularizedGammaQ because it should converge faster in this 165 // case. 166 ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations); 167 } else { 168 // calculate series 169 double n = 0.0; // current element index 170 double an = 1.0 / a; // n-th element in the series 171 double sum = an; // partial sum 172 while (Math.abs(an) > epsilon && n < maxIterations) { 173 // compute next element in the series 174 n = n + 1.0; 175 an = an * (x / (a + n)); 176 177 // update partial sum 178 sum = sum + an; 179 } 180 if (n >= maxIterations) { 181 throw new MaxIterationsExceededException(maxIterations); 182 } else { 183 ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * sum; 184 } 185 } 186 187 return ret; 188 } 189 190 /** 191 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x). 192 * 193 * @param a the a parameter. 194 * @param x the value. 195 * @return the regularized gamma function Q(a, x) 196 * @throws MathException if the algorithm fails to converge. 197 */ 198 public static double regularizedGammaQ(double a, double x) 199 throws MathException 200 { 201 return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); 202 } 203 204 /** 205 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x). 206 * 207 * The implementation of this method is based on: 208 * <ul> 209 * <li> 210 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 211 * Regularized Gamma Function</a>, equation (1).</li> 212 * <li> 213 * <a href=" http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/"> 214 * Regularized incomplete gamma function: Continued fraction representations (formula 06.08.10.0003)</a></li> 215 * </ul> 216 * 217 * @param a the a parameter. 218 * @param x the value. 219 * @param epsilon When the absolute value of the nth item in the 220 * series is less than epsilon the approximation ceases 221 * to calculate further elements in the series. 222 * @param maxIterations Maximum number of "iterations" to complete. 223 * @return the regularized gamma function P(a, x) 224 * @throws MathException if the algorithm fails to converge. 225 */ 226 public static double regularizedGammaQ(final double a, 227 double x, 228 double epsilon, 229 int maxIterations) 230 throws MathException 231 { 232 double ret; 233 234 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) { 235 ret = Double.NaN; 236 } else if (x == 0.0) { 237 ret = 1.0; 238 } else if (x < a || a < 1.0) { 239 // use regularizedGammaP because it should converge faster in this 240 // case. 241 ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations); 242 } else { 243 // create continued fraction 244 ContinuedFraction cf = new ContinuedFraction() { 245 246 @Override 247 protected double getA(int n, double x) { 248 return ((2.0 * n) + 1.0) - a + x; 249 } 250 251 @Override 252 protected double getB(int n, double x) { 253 return n * (a - n); 254 } 255 }; 256 257 ret = 1.0 / cf.evaluate(x, epsilon, maxIterations); 258 ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * ret; 259 } 260 261 return ret; 262 } 263 264 265 // limits for switching algorithm in digamma 266 /** C limit */ 267 private static final double C_LIMIT = 49; 268 /** S limit */ 269 private static final double S_LIMIT = 1e-5; 270 271 /** 272 * <p>Computes the digamma function of x.</p> 273 * 274 * <p>This is an independently written implementation of the algorithm described in 275 * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p> 276 * 277 * <p>Some of the constants have been changed to increase accuracy at the moderate expense 278 * of run-time. The result should be accurate to within 10^-8 absolute tolerance for 279 * x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p> 280 * 281 * <p>Performance for large negative values of x will be quite expensive (proportional to 282 * |x|). Accuracy for negative values of x should be about 10^-8 absolute for results 283 * less than 10^5 and 10^-8 relative for results larger than that.</p> 284 * 285 * @param x the argument 286 * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller 287 * @see <a href="http://en.wikipedia.org/wiki/Digamma_function"> Digamma at wikipedia </a> 288 * @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf"> Bernardo's original article </a> 289 * @since 2.0 290 */ 291 public static double digamma(double x) { 292 if (x > 0 && x <= S_LIMIT) { 293 // use method 5 from Bernardo AS103 294 // accurate to O(x) 295 return -GAMMA - 1 / x; 296 } 297 298 if (x >= C_LIMIT) { 299 // use method 4 (accurate to O(1/x^8) 300 double inv = 1 / (x * x); 301 // 1 1 1 1 302 // log(x) - --- - ------ + ------- - ------- 303 // 2 x 12 x^2 120 x^4 252 x^6 304 return Math.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252)); 305 } 306 307 return digamma(x + 1) - 1 / x; 308 } 309 310 /** 311 * <p>Computes the trigamma function of x. This function is derived by taking the derivative of 312 * the implementation of digamma.</p> 313 * 314 * @param x the argument 315 * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller 316 * @see <a href="http://en.wikipedia.org/wiki/Trigamma_function"> Trigamma at wikipedia </a> 317 * @see Gamma#digamma(double) 318 * @since 2.0 319 */ 320 public static double trigamma(double x) { 321 if (x > 0 && x <= S_LIMIT) { 322 return 1 / (x * x); 323 } 324 325 if (x >= C_LIMIT) { 326 double inv = 1 / (x * x); 327 // 1 1 1 1 1 328 // - + ---- + ---- - ----- + ----- 329 // x 2 3 5 7 330 // 2 x 6 x 30 x 42 x 331 return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42)); 332 } 333 334 return trigamma(x + 1) + 1 / (x * x); 335 } 336 }