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: 920558 $ $Date: 2010-03-08 17:57:32 -0500 (Mon, 08 Mar 2010) $ 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 // limits for switching algorithm in digamma 064 /** C limit. */ 065 private static final double C_LIMIT = 49; 066 067 /** S limit. */ 068 private static final double S_LIMIT = 1e-5; 069 070 /** 071 * Default constructor. Prohibit instantiation. 072 */ 073 private Gamma() { 074 super(); 075 } 076 077 /** 078 * Returns the natural logarithm of the gamma function Γ(x). 079 * 080 * The implementation of this method is based on: 081 * <ul> 082 * <li><a href="http://mathworld.wolfram.com/GammaFunction.html"> 083 * Gamma Function</a>, equation (28).</li> 084 * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html"> 085 * Lanczos Approximation</a>, equations (1) through (5).</li> 086 * <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on 087 * the computation of the convergent Lanczos complex Gamma approximation 088 * </a></li> 089 * </ul> 090 * 091 * @param x the value. 092 * @return log(Γ(x)) 093 */ 094 public static double logGamma(double x) { 095 double ret; 096 097 if (Double.isNaN(x) || (x <= 0.0)) { 098 ret = Double.NaN; 099 } else { 100 double g = 607.0 / 128.0; 101 102 double sum = 0.0; 103 for (int i = LANCZOS.length - 1; i > 0; --i) { 104 sum = sum + (LANCZOS[i] / (x + i)); 105 } 106 sum = sum + LANCZOS[0]; 107 108 double tmp = x + g + .5; 109 ret = ((x + .5) * Math.log(tmp)) - tmp + 110 HALF_LOG_2_PI + Math.log(sum / x); 111 } 112 113 return ret; 114 } 115 116 /** 117 * Returns the regularized gamma function P(a, x). 118 * 119 * @param a the a parameter. 120 * @param x the value. 121 * @return the regularized gamma function P(a, x) 122 * @throws MathException if the algorithm fails to converge. 123 */ 124 public static double regularizedGammaP(double a, double x) 125 throws MathException 126 { 127 return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); 128 } 129 130 131 /** 132 * Returns the regularized gamma function P(a, x). 133 * 134 * The implementation of this method is based on: 135 * <ul> 136 * <li> 137 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 138 * Regularized Gamma Function</a>, equation (1).</li> 139 * <li> 140 * <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html"> 141 * Incomplete Gamma Function</a>, equation (4).</li> 142 * <li> 143 * <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html"> 144 * Confluent Hypergeometric Function of the First Kind</a>, equation (1). 145 * </li> 146 * </ul> 147 * 148 * @param a the a parameter. 149 * @param x the value. 150 * @param epsilon When the absolute value of the nth item in the 151 * series is less than epsilon the approximation ceases 152 * to calculate further elements in the series. 153 * @param maxIterations Maximum number of "iterations" to complete. 154 * @return the regularized gamma function P(a, x) 155 * @throws MathException if the algorithm fails to converge. 156 */ 157 public static double regularizedGammaP(double a, 158 double x, 159 double epsilon, 160 int maxIterations) 161 throws MathException 162 { 163 double ret; 164 165 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) { 166 ret = Double.NaN; 167 } else if (x == 0.0) { 168 ret = 0.0; 169 } else if (x >= a + 1) { 170 // use regularizedGammaQ because it should converge faster in this 171 // case. 172 ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations); 173 } else { 174 // calculate series 175 double n = 0.0; // current element index 176 double an = 1.0 / a; // n-th element in the series 177 double sum = an; // partial sum 178 while (Math.abs(an/sum) > epsilon && n < maxIterations && sum < Double.POSITIVE_INFINITY) { 179 // compute next element in the series 180 n = n + 1.0; 181 an = an * (x / (a + n)); 182 183 // update partial sum 184 sum = sum + an; 185 } 186 if (n >= maxIterations) { 187 throw new MaxIterationsExceededException(maxIterations); 188 } else if (Double.isInfinite(sum)) { 189 ret = 1.0; 190 } else { 191 ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * sum; 192 } 193 } 194 195 return ret; 196 } 197 198 /** 199 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x). 200 * 201 * @param a the a parameter. 202 * @param x the value. 203 * @return the regularized gamma function Q(a, x) 204 * @throws MathException if the algorithm fails to converge. 205 */ 206 public static double regularizedGammaQ(double a, double x) 207 throws MathException 208 { 209 return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE); 210 } 211 212 /** 213 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x). 214 * 215 * The implementation of this method is based on: 216 * <ul> 217 * <li> 218 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html"> 219 * Regularized Gamma Function</a>, equation (1).</li> 220 * <li> 221 * <a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/"> 222 * Regularized incomplete gamma function: Continued fraction representations (formula 06.08.10.0003)</a></li> 223 * </ul> 224 * 225 * @param a the a parameter. 226 * @param x the value. 227 * @param epsilon When the absolute value of the nth item in the 228 * series is less than epsilon the approximation ceases 229 * to calculate further elements in the series. 230 * @param maxIterations Maximum number of "iterations" to complete. 231 * @return the regularized gamma function P(a, x) 232 * @throws MathException if the algorithm fails to converge. 233 */ 234 public static double regularizedGammaQ(final double a, 235 double x, 236 double epsilon, 237 int maxIterations) 238 throws MathException 239 { 240 double ret; 241 242 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) { 243 ret = Double.NaN; 244 } else if (x == 0.0) { 245 ret = 1.0; 246 } else if (x < a + 1.0) { 247 // use regularizedGammaP because it should converge faster in this 248 // case. 249 ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations); 250 } else { 251 // create continued fraction 252 ContinuedFraction cf = new ContinuedFraction() { 253 254 @Override 255 protected double getA(int n, double x) { 256 return ((2.0 * n) + 1.0) - a + x; 257 } 258 259 @Override 260 protected double getB(int n, double x) { 261 return n * (a - n); 262 } 263 }; 264 265 ret = 1.0 / cf.evaluate(x, epsilon, maxIterations); 266 ret = Math.exp(-x + (a * Math.log(x)) - logGamma(a)) * ret; 267 } 268 269 return ret; 270 } 271 272 273 /** 274 * <p>Computes the digamma function of x.</p> 275 * 276 * <p>This is an independently written implementation of the algorithm described in 277 * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p> 278 * 279 * <p>Some of the constants have been changed to increase accuracy at the moderate expense 280 * of run-time. The result should be accurate to within 10^-8 absolute tolerance for 281 * x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p> 282 * 283 * <p>Performance for large negative values of x will be quite expensive (proportional to 284 * |x|). Accuracy for negative values of x should be about 10^-8 absolute for results 285 * less than 10^5 and 10^-8 relative for results larger than that.</p> 286 * 287 * @param x the argument 288 * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller 289 * @see <a href="http://en.wikipedia.org/wiki/Digamma_function"> Digamma at wikipedia </a> 290 * @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf"> Bernardo's original article </a> 291 * @since 2.0 292 */ 293 public static double digamma(double x) { 294 if (x > 0 && x <= S_LIMIT) { 295 // use method 5 from Bernardo AS103 296 // accurate to O(x) 297 return -GAMMA - 1 / x; 298 } 299 300 if (x >= C_LIMIT) { 301 // use method 4 (accurate to O(1/x^8) 302 double inv = 1 / (x * x); 303 // 1 1 1 1 304 // log(x) - --- - ------ + ------- - ------- 305 // 2 x 12 x^2 120 x^4 252 x^6 306 return Math.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252)); 307 } 308 309 return digamma(x + 1) - 1 / x; 310 } 311 312 /** 313 * <p>Computes the trigamma function of x. This function is derived by taking the derivative of 314 * the implementation of digamma.</p> 315 * 316 * @param x the argument 317 * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller 318 * @see <a href="http://en.wikipedia.org/wiki/Trigamma_function"> Trigamma at wikipedia </a> 319 * @see Gamma#digamma(double) 320 * @since 2.0 321 */ 322 public static double trigamma(double x) { 323 if (x > 0 && x <= S_LIMIT) { 324 return 1 / (x * x); 325 } 326 327 if (x >= C_LIMIT) { 328 double inv = 1 / (x * x); 329 // 1 1 1 1 1 330 // - + ---- + ---- - ----- + ----- 331 // x 2 3 5 7 332 // 2 x 6 x 30 x 42 x 333 return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42)); 334 } 335 336 return trigamma(x + 1) + 1 / (x * x); 337 } 338 }