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 &#915;(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(&#915;(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    }