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.math3.special;
018    
019    import org.apache.commons.math3.exception.MaxCountExceededException;
020    import org.apache.commons.math3.exception.NumberIsTooLargeException;
021    import org.apache.commons.math3.exception.NumberIsTooSmallException;
022    import org.apache.commons.math3.util.ContinuedFraction;
023    import org.apache.commons.math3.util.FastMath;
024    
025    /**
026     * <p>
027     * This is a utility class that provides computation methods related to the
028     * &Gamma; (Gamma) family of functions.
029     * </p>
030     * <p>
031     * Implementation of {@link #invGamma1pm1(double)} and
032     * {@link #logGamma1p(double)} is based on the algorithms described in
033     * <ul>
034     * <li><a href="http://dx.doi.org/10.1145/22721.23109">Didonato and Morris
035     * (1986)</a>, <em>Computation of the Incomplete Gamma Function Ratios and
036     *     their Inverse</em>, TOMS 12(4), 377-393,</li>
037     * <li><a href="http://dx.doi.org/10.1145/131766.131776">Didonato and Morris
038     * (1992)</a>, <em>Algorithm 708: Significant Digit Computation of the
039     *     Incomplete Beta Function Ratios</em>, TOMS 18(3), 360-373,</li>
040     * </ul>
041     * and implemented in the
042     * <a href="http://www.dtic.mil/docs/citations/ADA476840">NSWC Library of Mathematical Functions</a>,
043     * available
044     * <a href="http://www.ualberta.ca/CNS/RESEARCH/Software/NumericalNSWC/site.html">here</a>.
045     * This library is "approved for public release", and the
046     * <a href="http://www.dtic.mil/dtic/pdf/announcements/CopyrightGuidance.pdf">Copyright guidance</a>
047     * indicates that unless otherwise stated in the code, all FORTRAN functions in
048     * this library are license free. Since no such notice appears in the code these
049     * functions can safely be ported to Commons-Math.
050     * </p>
051     *
052     * @version $Id: Gamma.java 1422313 2012-12-15 18:53:41Z psteitz $
053     */
054    public class Gamma {
055        /**
056         * <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
057         * @since 2.0
058         */
059        public static final double GAMMA = 0.577215664901532860606512090082;
060    
061        /**
062         * The value of the {@code g} constant in the Lanczos approximation, see
063         * {@link #lanczos(double)}.
064         * @since 3.1
065         */
066        public static final double LANCZOS_G = 607.0 / 128.0;
067    
068        /** Maximum allowed numerical error. */
069        private static final double DEFAULT_EPSILON = 10e-15;
070    
071        /** Lanczos coefficients */
072        private static final double[] LANCZOS = {
073            0.99999999999999709182,
074            57.156235665862923517,
075            -59.597960355475491248,
076            14.136097974741747174,
077            -0.49191381609762019978,
078            .33994649984811888699e-4,
079            .46523628927048575665e-4,
080            -.98374475304879564677e-4,
081            .15808870322491248884e-3,
082            -.21026444172410488319e-3,
083            .21743961811521264320e-3,
084            -.16431810653676389022e-3,
085            .84418223983852743293e-4,
086            -.26190838401581408670e-4,
087            .36899182659531622704e-5,
088        };
089    
090        /** Avoid repeated computation of log of 2 PI in logGamma */
091        private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
092    
093        /** The constant value of &radic;(2&pi;). */
094        private static final double SQRT_TWO_PI = 2.506628274631000502;
095    
096        // limits for switching algorithm in digamma
097        /** C limit. */
098        private static final double C_LIMIT = 49;
099    
100        /** S limit. */
101        private static final double S_LIMIT = 1e-5;
102    
103        /*
104         * Constants for the computation of double invGamma1pm1(double).
105         * Copied from DGAM1 in the NSWC library.
106         */
107    
108        /** The constant {@code A0} defined in {@code DGAM1}. */
109        private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;
110    
111        /** The constant {@code A1} defined in {@code DGAM1}. */
112        private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;
113    
114        /** The constant {@code B1} defined in {@code DGAM1}. */
115        private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;
116    
117        /** The constant {@code B2} defined in {@code DGAM1}. */
118        private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;
119    
120        /** The constant {@code B3} defined in {@code DGAM1}. */
121        private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;
122    
123        /** The constant {@code B4} defined in {@code DGAM1}. */
124        private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;
125    
126        /** The constant {@code B5} defined in {@code DGAM1}. */
127        private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;
128    
129        /** The constant {@code B6} defined in {@code DGAM1}. */
130        private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;
131    
132        /** The constant {@code B7} defined in {@code DGAM1}. */
133        private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;
134    
135        /** The constant {@code B8} defined in {@code DGAM1}. */
136        private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;
137    
138        /** The constant {@code P0} defined in {@code DGAM1}. */
139        private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;
140    
141        /** The constant {@code P1} defined in {@code DGAM1}. */
142        private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;
143    
144        /** The constant {@code P2} defined in {@code DGAM1}. */
145        private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;
146    
147        /** The constant {@code P3} defined in {@code DGAM1}. */
148        private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;
149    
150        /** The constant {@code P4} defined in {@code DGAM1}. */
151        private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;
152    
153        /** The constant {@code P5} defined in {@code DGAM1}. */
154        private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;
155    
156        /** The constant {@code P6} defined in {@code DGAM1}. */
157        private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;
158    
159        /** The constant {@code Q1} defined in {@code DGAM1}. */
160        private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;
161    
162        /** The constant {@code Q2} defined in {@code DGAM1}. */
163        private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;
164    
165        /** The constant {@code Q3} defined in {@code DGAM1}. */
166        private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;
167    
168        /** The constant {@code Q4} defined in {@code DGAM1}. */
169        private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;
170    
171        /** The constant {@code C} defined in {@code DGAM1}. */
172        private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;
173    
174        /** The constant {@code C0} defined in {@code DGAM1}. */
175        private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;
176    
177        /** The constant {@code C1} defined in {@code DGAM1}. */
178        private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;
179    
180        /** The constant {@code C2} defined in {@code DGAM1}. */
181        private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;
182    
183        /** The constant {@code C3} defined in {@code DGAM1}. */
184        private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;
185    
186        /** The constant {@code C4} defined in {@code DGAM1}. */
187        private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;
188    
189        /** The constant {@code C5} defined in {@code DGAM1}. */
190        private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;
191    
192        /** The constant {@code C6} defined in {@code DGAM1}. */
193        private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;
194    
195        /** The constant {@code C7} defined in {@code DGAM1}. */
196        private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;
197    
198        /** The constant {@code C8} defined in {@code DGAM1}. */
199        private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;
200    
201        /** The constant {@code C9} defined in {@code DGAM1}. */
202        private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;
203    
204        /** The constant {@code C10} defined in {@code DGAM1}. */
205        private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;
206    
207        /** The constant {@code C11} defined in {@code DGAM1}. */
208        private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;
209    
210        /** The constant {@code C12} defined in {@code DGAM1}. */
211        private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;
212    
213        /** The constant {@code C13} defined in {@code DGAM1}. */
214        private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;
215    
216        /**
217         * Default constructor.  Prohibit instantiation.
218         */
219        private Gamma() {}
220    
221        /**
222         * <p>
223         * Returns the value of log&nbsp;&Gamma;(x) for x&nbsp;&gt;&nbsp;0.
224         * </p>
225         * <p>
226         * For x &le; 8, the implementation is based on the double precision
227         * implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
228         * {@code DGAMLN}. For x &gt; 8, the implementation is based on
229         * </p>
230         * <ul>
231         * <li><a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma
232         *     Function</a>, equation (28).</li>
233         * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html">
234         *     Lanczos Approximation</a>, equations (1) through (5).</li>
235         * <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on
236         *     the computation of the convergent Lanczos complex Gamma
237         *     approximation</a></li>
238         * </ul>
239         *
240         * @param x Argument.
241         * @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
242         * {@code x <= 0.0}.
243         */
244        public static double logGamma(double x) {
245            double ret;
246    
247            if (Double.isNaN(x) || (x <= 0.0)) {
248                ret = Double.NaN;
249            } else if (x < 0.5) {
250                return logGamma1p(x) - FastMath.log(x);
251            } else if (x <= 2.5) {
252                return logGamma1p((x - 0.5) - 0.5);
253            } else if (x <= 8.0) {
254                final int n = (int) FastMath.floor(x - 1.5);
255                double prod = 1.0;
256                for (int i = 1; i <= n; i++) {
257                    prod *= x - i;
258                }
259                return logGamma1p(x - (n + 1)) + FastMath.log(prod);
260            } else {
261                double sum = lanczos(x);
262                double tmp = x + LANCZOS_G + .5;
263                ret = ((x + .5) * FastMath.log(tmp)) - tmp +
264                    HALF_LOG_2_PI + FastMath.log(sum / x);
265            }
266    
267            return ret;
268        }
269    
270        /**
271         * Returns the regularized gamma function P(a, x).
272         *
273         * @param a Parameter.
274         * @param x Value.
275         * @return the regularized gamma function P(a, x).
276         * @throws MaxCountExceededException if the algorithm fails to converge.
277         */
278        public static double regularizedGammaP(double a, double x) {
279            return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
280        }
281    
282        /**
283         * Returns the regularized gamma function P(a, x).
284         *
285         * The implementation of this method is based on:
286         * <ul>
287         *  <li>
288         *   <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
289         *   Regularized Gamma Function</a>, equation (1)
290         *  </li>
291         *  <li>
292         *   <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">
293         *   Incomplete Gamma Function</a>, equation (4).
294         *  </li>
295         *  <li>
296         *   <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html">
297         *   Confluent Hypergeometric Function of the First Kind</a>, equation (1).
298         *  </li>
299         * </ul>
300         *
301         * @param a the a parameter.
302         * @param x the value.
303         * @param epsilon When the absolute value of the nth item in the
304         * series is less than epsilon the approximation ceases to calculate
305         * further elements in the series.
306         * @param maxIterations Maximum number of "iterations" to complete.
307         * @return the regularized gamma function P(a, x)
308         * @throws MaxCountExceededException if the algorithm fails to converge.
309         */
310        public static double regularizedGammaP(double a,
311                                               double x,
312                                               double epsilon,
313                                               int maxIterations) {
314            double ret;
315    
316            if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
317                ret = Double.NaN;
318            } else if (x == 0.0) {
319                ret = 0.0;
320            } else if (x >= a + 1) {
321                // use regularizedGammaQ because it should converge faster in this
322                // case.
323                ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);
324            } else {
325                // calculate series
326                double n = 0.0; // current element index
327                double an = 1.0 / a; // n-th element in the series
328                double sum = an; // partial sum
329                while (FastMath.abs(an/sum) > epsilon &&
330                       n < maxIterations &&
331                       sum < Double.POSITIVE_INFINITY) {
332                    // compute next element in the series
333                    n = n + 1.0;
334                    an = an * (x / (a + n));
335    
336                    // update partial sum
337                    sum = sum + an;
338                }
339                if (n >= maxIterations) {
340                    throw new MaxCountExceededException(maxIterations);
341                } else if (Double.isInfinite(sum)) {
342                    ret = 1.0;
343                } else {
344                    ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;
345                }
346            }
347    
348            return ret;
349        }
350    
351        /**
352         * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
353         *
354         * @param a the a parameter.
355         * @param x the value.
356         * @return the regularized gamma function Q(a, x)
357         * @throws MaxCountExceededException if the algorithm fails to converge.
358         */
359        public static double regularizedGammaQ(double a, double x) {
360            return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
361        }
362    
363        /**
364         * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
365         *
366         * The implementation of this method is based on:
367         * <ul>
368         *  <li>
369         *   <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
370         *   Regularized Gamma Function</a>, equation (1).
371         *  </li>
372         *  <li>
373         *   <a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/">
374         *   Regularized incomplete gamma function: Continued fraction representations
375         *   (formula 06.08.10.0003)</a>
376         *  </li>
377         * </ul>
378         *
379         * @param a the a parameter.
380         * @param x the value.
381         * @param epsilon When the absolute value of the nth item in the
382         * series is less than epsilon the approximation ceases to calculate
383         * further elements in the series.
384         * @param maxIterations Maximum number of "iterations" to complete.
385         * @return the regularized gamma function P(a, x)
386         * @throws MaxCountExceededException if the algorithm fails to converge.
387         */
388        public static double regularizedGammaQ(final double a,
389                                               double x,
390                                               double epsilon,
391                                               int maxIterations) {
392            double ret;
393    
394            if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
395                ret = Double.NaN;
396            } else if (x == 0.0) {
397                ret = 1.0;
398            } else if (x < a + 1.0) {
399                // use regularizedGammaP because it should converge faster in this
400                // case.
401                ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);
402            } else {
403                // create continued fraction
404                ContinuedFraction cf = new ContinuedFraction() {
405    
406                    @Override
407                    protected double getA(int n, double x) {
408                        return ((2.0 * n) + 1.0) - a + x;
409                    }
410    
411                    @Override
412                    protected double getB(int n, double x) {
413                        return n * (a - n);
414                    }
415                };
416    
417                ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);
418                ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;
419            }
420    
421            return ret;
422        }
423    
424    
425        /**
426         * <p>Computes the digamma function of x.</p>
427         *
428         * <p>This is an independently written implementation of the algorithm described in
429         * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p>
430         *
431         * <p>Some of the constants have been changed to increase accuracy at the moderate expense
432         * of run-time.  The result should be accurate to within 10^-8 absolute tolerance for
433         * x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p>
434         *
435         * <p>Performance for large negative values of x will be quite expensive (proportional to
436         * |x|).  Accuracy for negative values of x should be about 10^-8 absolute for results
437         * less than 10^5 and 10^-8 relative for results larger than that.</p>
438         *
439         * @param x Argument.
440         * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller.
441         * @see <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma</a>
442         * @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Bernardo&apos;s original article </a>
443         * @since 2.0
444         */
445        public static double digamma(double x) {
446            if (x > 0 && x <= S_LIMIT) {
447                // use method 5 from Bernardo AS103
448                // accurate to O(x)
449                return -GAMMA - 1 / x;
450            }
451    
452            if (x >= C_LIMIT) {
453                // use method 4 (accurate to O(1/x^8)
454                double inv = 1 / (x * x);
455                //            1       1        1         1
456                // log(x) -  --- - ------ + ------- - -------
457                //           2 x   12 x^2   120 x^4   252 x^6
458                return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));
459            }
460    
461            return digamma(x + 1) - 1 / x;
462        }
463    
464        /**
465         * Computes the trigamma function of x.
466         * This function is derived by taking the derivative of the implementation
467         * of digamma.
468         *
469         * @param x Argument.
470         * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller
471         * @see <a href="http://en.wikipedia.org/wiki/Trigamma_function">Trigamma</a>
472         * @see Gamma#digamma(double)
473         * @since 2.0
474         */
475        public static double trigamma(double x) {
476            if (x > 0 && x <= S_LIMIT) {
477                return 1 / (x * x);
478            }
479    
480            if (x >= C_LIMIT) {
481                double inv = 1 / (x * x);
482                //  1    1      1       1       1
483                //  - + ---- + ---- - ----- + -----
484                //  x      2      3       5       7
485                //      2 x    6 x    30 x    42 x
486                return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));
487            }
488    
489            return trigamma(x + 1) + 1 / (x * x);
490        }
491    
492        /**
493         * <p>
494         * Returns the Lanczos approximation used to compute the gamma function.
495         * The Lanczos approximation is related to the Gamma function by the
496         * following equation
497         * <center>
498         * {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)
499         *                   * exp(-x - g - 0.5) * lanczos(x)},
500         * </center>
501         * where {@code g} is the Lanczos constant.
502         * </p>
503         *
504         * @param x Argument.
505         * @return The Lanczos approximation.
506         * @see <a href="http://mathworld.wolfram.com/LanczosApproximation.html">Lanczos Approximation</a>
507         * equations (1) through (5), and Paul Godfrey's
508         * <a href="http://my.fit.edu/~gabdo/gamma.txt">Note on the computation
509         * of the convergent Lanczos complex Gamma approximation</a>
510         * @since 3.1
511         */
512        public static double lanczos(final double x) {
513            double sum = 0.0;
514            for (int i = LANCZOS.length - 1; i > 0; --i) {
515                sum = sum + (LANCZOS[i] / (x + i));
516            }
517            return sum + LANCZOS[0];
518        }
519    
520        /**
521         * Returns the value of 1 / &Gamma;(1 + x) - 1 for -0&#46;5 &le; x &le;
522         * 1&#46;5. This implementation is based on the double precision
523         * implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
524         * {@code DGAM1}.
525         *
526         * @param x Argument.
527         * @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
528         * @throws NumberIsTooSmallException if {@code x < -0.5}
529         * @throws NumberIsTooLargeException if {@code x > 1.5}
530         * @since 3.1
531         */
532        public static double invGamma1pm1(final double x) {
533    
534            if (x < -0.5) {
535                throw new NumberIsTooSmallException(x, -0.5, true);
536            }
537            if (x > 1.5) {
538                throw new NumberIsTooLargeException(x, 1.5, true);
539            }
540    
541            final double ret;
542            final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
543            if (t < 0.0) {
544                final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;
545                double b = INV_GAMMA1P_M1_B8;
546                b = INV_GAMMA1P_M1_B7 + t * b;
547                b = INV_GAMMA1P_M1_B6 + t * b;
548                b = INV_GAMMA1P_M1_B5 + t * b;
549                b = INV_GAMMA1P_M1_B4 + t * b;
550                b = INV_GAMMA1P_M1_B3 + t * b;
551                b = INV_GAMMA1P_M1_B2 + t * b;
552                b = INV_GAMMA1P_M1_B1 + t * b;
553                b = 1.0 + t * b;
554    
555                double c = INV_GAMMA1P_M1_C13 + t * (a / b);
556                c = INV_GAMMA1P_M1_C12 + t * c;
557                c = INV_GAMMA1P_M1_C11 + t * c;
558                c = INV_GAMMA1P_M1_C10 + t * c;
559                c = INV_GAMMA1P_M1_C9 + t * c;
560                c = INV_GAMMA1P_M1_C8 + t * c;
561                c = INV_GAMMA1P_M1_C7 + t * c;
562                c = INV_GAMMA1P_M1_C6 + t * c;
563                c = INV_GAMMA1P_M1_C5 + t * c;
564                c = INV_GAMMA1P_M1_C4 + t * c;
565                c = INV_GAMMA1P_M1_C3 + t * c;
566                c = INV_GAMMA1P_M1_C2 + t * c;
567                c = INV_GAMMA1P_M1_C1 + t * c;
568                c = INV_GAMMA1P_M1_C + t * c;
569                if (x > 0.5) {
570                    ret = t * c / x;
571                } else {
572                    ret = x * ((c + 0.5) + 0.5);
573                }
574            } else {
575                double p = INV_GAMMA1P_M1_P6;
576                p = INV_GAMMA1P_M1_P5 + t * p;
577                p = INV_GAMMA1P_M1_P4 + t * p;
578                p = INV_GAMMA1P_M1_P3 + t * p;
579                p = INV_GAMMA1P_M1_P2 + t * p;
580                p = INV_GAMMA1P_M1_P1 + t * p;
581                p = INV_GAMMA1P_M1_P0 + t * p;
582    
583                double q = INV_GAMMA1P_M1_Q4;
584                q = INV_GAMMA1P_M1_Q3 + t * q;
585                q = INV_GAMMA1P_M1_Q2 + t * q;
586                q = INV_GAMMA1P_M1_Q1 + t * q;
587                q = 1.0 + t * q;
588    
589                double c = INV_GAMMA1P_M1_C13 + (p / q) * t;
590                c = INV_GAMMA1P_M1_C12 + t * c;
591                c = INV_GAMMA1P_M1_C11 + t * c;
592                c = INV_GAMMA1P_M1_C10 + t * c;
593                c = INV_GAMMA1P_M1_C9 + t * c;
594                c = INV_GAMMA1P_M1_C8 + t * c;
595                c = INV_GAMMA1P_M1_C7 + t * c;
596                c = INV_GAMMA1P_M1_C6 + t * c;
597                c = INV_GAMMA1P_M1_C5 + t * c;
598                c = INV_GAMMA1P_M1_C4 + t * c;
599                c = INV_GAMMA1P_M1_C3 + t * c;
600                c = INV_GAMMA1P_M1_C2 + t * c;
601                c = INV_GAMMA1P_M1_C1 + t * c;
602                c = INV_GAMMA1P_M1_C0 + t * c;
603    
604                if (x > 0.5) {
605                    ret = (t / x) * ((c - 0.5) - 0.5);
606                } else {
607                    ret = x * c;
608                }
609            }
610    
611            return ret;
612        }
613    
614        /**
615         * Returns the value of log &Gamma;(1 + x) for -0&#46;5 &le; x &le; 1&#46;5.
616         * This implementation is based on the double precision implementation in
617         * the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
618         *
619         * @param x Argument.
620         * @return The value of {@code log(Gamma(1 + x))}.
621         * @throws NumberIsTooSmallException if {@code x < -0.5}.
622         * @throws NumberIsTooLargeException if {@code x > 1.5}.
623         * @since 3.1
624         */
625        public static double logGamma1p(final double x)
626            throws NumberIsTooSmallException, NumberIsTooLargeException {
627    
628            if (x < -0.5) {
629                throw new NumberIsTooSmallException(x, -0.5, true);
630            }
631            if (x > 1.5) {
632                throw new NumberIsTooLargeException(x, 1.5, true);
633            }
634    
635            return -FastMath.log1p(invGamma1pm1(x));
636        }
637    
638    
639        /**
640         * Returns the value of Γ(x). Based on the <em>NSWC Library of
641         * Mathematics Subroutines</em> double precision implementation,
642         * {@code DGAMMA}.
643         *
644         * @param x Argument.
645         * @return the value of {@code Gamma(x)}.
646         * @since 3.1
647         */
648        public static double gamma(final double x) {
649    
650            if ((x == FastMath.rint(x)) && (x <= 0.0)) {
651                return Double.NaN;
652            }
653    
654            final double ret;
655            final double absX = FastMath.abs(x);
656            if (absX <= 20.0) {
657                if (x >= 1.0) {
658                    /*
659                     * From the recurrence relation
660                     * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
661                     * then
662                     * Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
663                     * where t = x - n. This means that t must satisfy
664                     * -0.5 <= t - 1 <= 1.5.
665                     */
666                    double prod = 1.0;
667                    double t = x;
668                    while (t > 2.5) {
669                        t = t - 1.0;
670                        prod *= t;
671                    }
672                    ret = prod / (1.0 + invGamma1pm1(t - 1.0));
673                } else {
674                    /*
675                     * From the recurrence relation
676                     * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
677                     * then
678                     * Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
679                     * which requires -0.5 <= x + n <= 1.5.
680                     */
681                    double prod = x;
682                    double t = x;
683                    while (t < -0.5) {
684                        t = t + 1.0;
685                        prod *= t;
686                    }
687                    ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
688                }
689            } else {
690                final double y = absX + LANCZOS_G + 0.5;
691                final double gammaAbs = SQRT_TWO_PI / x *
692                                        FastMath.pow(y, absX + 0.5) *
693                                        FastMath.exp(-y) * lanczos(absX);
694                if (x > 0.0) {
695                    ret = gammaAbs;
696                } else {
697                    /*
698                     * From the reflection formula
699                     * Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
700                     * and the recurrence relation
701                     * Gamma(1 - x) = -x * Gamma(-x),
702                     * it is found
703                     * Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
704                     */
705                    ret = -FastMath.PI /
706                          (x * FastMath.sin(FastMath.PI * x) * gammaAbs);
707                }
708            }
709            return ret;
710        }
711    }