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 */
017package org.apache.commons.math3.special;
018
019import org.apache.commons.math3.exception.MaxCountExceededException;
020import org.apache.commons.math3.exception.NumberIsTooLargeException;
021import org.apache.commons.math3.exception.NumberIsTooSmallException;
022import org.apache.commons.math3.util.ContinuedFraction;
023import 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 */
054public 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}