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 * Γ (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 √(2π). */
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 Γ(x) for x > 0.
224 * </p>
225 * <p>
226 * For x ≤ 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 > 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'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 / Γ(1 + x) - 1 for -0.5 ≤ x ≤
522 * 1.5. 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 Γ(1 + x) for -0.5 ≤ x ≤ 1.5.
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 }