source: src/main/java/agents/anac/y2019/harddealer/math3/special/Gamma.java

Last change on this file was 204, checked in by Katsuhide Fujita, 5 years ago

Fixed errors of ANAC2019 agents
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1/*
2 * Licensed to the Apache Software Foundation (ASF) under one or more
3 * contributor license agreements. See the NOTICE file distributed with
4 * this work for additional information regarding copyright ownership.
5 * The ASF licenses this file to You under the Apache License, Version 2.0
6 * (the "License"); you may not use this file except in compliance with
7 * the License. You may obtain a copy of the License at
8 *
9 * http://www.apache.org/licenses/LICENSE-2.0
10 *
11 * Unless required by applicable law or agreed to in writing, software
12 * distributed under the License is distributed on an "AS IS" BASIS,
13 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14 * See the License for the specific language governing permissions and
15 * limitations under the License.
16 */
17package agents.anac.y2019.harddealer.math3.special;
18
19import agents.anac.y2019.harddealer.math3.exception.MaxCountExceededException;
20import agents.anac.y2019.harddealer.math3.exception.NumberIsTooLargeException;
21import agents.anac.y2019.harddealer.math3.exception.NumberIsTooSmallException;
22import agents.anac.y2019.harddealer.math3.util.ContinuedFraction;
23import agents.anac.y2019.harddealer.math3.util.FastMath;
24
25/**
26 * <p>
27 * This is a utility class that provides computation methods related to the
28 * &Gamma; (Gamma) family of functions.
29 * </p>
30 * <p>
31 * Implementation of {@link #invGamma1pm1(double)} and
32 * {@link #logGamma1p(double)} is based on the algorithms described in
33 * <ul>
34 * <li><a href="http://dx.doi.org/10.1145/22721.23109">Didonato and Morris
35 * (1986)</a>, <em>Computation of the Incomplete Gamma Function Ratios and
36 * their Inverse</em>, TOMS 12(4), 377-393,</li>
37 * <li><a href="http://dx.doi.org/10.1145/131766.131776">Didonato and Morris
38 * (1992)</a>, <em>Algorithm 708: Significant Digit Computation of the
39 * Incomplete Beta Function Ratios</em>, TOMS 18(3), 360-373,</li>
40 * </ul>
41 * and implemented in the
42 * <a href="http://www.dtic.mil/docs/citations/ADA476840">NSWC Library of Mathematical Functions</a>,
43 * available
44 * <a href="http://www.ualberta.ca/CNS/RESEARCH/Software/NumericalNSWC/site.html">here</a>.
45 * This library is "approved for public release", and the
46 * <a href="http://www.dtic.mil/dtic/pdf/announcements/CopyrightGuidance.pdf">Copyright guidance</a>
47 * indicates that unless otherwise stated in the code, all FORTRAN functions in
48 * this library are license free. Since no such notice appears in the code these
49 * functions can safely be ported to Commons-Math.
50 * </p>
51 *
52 */
53public class Gamma {
54 /**
55 * <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a>
56 * @since 2.0
57 */
58 public static final double GAMMA = 0.577215664901532860606512090082;
59
60 /**
61 * The value of the {@code g} constant in the Lanczos approximation, see
62 * {@link #lanczos(double)}.
63 * @since 3.1
64 */
65 public static final double LANCZOS_G = 607.0 / 128.0;
66
67 /** Maximum allowed numerical error. */
68 private static final double DEFAULT_EPSILON = 10e-15;
69
70 /** Lanczos coefficients */
71 private static final double[] LANCZOS = {
72 0.99999999999999709182,
73 57.156235665862923517,
74 -59.597960355475491248,
75 14.136097974741747174,
76 -0.49191381609762019978,
77 .33994649984811888699e-4,
78 .46523628927048575665e-4,
79 -.98374475304879564677e-4,
80 .15808870322491248884e-3,
81 -.21026444172410488319e-3,
82 .21743961811521264320e-3,
83 -.16431810653676389022e-3,
84 .84418223983852743293e-4,
85 -.26190838401581408670e-4,
86 .36899182659531622704e-5,
87 };
88
89 /** Avoid repeated computation of log of 2 PI in logGamma */
90 private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
91
92 /** The constant value of &radic;(2&pi;). */
93 private static final double SQRT_TWO_PI = 2.506628274631000502;
94
95 // limits for switching algorithm in digamma
96 /** C limit. */
97 private static final double C_LIMIT = 49;
98
99 /** S limit. */
100 private static final double S_LIMIT = 1e-5;
101
102 /*
103 * Constants for the computation of double invGamma1pm1(double).
104 * Copied from DGAM1 in the NSWC library.
105 */
106
107 /** The constant {@code A0} defined in {@code DGAM1}. */
108 private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;
109
110 /** The constant {@code A1} defined in {@code DGAM1}. */
111 private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;
112
113 /** The constant {@code B1} defined in {@code DGAM1}. */
114 private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;
115
116 /** The constant {@code B2} defined in {@code DGAM1}. */
117 private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;
118
119 /** The constant {@code B3} defined in {@code DGAM1}. */
120 private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;
121
122 /** The constant {@code B4} defined in {@code DGAM1}. */
123 private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;
124
125 /** The constant {@code B5} defined in {@code DGAM1}. */
126 private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;
127
128 /** The constant {@code B6} defined in {@code DGAM1}. */
129 private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;
130
131 /** The constant {@code B7} defined in {@code DGAM1}. */
132 private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;
133
134 /** The constant {@code B8} defined in {@code DGAM1}. */
135 private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;
136
137 /** The constant {@code P0} defined in {@code DGAM1}. */
138 private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;
139
140 /** The constant {@code P1} defined in {@code DGAM1}. */
141 private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;
142
143 /** The constant {@code P2} defined in {@code DGAM1}. */
144 private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;
145
146 /** The constant {@code P3} defined in {@code DGAM1}. */
147 private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;
148
149 /** The constant {@code P4} defined in {@code DGAM1}. */
150 private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;
151
152 /** The constant {@code P5} defined in {@code DGAM1}. */
153 private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;
154
155 /** The constant {@code P6} defined in {@code DGAM1}. */
156 private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;
157
158 /** The constant {@code Q1} defined in {@code DGAM1}. */
159 private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;
160
161 /** The constant {@code Q2} defined in {@code DGAM1}. */
162 private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;
163
164 /** The constant {@code Q3} defined in {@code DGAM1}. */
165 private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;
166
167 /** The constant {@code Q4} defined in {@code DGAM1}. */
168 private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;
169
170 /** The constant {@code C} defined in {@code DGAM1}. */
171 private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;
172
173 /** The constant {@code C0} defined in {@code DGAM1}. */
174 private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;
175
176 /** The constant {@code C1} defined in {@code DGAM1}. */
177 private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;
178
179 /** The constant {@code C2} defined in {@code DGAM1}. */
180 private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;
181
182 /** The constant {@code C3} defined in {@code DGAM1}. */
183 private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;
184
185 /** The constant {@code C4} defined in {@code DGAM1}. */
186 private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;
187
188 /** The constant {@code C5} defined in {@code DGAM1}. */
189 private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;
190
191 /** The constant {@code C6} defined in {@code DGAM1}. */
192 private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;
193
194 /** The constant {@code C7} defined in {@code DGAM1}. */
195 private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;
196
197 /** The constant {@code C8} defined in {@code DGAM1}. */
198 private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;
199
200 /** The constant {@code C9} defined in {@code DGAM1}. */
201 private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;
202
203 /** The constant {@code C10} defined in {@code DGAM1}. */
204 private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;
205
206 /** The constant {@code C11} defined in {@code DGAM1}. */
207 private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;
208
209 /** The constant {@code C12} defined in {@code DGAM1}. */
210 private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;
211
212 /** The constant {@code C13} defined in {@code DGAM1}. */
213 private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;
214
215 /**
216 * Default constructor. Prohibit instantiation.
217 */
218 private Gamma() {}
219
220 /**
221 * <p>
222 * Returns the value of log&nbsp;&Gamma;(x) for x&nbsp;&gt;&nbsp;0.
223 * </p>
224 * <p>
225 * For x &le; 8, the implementation is based on the double precision
226 * implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
227 * {@code DGAMLN}. For x &gt; 8, the implementation is based on
228 * </p>
229 * <ul>
230 * <li><a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma
231 * Function</a>, equation (28).</li>
232 * <li><a href="http://mathworld.wolfram.com/LanczosApproximation.html">
233 * Lanczos Approximation</a>, equations (1) through (5).</li>
234 * <li><a href="http://my.fit.edu/~gabdo/gamma.txt">Paul Godfrey, A note on
235 * the computation of the convergent Lanczos complex Gamma
236 * approximation</a></li>
237 * </ul>
238 *
239 * @param x Argument.
240 * @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
241 * {@code x <= 0.0}.
242 */
243 public static double logGamma(double x) {
244 double ret;
245
246 if (Double.isNaN(x) || (x <= 0.0)) {
247 ret = Double.NaN;
248 } else if (x < 0.5) {
249 return logGamma1p(x) - FastMath.log(x);
250 } else if (x <= 2.5) {
251 return logGamma1p((x - 0.5) - 0.5);
252 } else if (x <= 8.0) {
253 final int n = (int) FastMath.floor(x - 1.5);
254 double prod = 1.0;
255 for (int i = 1; i <= n; i++) {
256 prod *= x - i;
257 }
258 return logGamma1p(x - (n + 1)) + FastMath.log(prod);
259 } else {
260 double sum = lanczos(x);
261 double tmp = x + LANCZOS_G + .5;
262 ret = ((x + .5) * FastMath.log(tmp)) - tmp +
263 HALF_LOG_2_PI + FastMath.log(sum / x);
264 }
265
266 return ret;
267 }
268
269 /**
270 * Returns the regularized gamma function P(a, x).
271 *
272 * @param a Parameter.
273 * @param x Value.
274 * @return the regularized gamma function P(a, x).
275 * @throws MaxCountExceededException if the algorithm fails to converge.
276 */
277 public static double regularizedGammaP(double a, double x) {
278 return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
279 }
280
281 /**
282 * Returns the regularized gamma function P(a, x).
283 *
284 * The implementation of this method is based on:
285 * <ul>
286 * <li>
287 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
288 * Regularized Gamma Function</a>, equation (1)
289 * </li>
290 * <li>
291 * <a href="http://mathworld.wolfram.com/IncompleteGammaFunction.html">
292 * Incomplete Gamma Function</a>, equation (4).
293 * </li>
294 * <li>
295 * <a href="http://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html">
296 * Confluent Hypergeometric Function of the First Kind</a>, equation (1).
297 * </li>
298 * </ul>
299 *
300 * @param a the a parameter.
301 * @param x the value.
302 * @param epsilon When the absolute value of the nth item in the
303 * series is less than epsilon the approximation ceases to calculate
304 * further elements in the series.
305 * @param maxIterations Maximum number of "iterations" to complete.
306 * @return the regularized gamma function P(a, x)
307 * @throws MaxCountExceededException if the algorithm fails to converge.
308 */
309 public static double regularizedGammaP(double a,
310 double x,
311 double epsilon,
312 int maxIterations) {
313 double ret;
314
315 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
316 ret = Double.NaN;
317 } else if (x == 0.0) {
318 ret = 0.0;
319 } else if (x >= a + 1) {
320 // use regularizedGammaQ because it should converge faster in this
321 // case.
322 ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);
323 } else {
324 // calculate series
325 double n = 0.0; // current element index
326 double an = 1.0 / a; // n-th element in the series
327 double sum = an; // partial sum
328 while (FastMath.abs(an/sum) > epsilon &&
329 n < maxIterations &&
330 sum < Double.POSITIVE_INFINITY) {
331 // compute next element in the series
332 n += 1.0;
333 an *= x / (a + n);
334
335 // update partial sum
336 sum += an;
337 }
338 if (n >= maxIterations) {
339 throw new MaxCountExceededException(maxIterations);
340 } else if (Double.isInfinite(sum)) {
341 ret = 1.0;
342 } else {
343 ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;
344 }
345 }
346
347 return ret;
348 }
349
350 /**
351 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
352 *
353 * @param a the a parameter.
354 * @param x the value.
355 * @return the regularized gamma function Q(a, x)
356 * @throws MaxCountExceededException if the algorithm fails to converge.
357 */
358 public static double regularizedGammaQ(double a, double x) {
359 return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
360 }
361
362 /**
363 * Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
364 *
365 * The implementation of this method is based on:
366 * <ul>
367 * <li>
368 * <a href="http://mathworld.wolfram.com/RegularizedGammaFunction.html">
369 * Regularized Gamma Function</a>, equation (1).
370 * </li>
371 * <li>
372 * <a href="http://functions.wolfram.com/GammaBetaErf/GammaRegularized/10/0003/">
373 * Regularized incomplete gamma function: Continued fraction representations
374 * (formula 06.08.10.0003)</a>
375 * </li>
376 * </ul>
377 *
378 * @param a the a parameter.
379 * @param x the value.
380 * @param epsilon When the absolute value of the nth item in the
381 * series is less than epsilon the approximation ceases to calculate
382 * further elements in the series.
383 * @param maxIterations Maximum number of "iterations" to complete.
384 * @return the regularized gamma function P(a, x)
385 * @throws MaxCountExceededException if the algorithm fails to converge.
386 */
387 public static double regularizedGammaQ(final double a,
388 double x,
389 double epsilon,
390 int maxIterations) {
391 double ret;
392
393 if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
394 ret = Double.NaN;
395 } else if (x == 0.0) {
396 ret = 1.0;
397 } else if (x < a + 1.0) {
398 // use regularizedGammaP because it should converge faster in this
399 // case.
400 ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);
401 } else {
402 // create continued fraction
403 ContinuedFraction cf = new ContinuedFraction() {
404
405 /** {@inheritDoc} */
406 @Override
407 protected double getA(int n, double x) {
408 return ((2.0 * n) + 1.0) - a + x;
409 }
410
411 /** {@inheritDoc} */
412 @Override
413 protected double getB(int n, double x) {
414 return n * (a - n);
415 }
416 };
417
418 ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);
419 ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;
420 }
421
422 return ret;
423 }
424
425
426 /**
427 * <p>Computes the digamma function of x.</p>
428 *
429 * <p>This is an independently written implementation of the algorithm described in
430 * Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.</p>
431 *
432 * <p>Some of the constants have been changed to increase accuracy at the moderate expense
433 * of run-time. The result should be accurate to within 10^-8 absolute tolerance for
434 * x >= 10^-5 and within 10^-8 relative tolerance for x > 0.</p>
435 *
436 * <p>Performance for large negative values of x will be quite expensive (proportional to
437 * |x|). Accuracy for negative values of x should be about 10^-8 absolute for results
438 * less than 10^5 and 10^-8 relative for results larger than that.</p>
439 *
440 * @param x Argument.
441 * @return digamma(x) to within 10-8 relative or absolute error whichever is smaller.
442 * @see <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma</a>
443 * @see <a href="http://www.uv.es/~bernardo/1976AppStatist.pdf">Bernardo&apos;s original article </a>
444 * @since 2.0
445 */
446 public static double digamma(double x) {
447 if (Double.isNaN(x) || Double.isInfinite(x)) {
448 return x;
449 }
450
451 if (x > 0 && x <= S_LIMIT) {
452 // use method 5 from Bernardo AS103
453 // accurate to O(x)
454 return -GAMMA - 1 / x;
455 }
456
457 if (x >= C_LIMIT) {
458 // use method 4 (accurate to O(1/x^8)
459 double inv = 1 / (x * x);
460 // 1 1 1 1
461 // log(x) - --- - ------ + ------- - -------
462 // 2 x 12 x^2 120 x^4 252 x^6
463 return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));
464 }
465
466 return digamma(x + 1) - 1 / x;
467 }
468
469 /**
470 * Computes the trigamma function of x.
471 * This function is derived by taking the derivative of the implementation
472 * of digamma.
473 *
474 * @param x Argument.
475 * @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller
476 * @see <a href="http://en.wikipedia.org/wiki/Trigamma_function">Trigamma</a>
477 * @see Gamma#digamma(double)
478 * @since 2.0
479 */
480 public static double trigamma(double x) {
481 if (Double.isNaN(x) || Double.isInfinite(x)) {
482 return x;
483 }
484
485 if (x > 0 && x <= S_LIMIT) {
486 return 1 / (x * x);
487 }
488
489 if (x >= C_LIMIT) {
490 double inv = 1 / (x * x);
491 // 1 1 1 1 1
492 // - + ---- + ---- - ----- + -----
493 // x 2 3 5 7
494 // 2 x 6 x 30 x 42 x
495 return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));
496 }
497
498 return trigamma(x + 1) + 1 / (x * x);
499 }
500
501 /**
502 * <p>
503 * Returns the Lanczos approximation used to compute the gamma function.
504 * The Lanczos approximation is related to the Gamma function by the
505 * following equation
506 * <center>
507 * {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)
508 * * exp(-x - g - 0.5) * lanczos(x)},
509 * </center>
510 * where {@code g} is the Lanczos constant.
511 * </p>
512 *
513 * @param x Argument.
514 * @return The Lanczos approximation.
515 * @see <a href="http://mathworld.wolfram.com/LanczosApproximation.html">Lanczos Approximation</a>
516 * equations (1) through (5), and Paul Godfrey's
517 * <a href="http://my.fit.edu/~gabdo/gamma.txt">Note on the computation
518 * of the convergent Lanczos complex Gamma approximation</a>
519 * @since 3.1
520 */
521 public static double lanczos(final double x) {
522 double sum = 0.0;
523 for (int i = LANCZOS.length - 1; i > 0; --i) {
524 sum += LANCZOS[i] / (x + i);
525 }
526 return sum + LANCZOS[0];
527 }
528
529 /**
530 * Returns the value of 1 / &Gamma;(1 + x) - 1 for -0&#46;5 &le; x &le;
531 * 1&#46;5. This implementation is based on the double precision
532 * implementation in the <em>NSWC Library of Mathematics Subroutines</em>,
533 * {@code DGAM1}.
534 *
535 * @param x Argument.
536 * @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
537 * @throws NumberIsTooSmallException if {@code x < -0.5}
538 * @throws NumberIsTooLargeException if {@code x > 1.5}
539 * @since 3.1
540 */
541 public static double invGamma1pm1(final double x) {
542
543 if (x < -0.5) {
544 throw new NumberIsTooSmallException(x, -0.5, true);
545 }
546 if (x > 1.5) {
547 throw new NumberIsTooLargeException(x, 1.5, true);
548 }
549
550 final double ret;
551 final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
552 if (t < 0.0) {
553 final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;
554 double b = INV_GAMMA1P_M1_B8;
555 b = INV_GAMMA1P_M1_B7 + t * b;
556 b = INV_GAMMA1P_M1_B6 + t * b;
557 b = INV_GAMMA1P_M1_B5 + t * b;
558 b = INV_GAMMA1P_M1_B4 + t * b;
559 b = INV_GAMMA1P_M1_B3 + t * b;
560 b = INV_GAMMA1P_M1_B2 + t * b;
561 b = INV_GAMMA1P_M1_B1 + t * b;
562 b = 1.0 + t * b;
563
564 double c = INV_GAMMA1P_M1_C13 + t * (a / b);
565 c = INV_GAMMA1P_M1_C12 + t * c;
566 c = INV_GAMMA1P_M1_C11 + t * c;
567 c = INV_GAMMA1P_M1_C10 + t * c;
568 c = INV_GAMMA1P_M1_C9 + t * c;
569 c = INV_GAMMA1P_M1_C8 + t * c;
570 c = INV_GAMMA1P_M1_C7 + t * c;
571 c = INV_GAMMA1P_M1_C6 + t * c;
572 c = INV_GAMMA1P_M1_C5 + t * c;
573 c = INV_GAMMA1P_M1_C4 + t * c;
574 c = INV_GAMMA1P_M1_C3 + t * c;
575 c = INV_GAMMA1P_M1_C2 + t * c;
576 c = INV_GAMMA1P_M1_C1 + t * c;
577 c = INV_GAMMA1P_M1_C + t * c;
578 if (x > 0.5) {
579 ret = t * c / x;
580 } else {
581 ret = x * ((c + 0.5) + 0.5);
582 }
583 } else {
584 double p = INV_GAMMA1P_M1_P6;
585 p = INV_GAMMA1P_M1_P5 + t * p;
586 p = INV_GAMMA1P_M1_P4 + t * p;
587 p = INV_GAMMA1P_M1_P3 + t * p;
588 p = INV_GAMMA1P_M1_P2 + t * p;
589 p = INV_GAMMA1P_M1_P1 + t * p;
590 p = INV_GAMMA1P_M1_P0 + t * p;
591
592 double q = INV_GAMMA1P_M1_Q4;
593 q = INV_GAMMA1P_M1_Q3 + t * q;
594 q = INV_GAMMA1P_M1_Q2 + t * q;
595 q = INV_GAMMA1P_M1_Q1 + t * q;
596 q = 1.0 + t * q;
597
598 double c = INV_GAMMA1P_M1_C13 + (p / q) * t;
599 c = INV_GAMMA1P_M1_C12 + t * c;
600 c = INV_GAMMA1P_M1_C11 + t * c;
601 c = INV_GAMMA1P_M1_C10 + t * c;
602 c = INV_GAMMA1P_M1_C9 + t * c;
603 c = INV_GAMMA1P_M1_C8 + t * c;
604 c = INV_GAMMA1P_M1_C7 + t * c;
605 c = INV_GAMMA1P_M1_C6 + t * c;
606 c = INV_GAMMA1P_M1_C5 + t * c;
607 c = INV_GAMMA1P_M1_C4 + t * c;
608 c = INV_GAMMA1P_M1_C3 + t * c;
609 c = INV_GAMMA1P_M1_C2 + t * c;
610 c = INV_GAMMA1P_M1_C1 + t * c;
611 c = INV_GAMMA1P_M1_C0 + t * c;
612
613 if (x > 0.5) {
614 ret = (t / x) * ((c - 0.5) - 0.5);
615 } else {
616 ret = x * c;
617 }
618 }
619
620 return ret;
621 }
622
623 /**
624 * Returns the value of log &Gamma;(1 + x) for -0&#46;5 &le; x &le; 1&#46;5.
625 * This implementation is based on the double precision implementation in
626 * the <em>NSWC Library of Mathematics Subroutines</em>, {@code DGMLN1}.
627 *
628 * @param x Argument.
629 * @return The value of {@code log(Gamma(1 + x))}.
630 * @throws NumberIsTooSmallException if {@code x < -0.5}.
631 * @throws NumberIsTooLargeException if {@code x > 1.5}.
632 * @since 3.1
633 */
634 public static double logGamma1p(final double x)
635 throws NumberIsTooSmallException, NumberIsTooLargeException {
636
637 if (x < -0.5) {
638 throw new NumberIsTooSmallException(x, -0.5, true);
639 }
640 if (x > 1.5) {
641 throw new NumberIsTooLargeException(x, 1.5, true);
642 }
643
644 return -FastMath.log1p(invGamma1pm1(x));
645 }
646
647
648 /**
649 * Returns the value of Γ(x). Based on the <em>NSWC Library of
650 * Mathematics Subroutines</em> double precision implementation,
651 * {@code DGAMMA}.
652 *
653 * @param x Argument.
654 * @return the value of {@code Gamma(x)}.
655 * @since 3.1
656 */
657 public static double gamma(final double x) {
658
659 if ((x == FastMath.rint(x)) && (x <= 0.0)) {
660 return Double.NaN;
661 }
662
663 final double ret;
664 final double absX = FastMath.abs(x);
665 if (absX <= 20.0) {
666 if (x >= 1.0) {
667 /*
668 * From the recurrence relation
669 * Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
670 * then
671 * Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
672 * where t = x - n. This means that t must satisfy
673 * -0.5 <= t - 1 <= 1.5.
674 */
675 double prod = 1.0;
676 double t = x;
677 while (t > 2.5) {
678 t -= 1.0;
679 prod *= t;
680 }
681 ret = prod / (1.0 + invGamma1pm1(t - 1.0));
682 } else {
683 /*
684 * From the recurrence relation
685 * Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
686 * then
687 * Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
688 * which requires -0.5 <= x + n <= 1.5.
689 */
690 double prod = x;
691 double t = x;
692 while (t < -0.5) {
693 t += 1.0;
694 prod *= t;
695 }
696 ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
697 }
698 } else {
699 final double y = absX + LANCZOS_G + 0.5;
700 final double gammaAbs = SQRT_TWO_PI / absX *
701 FastMath.pow(y, absX + 0.5) *
702 FastMath.exp(-y) * lanczos(absX);
703 if (x > 0.0) {
704 ret = gammaAbs;
705 } else {
706 /*
707 * From the reflection formula
708 * Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
709 * and the recurrence relation
710 * Gamma(1 - x) = -x * Gamma(-x),
711 * it is found
712 * Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
713 */
714 ret = -FastMath.PI /
715 (x * FastMath.sin(FastMath.PI * x) * gammaAbs);
716 }
717 }
718 return ret;
719 }
720}
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