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 }