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 * 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 */ 053public class Gamma { 054 /** 055 * <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant">Euler-Mascheroni constant</a> 056 * @since 2.0 057 */ 058 public static final double GAMMA = 0.577215664901532860606512090082; 059 060 /** 061 * The value of the {@code g} constant in the Lanczos approximation, see 062 * {@link #lanczos(double)}. 063 * @since 3.1 064 */ 065 public static final double LANCZOS_G = 607.0 / 128.0; 066 067 /** Maximum allowed numerical error. */ 068 private static final double DEFAULT_EPSILON = 10e-15; 069 070 /** Lanczos coefficients */ 071 private static final double[] LANCZOS = { 072 0.99999999999999709182, 073 57.156235665862923517, 074 -59.597960355475491248, 075 14.136097974741747174, 076 -0.49191381609762019978, 077 .33994649984811888699e-4, 078 .46523628927048575665e-4, 079 -.98374475304879564677e-4, 080 .15808870322491248884e-3, 081 -.21026444172410488319e-3, 082 .21743961811521264320e-3, 083 -.16431810653676389022e-3, 084 .84418223983852743293e-4, 085 -.26190838401581408670e-4, 086 .36899182659531622704e-5, 087 }; 088 089 /** Avoid repeated computation of log of 2 PI in logGamma */ 090 private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI); 091 092 /** The constant value of √(2π). */ 093 private static final double SQRT_TWO_PI = 2.506628274631000502; 094 095 // limits for switching algorithm in digamma 096 /** C limit. */ 097 private static final double C_LIMIT = 49; 098 099 /** 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 Γ(x) for x > 0. 223 * </p> 224 * <p> 225 * For x ≤ 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 > 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'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 / Γ(1 + x) - 1 for -0.5 ≤ x ≤ 531 * 1.5. 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 Γ(1 + x) for -0.5 ≤ x ≤ 1.5. 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}