# Module: Distribution::MathExtension::IncompleteGamma

Defined in:
lib/distribution/math_extension/incomplete_gamma.rb

## Constant Summary collapse

NMAX =
5000
SMALL =
Float::EPSILON**3
PG21 =

PolyGamma[2,1]

-2.404113806319188570799476 # PolyGamma[2,1]

## Class Method Summary collapse

• gamma_inc_a_gt_0.

• gamma_inc_CF.

• The dominant part, D(a,x) := x^a e^(-x) / Gamma(a+1) gamma_inc_D in GSL-1.9.

• gamma_inc_F_CF.

• The incomplete gamma function.

• gamma_inc_P_series.

• gamma_inc_Q_e.

• Uniform asymptotic for x near a, a and x large gamma_inc_Q_asymp_unif.

• This function does not exist in GSL, but is nonetheless GSL code.

• gamma_inc_Q_CF.

• gamma_inc_Q_large_x in GSL-1.9.

• gamma_inc_series.

• Unnormalized incomplete gamma function.

## Class Method Details

### .a_greater_than_0(a, x, with_error = false) ⇒ Object

gamma_inc_a_gt_0

 355 356 357 358 359 360 361 362 363 364 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 355 def a_greater_than_0(a, x, with_error = false) q = q(a, x, with_error) q, q_err = q if with_error g = Math.gamma(a) STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error g_err = Float::EPSILON result = g * q error = (g * q_err).abs + (g_err * q).abs if with_error with_error ? [result, error] : result end

### .continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_CF

 367 368 369 370 371 372 373 374 375 376 377 378 379 380 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 367 def continued_fraction(a, x, with_error = false) f = f_continued_fraction(a, x, with_error) f, f_error = f if with_error pre = Math.exp((a - 1.0) * Math.log(x) - x) STDERR.puts("Warning: Don't know error for Math.exp. Error will be incorrect") if with_error pre_error = Float::EPSILON result = f * pre if with_error error = (f_error * pre).abs + (f * pre_error) + (2.0 + a.abs) * Float::EPSILON * result.abs [result, error] else result end end

### .d(a, x, with_error = false) ⇒ Object

The dominant part, D(a,x) := x^a e^(-x) / Gamma(a+1) gamma_inc_D in GSL-1.9.

 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 38 def d(a, x, with_error = false) error = nil if a < 10.0 ln_a = Math.lgamma(a + 1.0).first lnr = a * Math.log(x) - x - ln_a result = Math.exp(lnr) error = 2.0 * Float::EPSILON * (lnr.abs + 1.0) + result.abs if with_error with_error ? [result, error] : result else ln_term = ln_term_error = nil if x < 0.5 * a u = x / a.to_f ln_u = Math.log(u) ln_term = ln_u - u + 1.0 ln_term_error = (ln_u.abs + u.abs + 1.0) * Float::EPSILON if with_error else mu = (x - a) / a.to_f ln_term = Log.log_1plusx_minusx(mu, with_error) ln_term, ln_term_error = ln_term if with_error end gstar = Gammastar.evaluate(a, with_error) gstar, gstar_error = gstar if with_error term1 = Math.exp(a * ln_term) / Math.sqrt(2.0 * Math::PI * a) result = term1 / gstar error = 2.0 * Float::EPSILON * ((a * ln_term).abs + 1.0) * result.abs + gstar_error / gstar.abs * result.abs if with_error with_error ? [result, error] : result end end

### .f_continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_F_CF

 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 231 def f_continued_fraction(a, x, with_error = false) hn = 1.0 # convergent cn = 1.0 / SMALL dn = 1.0 n = 2 2.upto(NMAX - 1).each do |n| an = n.odd? ? 0.5 * (n - 1) / x : (0.5 * n - a) / x dn = 1.0 + an * dn dn = SMALL if dn.abs < SMALL cn = 1.0 + an / cn cn = SMALL if cn.abs < SMALL dn = 1.0 / dn delta = cn * dn hn *= delta break if (delta - 1.0).abs < Float::EPSILON end if n == NMAX STDERR.puts('Error: n reached NMAX in f continued fraction') else with_error ? [hn, 2.0 * Float::EPSILON * hn.abs + Float::EPSILON * (2.0 + 0.5 * n) * hn.abs] : hn end end

### .p(a, x, with_error = false) ⇒ Object

The incomplete gamma function. gsl_sf_gamma_inc_P_e

 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 109 def p(a, x, with_error = false) fail(ArgumentError, 'Range Error: a must be positive, x must be non-negative') if a <= 0.0 || x < 0.0 if x == 0.0 return with_error ? [0.0, 0.0] : 0.0 elsif x < 20.0 || x < 0.5 * a return p_series(a, x, with_error) elsif a > 1e6 && (x - a) * (x - a) < a return q_asymptotic_uniform_complement a, x, with_error elsif a <= x if a > 0.2 * x return q_continued_fraction_complement(a, x, with_error) else return q_large_x_complement(a, x, with_error) end elsif (x - a) * (x - a) < a return q_asymptotic_uniform_complement a, x, with_error else return p_series(a, x, with_error) end end

### .p_series(a, x, with_error = false) ⇒ Object

gamma_inc_P_series

 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 68 def p_series(a, x, with_error = false) d = d(a, x, with_error) d, d_err = d if with_error sum = 1.0 term = 1.0 n = 1 1.upto(NMAX - 1) do |n| term *= x / (a + n).to_f sum += term break if (term / sum).abs < Float::EPSILON end result = d * sum if n == NMAX STDERR.puts('Error: n reached NMAX in p series') else return with_error ? [result, d_err * sum.abs + (1.0 + n) * Float::EPSILON * result.abs] : result end end

### .q(a, x, with_error = false) ⇒ Object

gamma_inc_Q_e

 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 131 def q(a, x, with_error = false) fail(ArgumentError, 'Range Error: a and x must be non-negative') if a < 0.0 || x < 0.0 if x == 0.0 return with_error ? [1.0, 0.0] : 1.0 elsif a == 0.0 return with_error ? [0.0, 0.0] : 0.0 elsif x <= 0.5 * a # If series is quick, do that. p = p_series(a, x, with_error) p, p_err = p if with_error result = 1.0 - p return with_error ? [result, p_err + 2.0 * Float::EPSILON * result.abs] : result elsif a >= 1.0e+06 && (x - a) * (x - a) < a # difficult asymptotic regime, only way to do this region return q_asymptotic_uniform(a, x, with_error) elsif a < 0.2 && x < 5.0 return q_series(a, x, with_error) elsif a <= x return x <= 1.0e+06 ? q_continued_fraction(a, x, with_error) : q_large_x(a, x, with_error) else if x > a - Math.sqrt(a) return q_continued_fraction(a, x, with_error) else p = p_series(a, x, with_error) p, p_err = p if with_error result = 1.0 - p return with_error ? [result, p_err + 2.0 * Float::EPSILON * result.abs] : result end end end

### .q_asymptotic_uniform(a, x, with_error = false) ⇒ Object

Uniform asymptotic for x near a, a and x large gamma_inc_Q_asymp_unif

 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 201 def q_asymptotic_uniform(a, x, with_error = false) rta = Math.sqrt(a) eps = (x - a).quo(a) ln_term = Log.log_1plusx_minusx(eps, with_error) ln_term, ln_term_err = ln_term if with_error eta = (eps >= 0 ? 1 : -1) * Math.sqrt(-2 * ln_term) erfc = Math.erfc_e(eta * rta / SQRT2, with_error) erfc, erfc_err = erfc if with_error c0 = c1 = nil if eps.abs < ROOT5_FLOAT_EPSILON c0 = -1.quo(3) + eps * (1.quo(12) - eps * (23.quo(540) - eps * (353.quo(12_960) - eps * 589.quo(30_240)))) c1 = -1.quo(540) - eps.quo(288) else rt_term = Math.sqrt(-2 * ln_term.quo(eps * eps)) lam = x.quo(a) c0 = (1 - 1 / rt_term) / eps c1 = -(eta**3 * (lam * lam + 10 * lam + 1) - 12 * eps**3).quo(12 * eta**3 * eps**3) end r = Math.exp(-0.5 * a * eta * eta) / (SQRT2 * SQRTPI * rta) * (c0 + c1.quo(a)) result = 0.5 * erfc + r with_error ? [result, Float::EPSILON + (r * 0.5 * a * eta * eta).abs + 0.5 * erfc_err + 2.0 * Float::EPSILON + result.abs] : result end

### .q_asymptotic_uniform_complement(a, x, with_error = false) ⇒ Object

This function does not exist in GSL, but is nonetheless GSL code. It's for calculating two specific ranges of p.

 90 91 92 93 94 95 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 90 def q_asymptotic_uniform_complement(a, x, with_error = false) q = q_asymptotic_uniform(a, x, with_error) q, q_err = q if with_error result = 1.0 - q with_error ? [result, q_err + 2.0 * Float::EPSILON * result.abs] : result end

### .q_continued_fraction(a, x, with_error = false) ⇒ Object

gamma_inc_Q_CF

 162 163 164 165 166 167 168 169 170 171 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 162 def q_continued_fraction(a, x, with_error = false) d = d(a, x, with_error) f = f_continued_fraction(a, x, with_error) if with_error [d.first * (a / x).to_f * f.first, d.last * ((a / x).to_f * f.first).abs + (d.first * a / x * f.last).abs] else d * (a / x).to_f * f end end

### .q_continued_fraction_complement(a, x, with_error = false) ⇒ Object

 97 98 99 100 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 97 def q_continued_fraction_complement(a, x, with_error = false) q = q_continued_fraction(a, x, with_error) with_error ? [1.0 - q.first, q.last + 2.0 * Float::EPSILON * (1.0 - q.first).abs] : 1.0 - q end

### .q_large_x(a, x, with_error = false) ⇒ Object

gamma_inc_Q_large_x in GSL-1.9

 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 174 def q_large_x(a, x, with_error = false) d = d(a, x, with_error) d, d_err = d if with_error sum = 1.0 term = 1.0 last = 1.0 n = 1 1.upto(NMAX - 1).each do |n| term *= (a - n) / x break if (term / last).abs > 1.0 break if (term / sum).abs < Float::EPSILON sum += term last = term end result = d * (a / x) * sum error = d_err * (a / x).abs * sum if with_error if n == NMAX STDERR.puts('Error: n reached NMAX in q_large_x') else return with_error ? [result, error] : result end end

### .q_large_x_complement(a, x, with_error = false) ⇒ Object

 102 103 104 105 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 102 def q_large_x_complement(a, x, with_error = false) q = q_large_x(a, x, with_error) with_error ? [1.0 - q.first, q.last + 2.0 * Float::EPSILON * (1.0 - q.first).abs] : 1.0 - q end

### .q_series(a, x, with_error = false) ⇒ Object

 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 255 def q_series(a, x, with_error = false) term1 = nil sum = nil term2 = nil begin lnx = Math.log(x) el = EULER + lnx c1 = -el c2 = Math::PI * Math::PI / 12.0 - 0.5 * el * el c3 = el * (Math::PI * Math::PI / 12.0 - el * el / 6.0) + PG21 / 6.0 c4 = -0.04166666666666666667 * (-1.758243446661483480 + lnx) * (-0.764428657272716373 + lnx) * (0.723980571623507657 + lnx) * (4.107554191916823640 + lnx) c5 = -0.0083333333333333333 * (-2.06563396085715900 + lnx) * (-1.28459889470864700 + lnx) * (-0.27583535756454143 + lnx) * (1.33677371336239618 + lnx) * (5.17537282427561550 + lnx) c6 = -0.0013888888888888889 * (-2.30814336454783200 + lnx) * (-1.65846557706987300 + lnx) * (-0.88768082560020400 + lnx) * (0.17043847751371778 + lnx) * (1.92135970115863890 + lnx) * (6.22578557795474900 + lnx) c7 = -0.00019841269841269841 (-2.5078657901291800 + lnx) * (-1.9478900888958200 + lnx) * (-1.3194837322612730 + lnx) * (-0.5281322700249279 + lnx) * (0.5913834939078759 + lnx) * (2.4876819633378140 + lnx) * (7.2648160783762400 + lnx) c8 = -0.00002480158730158730 * (-2.677341544966400 + lnx) * (-2.182810448271700 + lnx) * (-1.649350342277400 + lnx) * (-1.014099048290790 + lnx) * (-0.191366955370652 + lnx) * (0.995403817918724 + lnx) * (3.041323283529310 + lnx) * (8.295966556941250 + lnx) * c9 = -2.75573192239859e-6 * (-2.8243487670469080 + lnx) * (-2.3798494322701120 + lnx) * (-1.9143674728689960 + lnx) * (-1.3814529102920370 + lnx) * (-0.7294312810261694 + lnx) * (0.1299079285269565 + lnx) * (1.3873333251885240 + lnx) * (3.5857258865210760 + lnx) * (9.3214237073814600 + lnx) * c10 = -2.75573192239859e-7 * (-2.9540329644556910 + lnx) * (-2.5491366926991850 + lnx) * (-2.1348279229279880 + lnx) * (-1.6741881076349450 + lnx) * (-1.1325949616098420 + lnx) * (-0.4590034650618494 + lnx) * (0.4399352987435699 + lnx) * (1.7702236517651670 + lnx) * (4.1231539047474080 + lnx) * (10.342627908148680 + lnx) term1 = a * (c1 + a * (c2 + a * (c3 + a * (c4 + a * (c5 + a * (c6 + a * (c7 + a * (c8 + a * (c9 + a * c10))))))))) end n = 1 begin t = 1.0 sum = 1.0 1.upto(NMAX - 1).each do |n| t *= -x / (n + 1.0) sum += (a + 1.0) / (a + n + 1.0) * t break if (t / sum).abs < Float::EPSILON end end if n == NMAX STDERR.puts('Error: n reached NMAX in q_series') else term2 = (1.0 - term1) * a / (a + 1.0) * x * sum result = term1 + term2 with_error ? [result, Float::EPSILON * term1.abs + 2.0 * term2.abs + 2.0 * Float::EPSILON * result.abs] : result end end

### .series(a, x, with_error = false) ⇒ Object

gamma_inc_series

 345 346 347 348 349 350 351 352 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 345 def series(a, x, with_error = false) q = q_series(a, x, with_error) g = Math.gamma(a) STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error # When we get the error from Gamma, switch the comment on the next to lines # with_error ? [q.first*g.first, (q.first*g.last).abs + (q.last*g.first).abs + 2.0*Float::EPSILON*(q.first*g.first).abs] : q*g with_error ? [q.first * g, (q.first * Float::EPSILON).abs + (q.last * g.first).abs + 2.0 * Float::EPSILON(q.first * g).abs] : q * g end

### .unnormalized(a, x, with_error = false) ⇒ Object

Unnormalized incomplete gamma function. gsl_sf_gamma_inc_e

 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 # File 'lib/distribution/math_extension/incomplete_gamma.rb', line 384 def unnormalized(a, x, with_error = false) fail(ArgumentError, 'x cannot be negative') if x < 0.0 if x == 0.0 result = Math.gamma(a.to_f) STDERR.puts("Warning: Don't know error for Math.gamma. Error will be incorrect") if with_error return with_error ? [result, Float::EPSILON] : result elsif a == 0.0 return ExponentialIntegral.first_order(x.to_f, with_error) elsif a > 0.0 return a_greater_than_0(a.to_f, x.to_f, with_error) elsif x > 0.25 # continued fraction seems to fail for x too small return continued_fraction(a.to_f, x.to_f, with_error) elsif a.abs < 0.5 return series(a.to_f, x.to_f, with_error) else fa = a.floor.to_f da = a - fa g_da = da > 0.0 ? a_greater_than_0(da, x.to_f, with_error) : ExponentialIntegral.first_order(x.to_f, with_error) g_da, g_da_err = g_da if with_error alpha = da gax = g_da # Gamma(alpha-1,x) = 1/(alpha-1) (Gamma(a,x) - x^(alpha-1) e^-x) begin shift = Math.exp(-x + (alpha - 1.0) * Math.log(x)) gax = (gax - shift) / (alpha - 1.0) alpha -= 1.0 end while alpha > a result = gax return with_error ? [result, 2.0 * (1.0 + a.abs) * Float::EPSILON * gax.abs] : result end end