Module: Distribution::MathExtension::IncompleteGamma
- Defined in:
- lib/distribution/math_extension/incomplete_gamma.rb
Constant Summary collapse
- NMAX =
5000
- SMALL =
Float::EPSILON ** 3
- PG21 =
-2.404113806319188570799476 # PolyGamma[2,1]
Class Method Summary collapse
-
.a_greater_than_0(a, x, with_error = false) ⇒ Object
gamma_inc_a_gt_0.
-
.continued_fraction(a, x, with_error = false) ⇒ Object
gamma_inc_CF.
-
.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.
-
.f_continued_fraction(a, x, with_error = false) ⇒ Object
gamma_inc_F_CF.
-
.p(a, x, with_error = false) ⇒ Object
The incomplete gamma function.
-
.p_series(a, x, with_error = false) ⇒ Object
gamma_inc_P_series.
-
.q(a, x, with_error = false) ⇒ Object
gamma_inc_Q_e.
-
.q_asymptotic_uniform(a, x, with_error = false) ⇒ Object
Uniform asymptotic for x near a, a and x large gamma_inc_Q_asymp_unif.
-
.q_asymptotic_uniform_complement(a, x, with_error = false) ⇒ Object
This function does not exist in GSL, but is nonetheless GSL code.
-
.q_continued_fraction(a, x, with_error = false) ⇒ Object
gamma_inc_Q_CF.
- .q_continued_fraction_complement(a, x, with_error = false) ⇒ Object
-
.q_large_x(a, x, with_error = false) ⇒ Object
gamma_inc_Q_large_x in GSL-1.9.
- .q_large_x_complement(a, x, with_error = false) ⇒ Object
- .q_series(a, x, with_error = false) ⇒ Object
-
.series(a, x, with_error = false) ⇒ Object
gamma_inc_series.
-
.unnormalized(a, x, with_error = false) ⇒ Object
Unnormalized incomplete gamma function.
Class Method Details
.a_greater_than_0(a, x, with_error = false) ⇒ Object
gamma_inc_a_gt_0
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 356 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 368 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.
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 39 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 232 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 110 def p a,x,with_error=false raise(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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 69 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 132 def q a,x,with_error=false raise(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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 202 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(12960) - eps*589.quo(30240)))) 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.
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 91 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 return 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 163 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 98 def q_continued_fraction_complement a,x,with_error=false q = q_continued_fraction(a,x,with_error) return 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 175 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 103 def q_large_x_complement a,x,with_error=false q = q_large_x(a,x,with_error) return 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 256 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 346 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
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# File 'lib/distribution/math_extension/incomplete_gamma.rb', line 385 def unnormalized a,x,with_error = false raise(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 |