route/vendor/github.com/aclements/go-moremath/mathx/gamma.go

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// Copyright 2015 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package mathx
import "math"
// GammaInc returns the value of the incomplete gamma function (also
// known as the regularized gamma function):
//
// P(a, x) = 1 / Γ(a) * ∫₀ˣ exp(-t) t**(a-1) dt
func GammaInc(a, x float64) float64 {
// Based on Numerical Recipes in C, section 6.2.
if a <= 0 || x < 0 || math.IsNaN(a) || math.IsNaN(x) {
return math.NaN()
}
if x < a+1 {
// Use the series representation, which converges more
// rapidly in this range.
return gammaIncSeries(a, x)
} else {
// Use the continued fraction representation.
return 1 - gammaIncCF(a, x)
}
}
// GammaIncComp returns the complement of the incomplete gamma
// function 1 - GammaInc(a, x). This is more numerically stable for
// values near 0.
func GammaIncComp(a, x float64) float64 {
if a <= 0 || x < 0 || math.IsNaN(a) || math.IsNaN(x) {
return math.NaN()
}
if x < a+1 {
return 1 - gammaIncSeries(a, x)
} else {
return gammaIncCF(a, x)
}
}
func gammaIncSeries(a, x float64) float64 {
const maxIterations = 200
const epsilon = 3e-14
if x == 0 {
return 0
}
ap := a
del := 1 / a
sum := del
for n := 0; n < maxIterations; n++ {
ap++
del *= x / ap
sum += del
if math.Abs(del) < math.Abs(sum)*epsilon {
return sum * math.Exp(-x+a*math.Log(x)-lgamma(a))
}
}
panic("a too large; failed to converge")
}
func gammaIncCF(a, x float64) float64 {
const maxIterations = 200
const epsilon = 3e-14
raiseZero := func(z float64) float64 {
if math.Abs(z) < math.SmallestNonzeroFloat64 {
return math.SmallestNonzeroFloat64
}
return z
}
b := x + 1 - a
c := math.MaxFloat64
d := 1 / b
h := d
for i := 1; i <= maxIterations; i++ {
an := -float64(i) * (float64(i) - a)
b += 2
d = raiseZero(an*d + b)
c = raiseZero(b + an/c)
d = 1 / d
del := d * c
h *= del
if math.Abs(del-1) < epsilon {
return math.Exp(-x+a*math.Log(x)-lgamma(a)) * h
}
}
panic("a too large; failed to converge")
}