blob: 49f87225c8dcdad4959f34f87a736dd0e30aefb2 [file] [log] [blame]
// 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"
func lgamma(x float64) float64 {
y, _ := math.Lgamma(x)
return y
}
// Beta returns the value of the complete beta function B(a, b).
func Beta(a, b float64) float64 {
// B(x,y) = Γ(x)Γ(y) / Γ(x+y)
return math.Exp(lgamma(a) + lgamma(b) - lgamma(a+b))
}
// BetaInc returns the value of the regularized incomplete beta
// function Iₓ(a, b).
//
// This is not to be confused with the "incomplete beta function",
// which can be computed as BetaInc(x, a, b)*Beta(a, b).
//
// If x < 0 or x > 1, returns NaN.
func BetaInc(x, a, b float64) float64 {
// Based on Numerical Recipes in C, section 6.4. This uses the
// continued fraction definition of I:
//
// (xᵃ*(1-x)ᵇ)/(a*B(a,b)) * (1/(1+(d₁/(1+(d₂/(1+...))))))
//
// where B(a,b) is the beta function and
//
// d_{2m+1} = -(a+m)(a+b+m)x/((a+2m)(a+2m+1))
// d_{2m} = m(b-m)x/((a+2m-1)(a+2m))
if x < 0 || x > 1 {
return math.NaN()
}
bt := 0.0
if 0 < x && x < 1 {
// Compute the coefficient before the continued
// fraction.
bt = math.Exp(lgamma(a+b) - lgamma(a) - lgamma(b) +
a*math.Log(x) + b*math.Log(1-x))
}
if x < (a+1)/(a+b+2) {
// Compute continued fraction directly.
return bt * betacf(x, a, b) / a
} else {
// Compute continued fraction after symmetry transform.
return 1 - bt*betacf(1-x, b, a)/b
}
}
// betacf is the continued fraction component of the regularized
// incomplete beta function Iₓ(a, b).
func betacf(x, a, b 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
}
c := 1.0
d := 1 / raiseZero(1-(a+b)*x/(a+1))
h := d
for m := 1; m <= maxIterations; m++ {
mf := float64(m)
// Even step of the recurrence.
numer := mf * (b - mf) * x / ((a + 2*mf - 1) * (a + 2*mf))
d = 1 / raiseZero(1+numer*d)
c = raiseZero(1 + numer/c)
h *= d * c
// Odd step of the recurrence.
numer = -(a + mf) * (a + b + mf) * x / ((a + 2*mf) * (a + 2*mf + 1))
d = 1 / raiseZero(1+numer*d)
c = raiseZero(1 + numer/c)
hfac := d * c
h *= hfac
if math.Abs(hfac-1) < epsilon {
return h
}
}
panic("betainc: a or b too big; failed to converge")
}