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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
/*
* Cephes Math Library, Release 2.3: March, 1995
* Copyright 1984, 1995 by Stephen L. Moshier
*/
package cephes
import (
"math"
"gonum.org/v1/gonum/mathext/internal/gonum"
)
const (
maxGam = 171.624376956302725
big = 4.503599627370496e15
biginv = 2.22044604925031308085e-16
)
// Incbet computes the regularized incomplete beta function.
func Incbet(aa, bb, xx float64) float64 {
if aa <= 0 || bb <= 0 {
panic(paramOutOfBounds)
}
if xx <= 0 || xx >= 1 {
if xx == 0 {
return 0
}
if xx == 1 {
return 1
}
panic(paramOutOfBounds)
}
var flag int
if bb*xx <= 1 && xx <= 0.95 {
t := pseries(aa, bb, xx)
return transformT(t, flag)
}
w := 1 - xx
// Reverse a and b if x is greater than the mean.
var a, b, xc, x float64
if xx > aa/(aa+bb) {
flag = 1
a = bb
b = aa
xc = xx
x = w
} else {
a = aa
b = bb
xc = w
x = xx
}
if flag == 1 && (b*x) <= 1.0 && x <= 0.95 {
t := pseries(a, b, x)
return transformT(t, flag)
}
// Choose expansion for better convergence.
y := x*(a+b-2.0) - (a - 1.0)
if y < 0.0 {
w = incbcf(a, b, x)
} else {
w = incbd(a, b, x) / xc
}
// Multiply w by the factor
// x^a * (1-x)^b * Γ(a+b) / (a*Γ(a)*Γ(b))
var t float64
y = a * math.Log(x)
t = b * math.Log(xc)
if (a+b) < maxGam && math.Abs(y) < maxLog && math.Abs(t) < maxLog {
t = math.Pow(xc, b)
t *= math.Pow(x, a)
t /= a
t *= w
t *= 1.0 / gonum.Beta(a, b)
return transformT(t, flag)
}
// Resort to logarithms.
y += t - gonum.Lbeta(a, b)
y += math.Log(w / a)
if y < minLog {
t = 0.0
} else {
t = math.Exp(y)
}
return transformT(t, flag)
}
func transformT(t float64, flag int) float64 {
if flag == 1 {
if t <= machEp {
t = 1.0 - machEp
} else {
t = 1.0 - t
}
}
return t
}
// incbcf returns the incomplete beta integral evaluated by a continued fraction
// expansion.
func incbcf(a, b, x float64) float64 {
var xk, pk, pkm1, pkm2, qk, qkm1, qkm2 float64
var k1, k2, k3, k4, k5, k6, k7, k8 float64
var r, t, ans, thresh float64
var n int
k1 = a
k2 = a + b
k3 = a
k4 = a + 1.0
k5 = 1.0
k6 = b - 1.0
k7 = k4
k8 = a + 2.0
pkm2 = 0.0
qkm2 = 1.0
pkm1 = 1.0
qkm1 = 1.0
ans = 1.0
r = 1.0
thresh = 3.0 * machEp
for n = 0; n <= 300; n++ {
xk = -(x * k1 * k2) / (k3 * k4)
pk = pkm1 + pkm2*xk
qk = qkm1 + qkm2*xk
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
xk = (x * k5 * k6) / (k7 * k8)
pk = pkm1 + pkm2*xk
qk = qkm1 + qkm2*xk
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
if qk != 0 {
r = pk / qk
}
if r != 0 {
t = math.Abs((ans - r) / r)
ans = r
} else {
t = 1.0
}
if t < thresh {
return ans
}
k1 += 1.0
k2 += 1.0
k3 += 2.0
k4 += 2.0
k5 += 1.0
k6 -= 1.0
k7 += 2.0
k8 += 2.0
if (math.Abs(qk) + math.Abs(pk)) > big {
pkm2 *= biginv
pkm1 *= biginv
qkm2 *= biginv
qkm1 *= biginv
}
if (math.Abs(qk) < biginv) || (math.Abs(pk) < biginv) {
pkm2 *= big
pkm1 *= big
qkm2 *= big
qkm1 *= big
}
}
return ans
}
// incbd returns the incomplete beta integral evaluated by a continued fraction
// expansion.
func incbd(a, b, x float64) float64 {
var xk, pk, pkm1, pkm2, qk, qkm1, qkm2 float64
var k1, k2, k3, k4, k5, k6, k7, k8 float64
var r, t, ans, z, thresh float64
var n int
k1 = a
k2 = b - 1.0
k3 = a
k4 = a + 1.0
k5 = 1.0
k6 = a + b
k7 = a + 1.0
k8 = a + 2.0
pkm2 = 0.0
qkm2 = 1.0
pkm1 = 1.0
qkm1 = 1.0
z = x / (1.0 - x)
ans = 1.0
r = 1.0
thresh = 3.0 * machEp
for n = 0; n <= 300; n++ {
xk = -(z * k1 * k2) / (k3 * k4)
pk = pkm1 + pkm2*xk
qk = qkm1 + qkm2*xk
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
xk = (z * k5 * k6) / (k7 * k8)
pk = pkm1 + pkm2*xk
qk = qkm1 + qkm2*xk
pkm2 = pkm1
pkm1 = pk
qkm2 = qkm1
qkm1 = qk
if qk != 0 {
r = pk / qk
}
if r != 0 {
t = math.Abs((ans - r) / r)
ans = r
} else {
t = 1.0
}
if t < thresh {
return ans
}
k1 += 1.0
k2 -= 1.0
k3 += 2.0
k4 += 2.0
k5 += 1.0
k6 += 1.0
k7 += 2.0
k8 += 2.0
if (math.Abs(qk) + math.Abs(pk)) > big {
pkm2 *= biginv
pkm1 *= biginv
qkm2 *= biginv
qkm1 *= biginv
}
if (math.Abs(qk) < biginv) || (math.Abs(pk) < biginv) {
pkm2 *= big
pkm1 *= big
qkm2 *= big
qkm1 *= big
}
}
return ans
}
// pseries returns the incomplete beta integral evaluated by a power series. Use
// when b*x is small and x not too close to 1.
func pseries(a, b, x float64) float64 {
var s, t, u, v, n, t1, z, ai float64
ai = 1.0 / a
u = (1.0 - b) * x
v = u / (a + 1.0)
t1 = v
t = u
n = 2.0
s = 0.0
z = machEp * ai
for math.Abs(v) > z {
u = (n - b) * x / n
t *= u
v = t / (a + n)
s += v
n += 1.0
}
s += t1
s += ai
u = a * math.Log(x)
if (a+b) < maxGam && math.Abs(u) < maxLog {
t = 1.0 / gonum.Beta(a, b)
s = s * t * math.Pow(x, a)
} else {
t = -gonum.Lbeta(a, b) + u + math.Log(s)
if t < minLog {
s = 0.0
} else {
s = math.Exp(t)
}
}
return (s)
}
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