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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 2.4: March,1996
* Copyright 1984, 1996 by Stephen L. Moshier
*/
package cephes
import "math"
// Incbi computes the inverse of the regularized incomplete beta integral.
func Incbi(aa, bb, yy0 float64) float64 {
var a, b, y0, d, y, x, x0, x1, lgm, yp, di, dithresh, yl, yh, xt float64
var i, rflg, dir, nflg int
if yy0 <= 0 {
return (0.0)
}
if yy0 >= 1.0 {
return (1.0)
}
x0 = 0.0
yl = 0.0
x1 = 1.0
yh = 1.0
nflg = 0
if aa <= 1.0 || bb <= 1.0 {
dithresh = 1.0e-6
rflg = 0
a = aa
b = bb
y0 = yy0
x = a / (a + b)
y = Incbet(a, b, x)
goto ihalve
} else {
dithresh = 1.0e-4
}
// Approximation to inverse function
yp = -Ndtri(yy0)
if yy0 > 0.5 {
rflg = 1
a = bb
b = aa
y0 = 1.0 - yy0
yp = -yp
} else {
rflg = 0
a = aa
b = bb
y0 = yy0
}
lgm = (yp*yp - 3.0) / 6.0
x = 2.0 / (1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0))
d = yp*math.Sqrt(x+lgm)/x - (1.0/(2.0*b-1.0)-1.0/(2.0*a-1.0))*(lgm+5.0/6.0-2.0/(3.0*x))
d = 2.0 * d
if d < minLog {
// mtherr("incbi", UNDERFLOW)
x = 0
goto done
}
x = a / (a + b*math.Exp(d))
y = Incbet(a, b, x)
yp = (y - y0) / y0
if math.Abs(yp) < 0.2 {
goto newt
}
/* Resort to interval halving if not close enough. */
ihalve:
dir = 0
di = 0.5
for i = 0; i < 100; i++ {
if i != 0 {
x = x0 + di*(x1-x0)
if x == 1.0 {
x = 1.0 - machEp
}
if x == 0.0 {
di = 0.5
x = x0 + di*(x1-x0)
if x == 0.0 {
// mtherr("incbi", UNDERFLOW)
goto done
}
}
y = Incbet(a, b, x)
yp = (x1 - x0) / (x1 + x0)
if math.Abs(yp) < dithresh {
goto newt
}
yp = (y - y0) / y0
if math.Abs(yp) < dithresh {
goto newt
}
}
if y < y0 {
x0 = x
yl = y
if dir < 0 {
dir = 0
di = 0.5
} else if dir > 3 {
di = 1.0 - (1.0-di)*(1.0-di)
} else if dir > 1 {
di = 0.5*di + 0.5
} else {
di = (y0 - y) / (yh - yl)
}
dir += 1
if x0 > 0.75 {
if rflg == 1 {
rflg = 0
a = aa
b = bb
y0 = yy0
} else {
rflg = 1
a = bb
b = aa
y0 = 1.0 - yy0
}
x = 1.0 - x
y = Incbet(a, b, x)
x0 = 0.0
yl = 0.0
x1 = 1.0
yh = 1.0
goto ihalve
}
} else {
x1 = x
if rflg == 1 && x1 < machEp {
x = 0.0
goto done
}
yh = y
if dir > 0 {
dir = 0
di = 0.5
} else if dir < -3 {
di = di * di
} else if dir < -1 {
di = 0.5 * di
} else {
di = (y - y0) / (yh - yl)
}
dir -= 1
}
}
// mtherr("incbi", PLOSS)
if x0 >= 1.0 {
x = 1.0 - machEp
goto done
}
if x <= 0.0 {
// mtherr("incbi", UNDERFLOW)
x = 0.0
goto done
}
newt:
if nflg > 0 {
goto done
}
nflg = 1
lgm = lgam(a+b) - lgam(a) - lgam(b)
for i = 0; i < 8; i++ {
/* Compute the function at this point. */
if i != 0 {
y = Incbet(a, b, x)
}
if y < yl {
x = x0
y = yl
} else if y > yh {
x = x1
y = yh
} else if y < y0 {
x0 = x
yl = y
} else {
x1 = x
yh = y
}
if x == 1.0 || x == 0.0 {
break
}
/* Compute the derivative of the function at this point. */
d = (a-1.0)*math.Log(x) + (b-1.0)*math.Log(1.0-x) + lgm
if d < minLog {
goto done
}
if d > maxLog {
break
}
d = math.Exp(d)
/* Compute the step to the next approximation of x. */
d = (y - y0) / d
xt = x - d
if xt <= x0 {
y = (x - x0) / (x1 - x0)
xt = x0 + 0.5*y*(x-x0)
if xt <= 0.0 {
break
}
}
if xt >= x1 {
y = (x1 - x) / (x1 - x0)
xt = x1 - 0.5*y*(x1-x)
if xt >= 1.0 {
break
}
}
x = xt
if math.Abs(d/x) < 128.0*machEp {
goto done
}
}
/* Did not converge. */
dithresh = 256.0 * machEp
goto ihalve
done:
if rflg > 0 {
if x <= machEp {
x = 1.0 - machEp
} else {
x = 1.0 - x
}
}
return (x)
}
func lgam(a float64) float64 {
lg, _ := math.Lgamma(a)
return lg
}
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