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// Derived from SciPy's special/cephes/igami.c
// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/igami.c
// Made freely available by Stephen L. Moshier without support or guarantee.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Copyright ©1984, ©1987, ©1995 by Stephen L. Moshier
// Portions Copyright ©2017 The Gonum Authors. All rights reserved.
package cephes
import "math"
// IgamI computes the inverse of the incomplete Gamma function. That is, it
// returns the x such that:
//
// IgamC(a, x) = p
//
// The input argument a must be positive and p must be between 0 and 1
// inclusive or IgamI will panic. IgamI should return a positive number, but
// can return 0 even with non-zero y due to underflow.
func IgamI(a, p float64) float64 {
// Bound the solution
x0 := math.MaxFloat64
yl := 0.0
x1 := 0.0
yh := 1.0
dithresh := 5.0 * machEp
if p < 0 || p > 1 || a <= 0 {
panic(paramOutOfBounds)
}
if p == 0 {
return math.Inf(1)
}
if p == 1 {
return 0.0
}
// Starting with the approximate value
// x = a y^3
// where
// y = 1 - d - ndtri(p) sqrt(d)
// and
// d = 1/9a
// the routine performs up to 10 Newton iterations to find the root of
// IgamC(a, x) - p = 0
d := 1.0 / (9.0 * a)
y := 1.0 - d - Ndtri(p)*math.Sqrt(d)
x := a * y * y * y
lgm := lgam(a)
for i := 0; i < 10; i++ {
if x > x0 || x < x1 {
break
}
y = IgamC(a, x)
if y < yl || y > yh {
break
}
if y < p {
x0 = x
yl = y
} else {
x1 = x
yh = y
}
// Compute the derivative of the function at this point
d = (a-1)*math.Log(x) - x - lgm
if d < -maxLog {
break
}
d = -math.Exp(d)
// Compute the step to the next approximation of x
d = (y - p) / d
if math.Abs(d/x) < machEp {
return x
}
x = x - d
}
d = 0.0625
if x0 == math.MaxFloat64 {
if x <= 0 {
x = 1
}
for x0 == math.MaxFloat64 {
x = (1 + d) * x
y = IgamC(a, x)
if y < p {
x0 = x
yl = y
break
}
d = d + d
}
}
d = 0.5
dir := 0
for i := 0; i < 400; i++ {
x = x1 + d*(x0-x1)
y = IgamC(a, x)
lgm = (x0 - x1) / (x1 + x0)
if math.Abs(lgm) < dithresh {
break
}
lgm = (y - p) / p
if math.Abs(lgm) < dithresh {
break
}
if x <= 0 {
break
}
if y >= p {
x1 = x
yh = y
if dir < 0 {
dir = 0
d = 0.5
} else if dir > 1 {
d = 0.5*d + 0.5
} else {
d = (p - yl) / (yh - yl)
}
dir++
} else {
x0 = x
yl = y
if dir > 0 {
dir = 0
d = 0.5
} else if dir < -1 {
d = 0.5 * d
} else {
d = (p - yl) / (yh - yl)
}
dir--
}
}
return x
}
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