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// Derived from SciPy's special/c_misc/gammaincinv.c
// https://github.com/scipy/scipy/blob/master/scipy/special/c_misc/gammaincinv.c
// Copyright ©2017 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.
package mathext
import (
"math"
"gonum.org/v1/gonum/mathext/internal/cephes"
)
const (
allowedATol = 1e-306
allowedRTol = 1e-6
)
func gammaIncReg(x float64, params []float64) float64 {
return cephes.Igam(params[0], x) - params[1]
}
// gammaIncRegInv is the inverse of the regularized incomplete Gamma integral. That is, it
// returns x such that:
//
// Igam(a, x) = y
//
// The input argument a must be positive and y must be between 0 and 1
// inclusive or gammaIncRegInv will panic. gammaIncRegInv should return a
// positive number, but can return NaN if there is a failure to converge.
func gammaIncRegInv(a, y float64) float64 {
// For y not small, we just use
// IgamI(a, 1-y)
// (inverse of the complemented incomplete Gamma integral). For y small,
// however, 1-y is about 1, and we lose digits.
if a <= 0 || y <= 0 || y >= 0.25 {
return cephes.IgamI(a, 1-y)
}
lo := 0.0
flo := -y
hi := cephes.IgamI(a, 0.75)
fhi := 0.25 - y
params := []float64{a, y}
// Also, after we generate a small interval by bisection above, false
// position will do a large step from an interval of width ~1e-4 to ~1e-14
// in one step (a=10, x=0.05, but similar for other values).
result, bestX, _, errEst := falsePosition(lo, hi, flo, fhi, 2*machEp, 2*machEp, 1e-2*a, gammaIncReg, params)
if result == fSolveMaxIterations && errEst > allowedATol+allowedRTol*math.Abs(bestX) {
bestX = math.NaN()
}
return bestX
}
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