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Before the Digamma function could be quite slow for large negative arguments because it had to do use the recurrence relation thousands of times. This avoids that by instead using the reflection formula. It also adds some explicit checks for special cases (Inf, NaN, poles).
45 lines
1.1 KiB
Go
45 lines
1.1 KiB
Go
// Copyright ©2016 The Gonum Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package mathext
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import (
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"math"
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)
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// Digamma returns the logorithmic derivative of the gamma function at x.
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// ψ(x) = d/dx (Ln (Γ(x)).
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func Digamma(x float64) float64 {
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// This is adapted from
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// http://web.science.mq.edu.au/~mjohnson/code/digamma.c
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var result float64
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switch {
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case math.IsNaN(x), math.IsInf(x, 1):
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return x
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case math.IsInf(x, -1):
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return math.NaN()
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case x == 0:
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return math.Copysign(math.Inf(1), -x)
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case x < 0:
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if x == math.Floor(x) {
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return math.NaN()
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}
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// Reflection formula, http://dlmf.nist.gov/5.5#E4
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_, r := math.Modf(x)
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result = -math.Pi / math.Tan(math.Pi * r)
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x = 1 - x
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}
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for ; x < 7; x++ {
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// Recurrence relation, http://dlmf.nist.gov/5.5#E2
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result -= 1 / x
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}
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x -= 0.5
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xx := 1 / x
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xx2 := xx * xx
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xx4 := xx2 * xx2
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// Asymptotic expansion, http://dlmf.nist.gov/5.11#E2
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result += math.Log(x) + (1.0/24.0)*xx2 - (7.0/960.0)*xx4 + (31.0/8064.0)*xx4*xx2 - (127.0/30720.0)*xx4*xx4
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return result
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}
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