Porting cephes igami.c

internal/cephes/igami.go

@@ -0,0 +1,202 @@
+// 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 2.0:  April, 1987
+ * Copyright 1985, 1987 by Stephen L. Moshier
+ * Direct inquiries to 30 Frost Street, Cambridge, MA 02140
+ */
+
+package cephes
+
+import (
+	"math"
+)
+
+/*
+ * Adapted from scipy's cephes igami.c
+ */
+
+/*
+ *
+ *      Inverse of complemented incomplete Gamma integral
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * a, x, p float64
+ *
+ * x = IgamI(a, p)
+ *
+ * DESCRIPTION:
+ *
+ * Given p, the function finds x such that
+ *
+ *  IgamC(a, x) = p
+ *
+ * Starting with the approximate value
+ *
+ *         3
+ *  x = a t
+ *
+ *  where
+ *
+ *  t = 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.
+ *
+ * ACCURACY:
+ *
+ * Tested at random a, p in the intervals indicated.
+ *
+ *                a        p                      Relative error:
+ * arithmetic   domain   domain     # trials      peak         rms
+ *    IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
+ *    IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
+ *    IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14
+ */
+
+/*
+ * Cephes Math Library Release 2.3:  March, 1995
+ * Copyright 1984, 1987, 1995 by Stephen L. Moshier
+ */
+
+// IgamI calculates the inverse of complemented incomplete Gamma integral
+func IgamI(a, y0 float64) float64 {
+	// bound the solution
+	x0 := math.MaxFloat64
+	yl := 0.0
+	x1 := 0.0
+	yh := 1.0
+	dithresh := 5.0 * machEp
+
+	if y0 < 0 || y0 > 1 || a <= 0 {
+		panic("IgamI: Domain error")
+	}
+
+	if y0 == 0 {
+		return math.MaxFloat64
+	}
+
+	if y0 == 1 {
+		return 0.0
+	}
+
+	// approximation to inverse function
+	d := 1.0 / (9.0 * a)
+	y := 1.0 - d - Ndtri(y0)*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 < y0 {
+			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 - y0) / 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 < y0 {
+				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 - y0) / y0
+		if math.Abs(lgm) < dithresh {
+			break
+		}
+
+		if x <= 0 {
+			break
+		}
+
+		if y >= y0 {
+			x1 = x
+			yh = y
+			if dir < 0 {
+				dir = 0
+				d = 0.5
+			} else if dir > 1 {
+				d = 0.5*d + 0.5
+			} else {
+				d = (y0 - 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 = (y0 - yl) / (yh - yl)
+			}
+			dir--
+		}
+	}
+
+	if x == 0 {
+		panic("IgamI: Underflow error")
+	}
+
+	return x
+}