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Inverse error function and inverse complementary error function. 

JIRA: MATH-948

git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1456905 13f79535-47bb-0310-9956-ffa450edef68
This commit is contained in:
Luc Maisonobe 2013-03-15 11:37:35 +00:00
parent f636e77707
commit 747476596e
4 changed files with 169 additions and 0 deletions
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5
NOTICE.txt
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@ -6,6 +6,11 @@ The Apache Software Foundation (http://www.apache.org/).
===============================================================================
The inverse error function implementation in the Erf class is based on CUDA
code developed by Mike Giles, Oxford-Man Institute of Quantitative Finance,
and published in GPU Computing Gems, volume 2, 2010.
===============================================================================
The BracketFinder (package org.apache.commons.math3.optimization.univariate)
and PowellOptimizer (package org.apache.commons.math3.optimization.general)
classes are based on the Python code in module "optimize.py" (version 0.5)

4
src/changes/changes.xml
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@ -55,6 +55,10 @@ This is a minor release: It combines bug fixes and new features.
Changes to existing features were made in a backwards-compatible
way such as to allow drop-in replacement of the v3.1[.1] JAR file.
">
<action dev="luc" type="add" issue="MATH-948" >
Implementations for inverse error function and inverse complementary
error functions have been added.
</action>
<action dev="luc" type="fix" issue="MATH-580" >
Extended ranges for FastMath performance tests.
</action>

114
src/main/java/org/apache/commons/math3/special/Erf.java
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@ -126,5 +126,119 @@ public class Erf {
erfc(x1) - erfc(x2) :
erf(x2) - erf(x1);
}
/**
* Returns the inverse erf.
* <p>
* This implementation is described in the paper:
* <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
* the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
* which was published in GPU Computing Gems, volume 2, 2010.
* The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
* </p>
* @param x the value
* @return t such that x = erf(t)
* @since 3.2
*/
public static double erfInv(final double x) {
// beware that the logarithm argument must be
// commputed as (1.0 - x) * (1.0 + x),
// it must NOT be simplified as 1.0 - x * x as this
// would induce rounding errors near the boundaries +/-1
double w = - FastMath.log((1.0 - x) * (1.0 + x));
double p;
if (w < 6.25) {
w = w - 3.125;
p = -3.6444120640178196996e-21;
p = -1.685059138182016589e-19 + p * w;
p = 1.2858480715256400167e-18 + p * w;
p = 1.115787767802518096e-17 + p * w;
p = -1.333171662854620906e-16 + p * w;
p = 2.0972767875968561637e-17 + p * w;
p = 6.6376381343583238325e-15 + p * w;
p = -4.0545662729752068639e-14 + p * w;
p = -8.1519341976054721522e-14 + p * w;
p = 2.6335093153082322977e-12 + p * w;
p = -1.2975133253453532498e-11 + p * w;
p = -5.4154120542946279317e-11 + p * w;
p = 1.051212273321532285e-09 + p * w;
p = -4.1126339803469836976e-09 + p * w;
p = -2.9070369957882005086e-08 + p * w;
p = 4.2347877827932403518e-07 + p * w;
p = -1.3654692000834678645e-06 + p * w;
p = -1.3882523362786468719e-05 + p * w;
p = 0.0001867342080340571352 + p * w;
p = -0.00074070253416626697512 + p * w;
p = -0.0060336708714301490533 + p * w;
p = 0.24015818242558961693 + p * w;
p = 1.6536545626831027356 + p * w;
} else if (w < 16.0) {
w = FastMath.sqrt(w) - 3.25;
p = 2.2137376921775787049e-09;
p = 9.0756561938885390979e-08 + p * w;
p = -2.7517406297064545428e-07 + p * w;
p = 1.8239629214389227755e-08 + p * w;
p = 1.5027403968909827627e-06 + p * w;
p = -4.013867526981545969e-06 + p * w;
p = 2.9234449089955446044e-06 + p * w;
p = 1.2475304481671778723e-05 + p * w;
p = -4.7318229009055733981e-05 + p * w;
p = 6.8284851459573175448e-05 + p * w;
p = 2.4031110387097893999e-05 + p * w;
p = -0.0003550375203628474796 + p * w;
p = 0.00095328937973738049703 + p * w;
p = -0.0016882755560235047313 + p * w;
p = 0.0024914420961078508066 + p * w;
p = -0.0037512085075692412107 + p * w;
p = 0.005370914553590063617 + p * w;
p = 1.0052589676941592334 + p * w;
p = 3.0838856104922207635 + p * w;
} else if (!Double.isInfinite(w)) {
w = FastMath.sqrt(w) - 5.0;
p = -2.7109920616438573243e-11;
p = -2.5556418169965252055e-10 + p * w;
p = 1.5076572693500548083e-09 + p * w;
p = -3.7894654401267369937e-09 + p * w;
p = 7.6157012080783393804e-09 + p * w;
p = -1.4960026627149240478e-08 + p * w;
p = 2.9147953450901080826e-08 + p * w;
p = -6.7711997758452339498e-08 + p * w;
p = 2.2900482228026654717e-07 + p * w;
p = -9.9298272942317002539e-07 + p * w;
p = 4.5260625972231537039e-06 + p * w;
p = -1.9681778105531670567e-05 + p * w;
p = 7.5995277030017761139e-05 + p * w;
p = -0.00021503011930044477347 + p * w;
p = -0.00013871931833623122026 + p * w;
p = 1.0103004648645343977 + p * w;
p = 4.8499064014085844221 + p * w;
} else {
// this branch does not appears in the original code, it
// was added because the previous branch does not handle
// x = +/-1 correctly. In this case, w is positive infinity
// and as the first coefficient (-2.71e-11) is negative.
// Once the first multiplication is done, p becomes negative
// infinity and remains so throughout the polynomial evaluation.
// So the branch above incorrectly returns negative infinity
// instead of the correct positive infinity.
p = Double.POSITIVE_INFINITY;
}
return p * x;
}
/**
* Returns the inverse erfc.
* @param x the value
* @return t such that x = erfc(t)
* @since 3.2
*/
public static double erfcInv(final double x) {
return erfInv(1 - x);
}
}

46
src/test/java/org/apache/commons/math3/special/ErfTest.java
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@ -213,4 +213,50 @@ public class ErfTest {
}
}
}
@Test
public void testErfInvNaN() {
Assert.assertTrue(Double.isNaN(Erf.erfInv(-1.001)));
Assert.assertTrue(Double.isNaN(Erf.erfInv(+1.001)));
}
@Test
public void testErfInvInfinite() {
Assert.assertTrue(Double.isInfinite(Erf.erfInv(-1)));
Assert.assertTrue(Erf.erfInv(-1) < 0);
Assert.assertTrue(Double.isInfinite(Erf.erfInv(+1)));
Assert.assertTrue(Erf.erfInv(+1) > 0);
}
@Test
public void testErfInv() {
for (double x = -5.9; x < 5.9; x += 0.01) {
final double y = Erf.erf(x);
final double dydx = 2 * FastMath.exp(-x * x) / FastMath.sqrt(FastMath.PI);
Assert.assertEquals(x, Erf.erfInv(y), 1.0e-15 / dydx);
}
}
@Test
public void testErfcInvNaN() {
Assert.assertTrue(Double.isNaN(Erf.erfcInv(-0.001)));
Assert.assertTrue(Double.isNaN(Erf.erfcInv(+2.001)));
}
@Test
public void testErfcInvInfinite() {
Assert.assertTrue(Double.isInfinite(Erf.erfcInv(-0)));
Assert.assertTrue(Erf.erfcInv( 0) > 0);
Assert.assertTrue(Double.isInfinite(Erf.erfcInv(+2)));
Assert.assertTrue(Erf.erfcInv(+2) < 0);
}
@Test
public void testErfcInv() {
for (double x = -5.85; x < 5.9; x += 0.01) {
final double y = Erf.erfc(x);
final double dydxAbs = 2 * FastMath.exp(-x * x) / FastMath.sqrt(FastMath.PI);
Assert.assertEquals(x, Erf.erfcInv(y), 1.0e-15 / dydxAbs);
}
}
}