MATH-1416: Delete functionality now in "Commons Numbers".
This commit is contained in:
Gilles
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7f74708201
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b81be1fea3
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.special;
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import org.apache.commons.math4.util.FastMath;
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import org.apache.commons.numbers.gamma.RegularizedGamma;
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/**
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* This is a utility class that provides computation methods related to the
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* error functions.
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*/
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public class Erf {
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/**
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* The number {@code X_CRIT} is used by {@link #erf(double, double)} internally.
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* This number solves {@code erf(x)=0.5} within 1ulp.
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* More precisely, the current implementations of
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* {@link #erf(double)} and {@link #erfc(double)} satisfy:<br>
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* {@code erf(X_CRIT) < 0.5},<br>
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* {@code erf(Math.nextUp(X_CRIT) > 0.5},<br>
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* {@code erfc(X_CRIT) = 0.5}, and<br>
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* {@code erfc(Math.nextUp(X_CRIT) < 0.5}
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*/
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private static final double X_CRIT = 0.4769362762044697;
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/**
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* Default constructor. Prohibit instantiation.
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*/
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private Erf() {}
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/**
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* Returns the error function.
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*
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* <p>erf(x) = 2/√π <sub>0</sub>∫<sup>x</sup> e<sup>-t<span style="position: relative; top: -.5em">2</span></sup>dt </p>
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*
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* <p>This implementation computes erf(x) using the
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* {@link RegularizedGamma.P.value(double, double, double, int) regularized gamma function},
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* following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p>
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*
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* <p>The value returned is always between -1 and 1 (inclusive).
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* If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
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* either 1 or -1 as a double, so the appropriate extreme value is returned.
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* </p>
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*
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* @param x the value.
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* @return the error function erf(x)
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* @throws org.apache.commons.math4.exception.MaxCountExceededException
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* if the algorithm fails to converge.
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* @see RegularizedGamma.P#value(double, double, double, int)
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*/
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public static double erf(double x) {
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if (FastMath.abs(x) > 40) {
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return x > 0 ? 1 : -1;
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}
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final double ret = RegularizedGamma.P.value(0.5, x * x, 1.0e-15, 10000);
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return x < 0 ? -ret : ret;
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}
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/**
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* Returns the complementary error function.
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*
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* <p>erfc(x) = 2/√π <sub>x</sub>∫<sup>∞</sup> e<sup>-t<span style="position: relative; top: -.5em">2</span></sup>dt
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* <br>
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* = 1 - {@link #erf(double) erf(x)} </p>
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*
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* <p>This implementation computes erfc(x) using the
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* {@link RegularizedGamma.Q#value(double, double, double, int) regularized gamma function},
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* following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p>
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*
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* <p>The value returned is always between 0 and 2 (inclusive).
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* If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from
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* either 0 or 2 as a double, so the appropriate extreme value is returned.
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* </p>
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*
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* @param x the value
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* @return the complementary error function erfc(x)
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* @throws org.apache.commons.math4.exception.MaxCountExceededException
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* if the algorithm fails to converge.
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* @see RegularizedGamma.Q#value(double, double, double, int)
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* @since 2.2
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*/
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public static double erfc(double x) {
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if (FastMath.abs(x) > 40) {
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return x > 0 ? 0 : 2;
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}
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final double ret = RegularizedGamma.Q.value(0.5, x * x, 1.0e-15, 10000);
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return x < 0 ? 2 - ret : ret;
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}
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/**
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* Returns the difference between erf(x1) and erf(x2).
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*
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* The implementation uses either erf(double) or erfc(double)
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* depending on which provides the most precise result.
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*
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* @param x1 the first value
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* @param x2 the second value
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* @return erf(x2) - erf(x1)
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*/
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public static double erf(double x1, double x2) {
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if(x1 > x2) {
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return -erf(x2, x1);
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}
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return
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x1 < -X_CRIT ?
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x2 < 0.0 ?
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erfc(-x2) - erfc(-x1) :
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erf(x2) - erf(x1) :
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x2 > X_CRIT && x1 > 0.0 ?
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erfc(x1) - erfc(x2) :
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erf(x2) - erf(x1);
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}
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/**
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* Returns the inverse erf.
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* <p>
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* This implementation is described in the paper:
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* <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
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* the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
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* which was published in GPU Computing Gems, volume 2, 2010.
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* The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
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* </p>
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* @param x the value
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* @return t such that x = erf(t)
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* @since 3.2
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*/
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public static double erfInv(final double x) {
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// beware that the logarithm argument must be
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// commputed as (1.0 - x) * (1.0 + x),
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// it must NOT be simplified as 1.0 - x * x as this
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// would induce rounding errors near the boundaries +/-1
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double w = - FastMath.log((1.0 - x) * (1.0 + x));
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double p;
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if (w < 6.25) {
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w -= 3.125;
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p = -3.6444120640178196996e-21;
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p = -1.685059138182016589e-19 + p * w;
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p = 1.2858480715256400167e-18 + p * w;
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p = 1.115787767802518096e-17 + p * w;
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p = -1.333171662854620906e-16 + p * w;
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p = 2.0972767875968561637e-17 + p * w;
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p = 6.6376381343583238325e-15 + p * w;
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p = -4.0545662729752068639e-14 + p * w;
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p = -8.1519341976054721522e-14 + p * w;
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p = 2.6335093153082322977e-12 + p * w;
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p = -1.2975133253453532498e-11 + p * w;
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p = -5.4154120542946279317e-11 + p * w;
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p = 1.051212273321532285e-09 + p * w;
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p = -4.1126339803469836976e-09 + p * w;
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p = -2.9070369957882005086e-08 + p * w;
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p = 4.2347877827932403518e-07 + p * w;
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p = -1.3654692000834678645e-06 + p * w;
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p = -1.3882523362786468719e-05 + p * w;
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p = 0.0001867342080340571352 + p * w;
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p = -0.00074070253416626697512 + p * w;
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p = -0.0060336708714301490533 + p * w;
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p = 0.24015818242558961693 + p * w;
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p = 1.6536545626831027356 + p * w;
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} else if (w < 16.0) {
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w = FastMath.sqrt(w) - 3.25;
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p = 2.2137376921775787049e-09;
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p = 9.0756561938885390979e-08 + p * w;
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p = -2.7517406297064545428e-07 + p * w;
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p = 1.8239629214389227755e-08 + p * w;
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p = 1.5027403968909827627e-06 + p * w;
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p = -4.013867526981545969e-06 + p * w;
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p = 2.9234449089955446044e-06 + p * w;
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p = 1.2475304481671778723e-05 + p * w;
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p = -4.7318229009055733981e-05 + p * w;
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p = 6.8284851459573175448e-05 + p * w;
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p = 2.4031110387097893999e-05 + p * w;
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p = -0.0003550375203628474796 + p * w;
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p = 0.00095328937973738049703 + p * w;
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p = -0.0016882755560235047313 + p * w;
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p = 0.0024914420961078508066 + p * w;
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p = -0.0037512085075692412107 + p * w;
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p = 0.005370914553590063617 + p * w;
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p = 1.0052589676941592334 + p * w;
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p = 3.0838856104922207635 + p * w;
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} else if (!Double.isInfinite(w)) {
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w = FastMath.sqrt(w) - 5.0;
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p = -2.7109920616438573243e-11;
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p = -2.5556418169965252055e-10 + p * w;
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p = 1.5076572693500548083e-09 + p * w;
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p = -3.7894654401267369937e-09 + p * w;
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p = 7.6157012080783393804e-09 + p * w;
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p = -1.4960026627149240478e-08 + p * w;
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p = 2.9147953450901080826e-08 + p * w;
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p = -6.7711997758452339498e-08 + p * w;
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p = 2.2900482228026654717e-07 + p * w;
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p = -9.9298272942317002539e-07 + p * w;
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p = 4.5260625972231537039e-06 + p * w;
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p = -1.9681778105531670567e-05 + p * w;
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p = 7.5995277030017761139e-05 + p * w;
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p = -0.00021503011930044477347 + p * w;
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p = -0.00013871931833623122026 + p * w;
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p = 1.0103004648645343977 + p * w;
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p = 4.8499064014085844221 + p * w;
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} else {
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// this branch does not appears in the original code, it
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// was added because the previous branch does not handle
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// x = +/-1 correctly. In this case, w is positive infinity
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// and as the first coefficient (-2.71e-11) is negative.
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// Once the first multiplication is done, p becomes negative
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// infinity and remains so throughout the polynomial evaluation.
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// So the branch above incorrectly returns negative infinity
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// instead of the correct positive infinity.
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p = Double.POSITIVE_INFINITY;
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}
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return p * x;
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}
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/**
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* Returns the inverse erfc.
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* @param x the value
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* @return t such that x = erfc(t)
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* @since 3.2
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*/
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public static double erfcInv(final double x) {
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return erfInv(1 - x);
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}
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}
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@ -1,261 +0,0 @@
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/*
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* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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package org.apache.commons.math4.special;
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import org.apache.commons.math4.TestUtils;
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import org.apache.commons.math4.special.Erf;
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import org.apache.commons.math4.util.FastMath;
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import org.junit.Test;
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import org.junit.Assert;
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/**
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*/
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public class ErfTest {
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@Test
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public void testErf0() {
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double actual = Erf.erf(0.0);
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double expected = 0.0;
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Assert.assertEquals(expected, actual, 1.0e-15);
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Assert.assertEquals(1 - expected, Erf.erfc(0.0), 1.0e-15);
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}
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@Test
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public void testErf1960() {
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double x = 1.960 / FastMath.sqrt(2.0);
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double actual = Erf.erf(x);
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double expected = 0.95;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(x), 1.0e-15);
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actual = Erf.erf(-x);
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expected = -expected;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(-x), 1.0e-15);
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}
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@Test
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public void testErf2576() {
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double x = 2.576 / FastMath.sqrt(2.0);
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double actual = Erf.erf(x);
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double expected = 0.99;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(x), 1e-15);
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actual = Erf.erf(-x);
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expected = -expected;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(-x), 1.0e-15);
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}
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@Test
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public void testErf2807() {
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double x = 2.807 / FastMath.sqrt(2.0);
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double actual = Erf.erf(x);
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double expected = 0.995;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(x), 1.0e-15);
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actual = Erf.erf(-x);
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expected = -expected;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - actual, Erf.erfc(-x), 1.0e-15);
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}
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@Test
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public void testErf3291() {
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double x = 3.291 / FastMath.sqrt(2.0);
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double actual = Erf.erf(x);
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double expected = 0.999;
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Assert.assertEquals(expected, actual, 1.0e-5);
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Assert.assertEquals(1 - expected, Erf.erfc(x), 1.0e-5);
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actual = Erf.erf(-x);
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expected = -expected;
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Assert.assertEquals(expected, actual, 1.0e-5);
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||||||
Assert.assertEquals(1 - expected, Erf.erfc(-x), 1.0e-5);
|
|
||||||
}
|
|
||||||
|
|
||||||
/**
|
|
||||||
* MATH-301, MATH-456
|
|
||||||
*/
|
|
||||||
@Test
|
|
||||||
public void testLargeValues() {
|
|
||||||
for (int i = 1; i < 200; i*=10) {
|
|
||||||
double result = Erf.erf(i);
|
|
||||||
Assert.assertFalse(Double.isNaN(result));
|
|
||||||
Assert.assertTrue(result > 0 && result <= 1);
|
|
||||||
result = Erf.erf(-i);
|
|
||||||
Assert.assertFalse(Double.isNaN(result));
|
|
||||||
Assert.assertTrue(result >= -1 && result < 0);
|
|
||||||
result = Erf.erfc(i);
|
|
||||||
Assert.assertFalse(Double.isNaN(result));
|
|
||||||
Assert.assertTrue(result >= 0 && result < 1);
|
|
||||||
result = Erf.erfc(-i);
|
|
||||||
Assert.assertFalse(Double.isNaN(result));
|
|
||||||
Assert.assertTrue(result >= 1 && result <= 2);
|
|
||||||
}
|
|
||||||
Assert.assertEquals(-1, Erf.erf(Double.NEGATIVE_INFINITY), 0);
|
|
||||||
Assert.assertEquals(1, Erf.erf(Double.POSITIVE_INFINITY), 0);
|
|
||||||
Assert.assertEquals(2, Erf.erfc(Double.NEGATIVE_INFINITY), 0);
|
|
||||||
Assert.assertEquals(0, Erf.erfc(Double.POSITIVE_INFINITY), 0);
|
|
||||||
}
|
|
||||||
|
|
||||||
/**
|
|
||||||
* Compare Erf.erf against reference values computed using GCC 4.2.1 (Apple OSX packaged version)
|
|
||||||
* erfl (extended precision erf).
|
|
||||||
*/
|
|
||||||
@Test
|
|
||||||
public void testErfGnu() {
|
|
||||||
final double tol = 1E-15;
|
|
||||||
final double[] gnuValues = new double[] {-1, -1, -1, -1, -1,
|
|
||||||
-1, -1, -1, -0.99999999999999997848,
|
|
||||||
-0.99999999999999264217, -0.99999999999846254017, -0.99999999980338395581, -0.99999998458274209971,
|
|
||||||
-0.9999992569016276586, -0.99997790950300141459, -0.99959304798255504108, -0.99532226501895273415,
|
|
||||||
-0.96610514647531072711, -0.84270079294971486948, -0.52049987781304653809, 0,
|
|
||||||
0.52049987781304653809, 0.84270079294971486948, 0.96610514647531072711, 0.99532226501895273415,
|
|
||||||
0.99959304798255504108, 0.99997790950300141459, 0.9999992569016276586, 0.99999998458274209971,
|
|
||||||
0.99999999980338395581, 0.99999999999846254017, 0.99999999999999264217, 0.99999999999999997848,
|
|
||||||
1, 1, 1, 1,
|
|
||||||
1, 1, 1, 1};
|
|
||||||
double x = -10d;
|
|
||||||
for (int i = 0; i < 41; i++) {
|
|
||||||
Assert.assertEquals(gnuValues[i], Erf.erf(x), tol);
|
|
||||||
x += 0.5d;
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/**
|
|
||||||
* Compare Erf.erfc against reference values computed using GCC 4.2.1 (Apple OSX packaged version)
|
|
||||||
* erfcl (extended precision erfc).
|
|
||||||
*/
|
|
||||||
@Test
|
|
||||||
public void testErfcGnu() {
|
|
||||||
final double tol = 1E-15;
|
|
||||||
final double[] gnuValues = new double[] { 2, 2, 2, 2, 2,
|
|
||||||
2, 2, 2, 1.9999999999999999785,
|
|
||||||
1.9999999999999926422, 1.9999999999984625402, 1.9999999998033839558, 1.9999999845827420998,
|
|
||||||
1.9999992569016276586, 1.9999779095030014146, 1.9995930479825550411, 1.9953222650189527342,
|
|
||||||
1.9661051464753107271, 1.8427007929497148695, 1.5204998778130465381, 1,
|
|
||||||
0.47950012218695346194, 0.15729920705028513051, 0.033894853524689272893, 0.0046777349810472658333,
|
|
||||||
0.00040695201744495893941, 2.2090496998585441366E-05, 7.4309837234141274516E-07, 1.5417257900280018858E-08,
|
|
||||||
1.966160441542887477E-10, 1.5374597944280348501E-12, 7.3578479179743980661E-15, 2.1519736712498913103E-17,
|
|
||||||
3.8421483271206474691E-20, 4.1838256077794144006E-23, 2.7766493860305691016E-26, 1.1224297172982927079E-29,
|
|
||||||
2.7623240713337714448E-33, 4.1370317465138102353E-37, 3.7692144856548799402E-41, 2.0884875837625447567E-45};
|
|
||||||
double x = -10d;
|
|
||||||
for (int i = 0; i < 41; i++) {
|
|
||||||
Assert.assertEquals(gnuValues[i], Erf.erfc(x), tol);
|
|
||||||
x += 0.5d;
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/**
|
|
||||||
* Tests erfc against reference data computed using Maple reported in Marsaglia, G,,
|
|
||||||
* "Evaluating the Normal Distribution," Journal of Statistical Software, July, 2004.
|
|
||||||
* http//www.jstatsoft.org/v11/a05/paper
|
|
||||||
*/
|
|
||||||
@Test
|
|
||||||
public void testErfcMaple() {
|
|
||||||
double[][] ref = new double[][]
|
|
||||||
{{0.1, 4.60172162722971e-01},
|
|
||||||
{1.2, 1.15069670221708e-01},
|
|
||||||
{2.3, 1.07241100216758e-02},
|
|
||||||
{3.4, 3.36929265676881e-04},
|
|
||||||
{4.5, 3.39767312473006e-06},
|
|
||||||
{5.6, 1.07175902583109e-08},
|
|
||||||
{6.7, 1.04209769879652e-11},
|
|
||||||
{7.8, 3.09535877195870e-15},
|
|
||||||
{8.9, 2.79233437493966e-19},
|
|
||||||
{10.0, 7.61985302416053e-24},
|
|
||||||
{11.1, 6.27219439321703e-29},
|
|
||||||
{12.2, 1.55411978638959e-34},
|
|
||||||
{13.3, 1.15734162836904e-40},
|
|
||||||
{14.4, 2.58717592540226e-47},
|
|
||||||
{15.5, 1.73446079179387e-54},
|
|
||||||
{16.6, 3.48454651995041e-62}
|
|
||||||
};
|
|
||||||
for (int i = 0; i < 15; i++) {
|
|
||||||
final double result = 0.5*Erf.erfc(ref[i][0]/FastMath.sqrt(2));
|
|
||||||
Assert.assertEquals(ref[i][1], result, 1E-15);
|
|
||||||
TestUtils.assertRelativelyEquals(ref[i][1], result, 1E-13);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
/**
|
|
||||||
* Test the implementation of Erf.erf(double, double) for consistency with results
|
|
||||||
* obtained from Erf.erf(double) and Erf.erfc(double).
|
|
||||||
*/
|
|
||||||
@Test
|
|
||||||
public void testTwoArgumentErf() {
|
|
||||||
double[] xi = new double[]{-2.0, -1.0, -0.9, -0.1, 0.0, 0.1, 0.9, 1.0, 2.0};
|
|
||||||
for(double x1 : xi) {
|
|
||||||
for(double x2 : xi) {
|
|
||||||
double a = Erf.erf(x1, x2);
|
|
||||||
double b = Erf.erf(x2) - Erf.erf(x1);
|
|
||||||
double c = Erf.erfc(x1) - Erf.erfc(x2);
|
|
||||||
Assert.assertEquals(a, b, 1E-15);
|
|
||||||
Assert.assertEquals(a, c, 1E-15);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
}
|
|
||||||
|
|
||||||
@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);
|
|
||||||
}
|
|
||||||
}
|
|
||||||
}
|
|
||||||
Loading…
Reference in New Issue