001 /* 002 * Licensed to the Apache Software Foundation (ASF) under one or more 003 * contributor license agreements. See the NOTICE file distributed with 004 * this work for additional information regarding copyright ownership. 005 * The ASF licenses this file to You under the Apache License, Version 2.0 006 * (the "License"); you may not use this file except in compliance with 007 * the License. You may obtain a copy of the License at 008 * 009 * http://www.apache.org/licenses/LICENSE-2.0 010 * 011 * Unless required by applicable law or agreed to in writing, software 012 * distributed under the License is distributed on an "AS IS" BASIS, 013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 014 * See the License for the specific language governing permissions and 015 * limitations under the License. 016 */ 017 package org.apache.commons.math3.special; 018 019 import org.apache.commons.math3.util.FastMath; 020 021 /** 022 * This is a utility class that provides computation methods related to the 023 * error functions. 024 * 025 * @version $Id: Erf.java 1416643 2012-12-03 19:37:14Z tn $ 026 */ 027 public class Erf { 028 029 /** 030 * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally. 031 * This number solves {@code erf(x)=0.5} within 1ulp. 032 * More precisely, the current implementations of 033 * {@link #erf(double)} and {@link #erfc(double)} satisfy:<br/> 034 * {@code erf(X_CRIT) < 0.5},<br/> 035 * {@code erf(Math.nextUp(X_CRIT) > 0.5},<br/> 036 * {@code erfc(X_CRIT) = 0.5}, and<br/> 037 * {@code erfc(Math.nextUp(X_CRIT) < 0.5} 038 */ 039 private static final double X_CRIT = 0.4769362762044697; 040 041 /** 042 * Default constructor. Prohibit instantiation. 043 */ 044 private Erf() {} 045 046 /** 047 * Returns the error function. 048 * 049 * <p>erf(x) = 2/√π <sub>0</sub>∫<sup>x</sup> e<sup>-t<sup>2</sup></sup>dt </p> 050 * 051 * <p>This implementation computes erf(x) using the 052 * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function}, 053 * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</p> 054 * 055 * <p>The value returned is always between -1 and 1 (inclusive). 056 * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from 057 * either 1 or -1 as a double, so the appropriate extreme value is returned. 058 * </p> 059 * 060 * @param x the value. 061 * @return the error function erf(x) 062 * @throws org.apache.commons.math3.exception.MaxCountExceededException 063 * if the algorithm fails to converge. 064 * @see Gamma#regularizedGammaP(double, double, double, int) 065 */ 066 public static double erf(double x) { 067 if (FastMath.abs(x) > 40) { 068 return x > 0 ? 1 : -1; 069 } 070 final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000); 071 return x < 0 ? -ret : ret; 072 } 073 074 /** 075 * Returns the complementary error function. 076 * 077 * <p>erfc(x) = 2/√π <sub>x</sub>∫<sup>∞</sup> e<sup>-t<sup>2</sup></sup>dt 078 * <br/> 079 * = 1 - {@link #erf(double) erf(x)} </p> 080 * 081 * <p>This implementation computes erfc(x) using the 082 * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function}, 083 * following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</p> 084 * 085 * <p>The value returned is always between 0 and 2 (inclusive). 086 * If {@code abs(x) > 40}, then {@code erf(x)} is indistinguishable from 087 * either 0 or 2 as a double, so the appropriate extreme value is returned. 088 * </p> 089 * 090 * @param x the value 091 * @return the complementary error function erfc(x) 092 * @throws org.apache.commons.math3.exception.MaxCountExceededException 093 * if the algorithm fails to converge. 094 * @see Gamma#regularizedGammaQ(double, double, double, int) 095 * @since 2.2 096 */ 097 public static double erfc(double x) { 098 if (FastMath.abs(x) > 40) { 099 return x > 0 ? 0 : 2; 100 } 101 final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000); 102 return x < 0 ? 2 - ret : ret; 103 } 104 105 /** 106 * Returns the difference between erf(x1) and erf(x2). 107 * 108 * The implementation uses either erf(double) or erfc(double) 109 * depending on which provides the most precise result. 110 * 111 * @param x1 the first value 112 * @param x2 the second value 113 * @return erf(x2) - erf(x1) 114 */ 115 public static double erf(double x1, double x2) { 116 if(x1 > x2) { 117 return -erf(x2, x1); 118 } 119 120 return 121 x1 < -X_CRIT ? 122 x2 < 0.0 ? 123 erfc(-x2) - erfc(-x1) : 124 erf(x2) - erf(x1) : 125 x2 > X_CRIT && x1 > 0.0 ? 126 erfc(x1) - erfc(x2) : 127 erf(x2) - erf(x1); 128 } 129 } 130