【0】README

0.1）本文部分文字描述转自或译自 https://en.wikipedia.org/wiki/Simpson%27s_rule和 https://en.wikipedia.org/wiki/Numerical_integration#Methods_for_one-dimensional_integrals；旨在利用java求积分；（定积分和无穷限积分）

0.2）you can also refer to this link for source code: https://github.com/pacosonTang/postgraduate-research/tree/master/integration

0.3）o m g. CSDN编辑器掉链子，无法正常显示source code, 大家凑合着看吧。oh.

【1】求定积分

1）intro：由 wikepedia 上关于 辛普森法的intro 以及 《高等数学第6版上册同济版》p229 关于定积分的近似计算中提到的辛普森法，本文求定积分的方法采用了辛普森近似法；

2）下面引用《高等数学第6版上册同济版》p229 关于辛普森法的描述

3）计算函数定积分的源代码如下：

// compute the numeric integration.
public class Integration {public Integration(){}// apply simpson rule to approximately compute the integration. public double simpsonRule(double upper, double lower, int n, Function df) {double result = 0;double unit = (upper-lower)/n;double factor1 = unit / 3;double[] x = new double[n+1];for (int i = 0; i < x.length; i++) {x[i] = lower + unit*i;}for (int i = 0; i < x.length; i++) {if(i==0 || i==x.length-1) {result += df.fun(x[i]);}else if(i%2 == 0) { // if i is even num.result += 2*df.fun(x[i]);}else { // if i is odd num.  result += 4*df.fun(x[i]);}}                result *= factor1;return result;}// compute the standard normal distribution integration// refer to the integration table in p382 of "probability and statistics" from ZheJiang University.public double stdGaussValue(double realUpper) {Integration integration = new Integration();double upper = 1.0;double lower = 0.0;int n = 200; // splited into 200 subintervals.// double realUpper = 0.03;if(realUpper >= 5.0) {return 1.0;}double result = integration.simpsonRule(upper, lower, n, new Function() {@Overridepublic double fun(double x) {if(x==0) {return 0;}double t =  realUpper-(1-x)/x;return Math.pow(Math.E, -0.5*t*t) / (x*x);         }});result /= Math.pow(2*Math.PI, 0.5);result = new BigDecimal(result).setScale(6, RoundingMode.HALF_UP).doubleValue(); // save 6 decimal places.return result;}
}

public class IntegrationTest { //test case.public static void main(String[] args) {Integration integration = new Integration();double result = integration.stdGaussValue(4.42);System.out.println(result);}public static void main3(String[] args) {Integration integration = new Integration();double upper = 1.0;double lower = 0.0;int n = 50;double realUpper = 0.39;double result = integration.simpsonRule(upper, lower, n, new Function() {@Overridepublic double fun(double x) {if(x==0) {return 0;}double t =  realUpper-(1-x)/x;return Math.pow(Math.E, -0.5*t*t) / (x*x);           }});result /= Math.pow(2*Math.PI, 0.5);result = new BigDecimal(result).setScale(4, RoundingMode.HALF_UP).doubleValue();       System.out.println(result);}public static void main2(String[] args) {Integration integration = new Integration();double upper = 1.0;double lower = 0.0;int n = 10;double result = integration.simpsonRule(upper, lower, n, new Function() {@Overridepublic double fun(double x) {return Math.pow(Math.E, -x*x/2);         }});result /= Math.pow(2*Math.PI, 0.5);System.out.println(result);BigDecimal decimal = new BigDecimal(result).setScale(4, RoundingMode.HALF_UP);result = Double.valueOf(decimal.toString());System.out.println(result);}public static void main1(String[] args) {Integration integration = new Integration();double upper = 1.0;double lower = 0;int n = 10;double result = integration.simpsonRule(upper, lower, n, new Function() {@Overridepublic double fun(double x) {return 4 / (1+Math.pow(x,2.0));            }});System.out.println(result);}
}

Attention）

【2】求无穷限积分（本文以求标准正态分布的无穷下限反常积分为例）

1）求无穷限积分是基于定积分的；如何求定积分，本文在章节【1】中已经讲了；

2）所以标准正态分布的无穷下限反常积分函数可转化为：

3）计算标准正态分布无穷下限积分的测试用例如上所示。

Attention）

A1）上述求标准正态分布无穷下限积分的代码对realUpper 有要求，小于等于5.0；因为当realUpper>5的话，其value=1了；

A2）需要求标准正态分布的下限积分时，强烈建议使用 integration.stdGaussValue() 其精度要高些。