利用java求积分（定积分和无穷限积分）

Posted 2023-02-01 PacosonSWJTU

【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() 
					@Override
					public 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() 
					@Override
					public 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() 
					@Override
					public 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() 
					@Override
					public double fun(double x) 
						return 4 / (1+Math.pow(x,2.0));			
					
				);
		System.out.println(result);
	

Attention）

A1）以上测试用例中涉及到的积分函数来自 《高等数学第6版上册同济版》p230的例2； A2）定积分表达式为

【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。