93 lines
3.1 KiB
Java
93 lines
3.1 KiB
Java
package Maths;
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import java.util.TreeMap;
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public class SimpsonIntegration{
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/*
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* Calculate definite integrals by using Composite Simpson's rule.
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* Wiki: https://en.wikipedia.org/wiki/Simpson%27s_rule#Composite_Simpson's_rule
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* Given f a function and an even number N of intervals that divide the integration interval e.g. [a, b],
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* we calculate the step h = (b-a)/N and create a table that contains all the x points of
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* the real axis xi = x0 + i*h and the value f(xi) that corresponds to these xi.
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*
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* To evaluate the integral i use the formula below:
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* I = h/3 * {f(x0) + 4*f(x1) + 2*f(x2) + 4*f(x3) + ... + 2*f(xN-2) + 4*f(xN-1) + f(xN)}
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*
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*/
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public static void main(String[] args) {
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SimpsonIntegration integration = new SimpsonIntegration();
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// Give random data for the example purposes
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int N = 16;
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double a = 1;
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double b = 3;
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// Check so that N is even
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if(N%2 != 0){
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System.out.println("N must be even number for Simpsons method. Aborted");
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System.exit(1);
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}
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// Calculate step h and evaluate the integral
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double h = (b-a) / (double) N;
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double integralEvaluation = integration.simpsonsMethod(N, h, a);
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System.out.println("The integral is equal to: " + integralEvaluation);
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}
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/*
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* @param N: Number of intervals (must be even number N=2*k)
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* @param h: Step h = (b-a)/N
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* @param a: Starting point of the interval
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* @param b: Ending point of the interval
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*
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* The interpolation points xi = x0 + i*h are stored the treeMap data
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*
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* @return result of the integral evaluation
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*/
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public double simpsonsMethod(int N, double h, double a){
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TreeMap<Integer, Double> data = new TreeMap<>(); // Key: i, Value: f(xi)
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double temp;
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double xi = a; // Initialize the variable xi = x0 + 0*h
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// Create the table of xi and yi points
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for(int i=0; i<=N; i++){
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temp = f(xi); // Get the value of the function at that point
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data.put(i, temp);
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xi += h; // Increase the xi to the next point
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}
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// Apply the formula
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double integralEvaluation = 0;
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for(int i=0; i<data.size(); i++){
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if(i == 0 || i == data.size()-1) {
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integralEvaluation += data.get(i);
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System.out.println("Multiply f(x" + i + ") by 1");
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}
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else if(i%2 == 1) {
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integralEvaluation += (double) 4 * data.get(i);
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System.out.println("Multiply f(x" + i + ") by 4");
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}
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else {
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integralEvaluation += (double) 2 * data.get(i);
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System.out.println("Multiply f(x" + i + ") by 2");
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}
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}
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// Multiply by h/3
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integralEvaluation = h/3 * integralEvaluation;
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// Return the result
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return integralEvaluation;
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}
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// Sample function f
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// Function f(x) = e^(-x) * (4 - x^2)
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public double f(double x){
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return Math.exp(-x) * (4 - Math.pow(x, 2));
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// return Math.sqrt(x);
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}
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}
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