JavaAlgorithms/Maths/SimpsonIntegration.java

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