SimpsonIntegration.java

packagecom.thealgorithms.maths;

importjava.util.TreeMap;

publicclassSimpsonIntegration {

/*
 * 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)}
 *
 */
publicstaticvoidmain(String[] args) {
SimpsonIntegrationintegration = newSimpsonIntegration();

// Give random data for the example purposes
intn = 16;
doublea = 1;
doubleb = 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
doubleh = (b - a) / (double) n;
doubleintegralEvaluation = 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
 */
publicdoublesimpsonsMethod(intn, doubleh, doublea) {
TreeMap<Integer, Double> data = newTreeMap<>(); // Key: i, Value: f(xi)
doubletemp;
doublexi = a; // Initialize the variable xi = x0 + 0*h

// Create the table of xi and yi points
for (inti = 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
doubleintegralEvaluation = 0;
for (inti = 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");
 } elseif (i % 2 != 0) {
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
returnintegralEvaluation;
 }

// Sample function f
// Function f(x) = e^(-x) * (4 - x^2)
publicdoublef(doublex) {
returnMath.exp(-x) * (4 - Math.pow(x, 2));
// return Math.sqrt(x);
 }
}