Computer Programming

Here's a simple implementation of Milne's and Adams' predictor-corrector methods in C++.

#include <iostream>
#include <vector>
#include <functional>

double f(double t, double y) { 
    return t + y; // Example ODE: dy/dt = t + y 
}

void adamsPredictorCorrector(double h, double t0, double y0, int n) {
    std::vector<double> y(n + 1);
    std::vector<double> t(n + 1);

    t[0] = t0;
    y[0] = y0;

    // Initial values (can use Euler or RK4 for the first few steps)
    for (int i = 1; i <= 3; ++i) {
        t[i] = t0 + i * h;
        y[i] = y[i - 1] + h * f(t[i - 1], y[i - 1]);
    }

    for (int i = 3; i < n; ++i) {
        // Predictor
        double y_predictor = y[i] + (h / 24) * (9 * f(t[i], y[i]) - 19 * f(t[i - 1], y[i - 1]) + 5 * f(t[i - 2], y[i - 2]) - f(t[i - 3], y[i - 3]));
        // Corrector
        y[i + 1] = y[i] + (h / 24) * (f(t[i + 1], y_predictor) + 19 * f(t[i], y[i]) - 5 * f(t[i - 1], y[i - 1]) + f(t[i - 2], y[i - 2]));
    }

    for (int i = 0; i <= n; ++i) {
        std::cout << "t = " << t[i] << ", y = " << y[i] << std::endl;
    }
}

int main() {
    double h = 0.1; // Step size
    double t0 = 0;  // Initial time
    double y0 = 1;  // Initial value of y
    int n = 10;     // Number of steps

    adamsPredictorCorrector(h, t0, y0, n);
    return 0;
}

This code provides a basic example of implementing the Adams-Bashforth predictor and Adams-Moulton corrector methods. You can modify the function f to fit your specific ODE.

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