计算方法非线性方程求根

本文最后更新于：2024年3月18日 凌晨

计算方法非线性方程求根

迭代法

算法

代码实现

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public class 迭代法 {
    public static void main(String[] args) {
        int n = 11;
        double x0 = 1.5;
        double x1;
        double[] x = new double[n];
        x[0] = x0;
        // 方程（x+1)^(1/3)
        for (int i = 1; i < n; i++) {
            x1 = f(x0);
            x0 = x1;
            x[i] = x0;
        }
        for (int i = 0; i < x.length; i++) {
            System.out.println(i + ": " + x[i]);
        }
    }

    static double f(double x) {
        double y = Math.pow(x + 1, 1.0 / 3);
        return y;
    }

牛顿迭代法

算法

代码实现

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public class 牛顿迭代法 {
    public static void main(String[] args) {
        double x0 = 1;
        double x1;
        // 方程x^3-7.7x^2+19.2x-15.3
        int i = 1;
        while (Math.abs(f(x0)) > 10e-6) {
            x1 = x0 - f(x0) / f2(x0);
            x0 = x1;
            System.out.print(i + ": x=" + x0);
            System.out.println("      y=" + (Math.abs(Math.pow(x0, 3) - 7.7 * Math.pow(x0, 2) + 19.2 * x0 - 15.3)));
            i++;
        }
    }

    static double f(double x) {
        double y = Math.pow(x, 3) - 7.7 * Math.pow(x, 2) + 19.2 * x - 15.3;
        return y;
    }

    static double f2(double x) {
        double y = 3 * Math.pow(x, 2) - 15.4 * x + 19.2;
        return y;
    }
}

牛顿下山法

算法

正割法

算法

代码实现

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public class 正割法 {
    public static void main(String[] args) {
        double x0 = 0;
        // 方程x^3-7.7x^2+19.2x-15.3
        double x1 = x0 - f(x0) / f2(x0);
        double x2;
        int i = 1;
        while (Math.abs(f(x1)) > 10e-6) {
            x2 = x1 - (f(x1) * (x1 - x0)) / (f(x1) - f(x0));
            x0 = x1;
            x1 = x2;
            System.out.print(i + ": x=" + x1);
            System.out.println("      y=" + (f(x1)));
            i++;
        }
    }

    static double f(double x) {
        double y = Math.pow(x, 3) - 7.7 * Math.pow(x, 2) + 19.2 * x - 15.3;
        return y;
    }

    static double f2(double x) {
        double y = 3 * Math.pow(x, 2) - 15.4 * x + 19.2;
        return y;
    }

二分法

代码实现

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    public class 二分法 {
    public static void main(String[] args) {
        int n = 20;
        double low = -100;
        double high = 100;
        double mid = -1;
        int i;
        // 方程2x+1=0
        if (f(low) * f(high) < 0) {
            for (i = 0; i < n; i++) {
                mid = (low + high) / 2;
                if (f(mid) * f(low) < 0) {
                    high = mid;
                } else {
                    low = mid;
                }
                System.out.println(i + " " + "  x = " + mid);
            }
            System.out.println(i + "  x = " + mid);
        } else {
            System.out.println("函数在该定义域内无根或根在边界上!");
        }
    }

    static double f(double x) {
        double y = 2 * x + 1;
        return y;
    }
}