Java实现三次样条插值方法

三次样条插值方法是一种用于构造一条光滑曲线的插值方法。下面是Java实现三次样条插值方法的代码：

public class CubicSplineInterpolation {
    private double[] x; // 插值节点的x坐标
    private double[] y; // 插值节点的y坐标
    private double[] h; // 差分节点的距离
    private double[] a; // 三次多项式的系数
    private double[] b;
    private double[] c;
    private double[] d;

    public CubicSplineInterpolation(double[] x, double[] y) {
        this.x = x;
        this.y = y;
        int n = x.length - 1;
        h = new double[n];
        for (int i = 0; i < n; i++) {
            h[i] = x[i + 1] - x[i];
        }
        a = new double[n + 1];
        b = new double[n];
        c = new double[n + 1];
        d = new double[n];
        for (int i = 1; i < n; i++) {
            b[i] = 2 * (h[i - 1] + h[i]);
            a[i] = h[i - 1] / b[i];
            c[i] = h[i] / b[i];
            d[i] = 6 * ((y[i + 1] - y[i]) / h[i] - (y[i] - y[i - 1]) / h[i - 1]) / b[i];
        }
        solve();
    }

    private void solve() {
        int n = x.length - 1;
        double[] l = new double[n + 1];
        double[] u = new double[n + 1];
        double[] z = new double[n + 1];
        l[1] = 1;
        u[1] = z[1] = 0;
        for (int i = 2; i < n; i++) {
            l[i] = 2 * (x[i + 1] - x[i - 1]) - h[i - 1] * u[i - 1];
            u[i] = h[i] / l[i];
            z[i] = (d[i] - h[i - 1] * z[i - 1]) / l[i];
        }
        l[n] = 1;
        z[n] = c[n] = 0;
        for (int i = n - 1; i >= 1; i--) {
            c[i] = z[i] - u[i] * c[i + 1];
            b[i] = (y[i + 1] - y[i]) / h[i] - h[i] * (c[i + 1] + 2 * c[i]) / 3;
            d[i] = (c[i + 1] - c[i]) / (3 * h[i]);
        }
    }

    public double interpolate(double t) {
        int i = 0;
        int j = x.length - 1;
        while (i < j) {
            int m = (i + j) / 2;
            if (x[m] < t) {
                i = m + 1;
            } else {
                j = m;
            }
        }
        if (i == 0) {
            i = 1;
        }
        if (i == x.length) {
            i = x.length - 1;
        }
        double dx = t - x[i];
        return y[i] + b[i] * dx + c[i] * dx * dx + d[i] * dx * dx * dx;
    }
}

代码中，`x`和`y`分别是插值节点的x坐标和y坐标，`h`是差分节点的距离，`a`、`b`、`c`和`d`是三次多项式的系数。在构造函数中，我们首先计算了差分节点的距离，并计算了三次多项式的系数。然后调用`solve`方法解出`c`和`d`的值。最后，我们实现了`interpolate`方法，它接受一个参数`t`，返回在`t`处的插值结果。