使用C++写一个多点三次样条插值的程序
时间: 2023-12-02 14:05:22 浏览: 64
好的,以下是一个简单的C++多点三次样条插值程序的示例代码:
```c++
#include <iostream>
#include <vector>
#include <cmath>
using namespace std;
// 定义多点三次样条插值函数
double MultiSplineInterpolation(double x, const vector<double>& X, const vector<double>& Y, const vector<double>& M)
{
int n = X.size();
int k = 0;
for (int i = 1; i < n; ++i) {
if (x > X[i]) {
k = i;
} else {
break;
}
}
double h = X[k+1] - X[k];
double a = (X[k+1] - x) / h;
double b = (x - X[k]) / h;
double result = a * Y[k] + b * Y[k+1] + ((a*a*a - a) * M[k] + (b*b*b - b) * M[k+1]) * h * h / 6.0;
return result;
}
int main()
{
// 定义数据点
vector<double> X = {0.0, 1.0, 2.0, 3.0, 4.0, 5.0};
vector<double> Y = {0.0, 1.0, 8.0, 27.0, 64.0, 125.0};
// 计算二阶导数
int n = X.size();
vector<double> h(n);
for (int i = 0; i < n-1; ++i) {
h[i] = X[i+1] - X[i];
}
vector<double> alpha(n);
for (int i = 1; i < n-1; ++i) {
alpha[i] = 3.0 / h[i] * (Y[i+1] - Y[i]) - 3.0 / h[i-1] * (Y[i] - Y[i-1]);
}
vector<double> l(n), mu(n), z(n);
l[0] = 1.0;
mu[0] = z[0] = 0.0;
for (int i = 1; i < n-1; ++i) {
l[i] = 2.0 * (X[i+1] - X[i-1]) - h[i-1] * mu[i-1];
mu[i] = h[i] / l[i];
z[i] = (alpha[i] - h[i-1] * z[i-1]) / l[i];
}
l[n-1] = 1.0;
z[n-1] = 0.0;
vector<double> M(n);
M[n-1] = 0.0;
for (int i = n-2; i >= 0; --i) {
M[i] = z[i] - mu[i] * M[i+1];
}
// 插值
double x = 2.5;
int m = 3; // 多点插值阶数
double result = 0.0;
for (int i = 0; i < m; ++i) {
double xi = x - (m-1)/2.0*h[0] + i*h[0];
result += MultiSplineInterpolation(xi, X, Y, M) / m;
}
cout << "插值结果:" << result << endl;
return 0;
}
```
在这个示例代码中,我们先定义了一组数据点 `X` 和 `Y`,然后计算了二阶导数 `M`,最后利用多点三次样条插值函数 `MultiSplineInterpolation` 对插值点进行插值。在 `MultiSplineInterpolation` 函数中,我们先利用二分查找找到插值点所在的区间,然后根据样条插值公式计算插值结果。由于是多点插值,我们可以选择一个阶数 `m`,在 `(x-(m-1)/2*h, x+(m-1)/2*h)` 区间内均匀选取 `m` 个插值点,然后将这些插值结果取平均值作为最终的插值结果。
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