用c++写一个计算迭代最小二乘法

以下是用C++编写的计算迭代最小二乘法的示例代码：

#include <iostream>
#include <cmath>
#include <vector>
using namespace std;

const double EPS = 1e-8; // 精度要求

// 计算向量的模长
double norm(vector<double>& x) {
    double res = 0;
    for (int i = 0; i < x.size(); i++) {
        res += x[i] * x[i];
    }
    return sqrt(res);
}

// 计算向量的点积
double dot(vector<double>& x, vector<double>& y) {
    double res = 0;
    for (int i = 0; i < x.size(); i++) {
        res += x[i] * y[i];
    }
    return res;
}

// 计算矩阵的转置
vector<vector<double>> transpose(vector<vector<double>>& A) {
    int m = A.size(), n = A[0].size();
    vector<vector<double>> res(n, vector<double>(m));
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < n; j++) {
            res[j][i] = A[i][j];
        }
    }
    return res;
}

// 计算矩阵的乘积
vector<vector<double>> multiply(vector<vector<double>>& A, vector<vector<double>>& B) {
    int m = A.size(), n = A[0].size(), p = B[0].size();
    vector<vector<double>> res(m, vector<double>(p));
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < p; j++) {
            for (int k = 0; k < n; k++) {
                res[i][j] += A[i][k] * B[k][j];
            }
        }
    }
    return res;
}

// 计算迭代最小二乘法
vector<double> iterative_least_squares(vector<vector<double>>& A, vector<double>& b, int max_iters) {
    int m = A.size(), n = A[0].size();
    vector<double> x(n), r(m), p(m), Ap(m);
    vector<vector<double>> AtA = multiply(transpose(A), A);

    // 初始化x
    for (int i = 0; i < n; i++) {
        x[i] = 0;
    }

    for (int k = 0; k < max_iters; k++) {
        // 计算残差r = b - Ax
        for (int i = 0; i < m; i++) {
            r[i] = b[i] - dot(A[i], x);
        }

        // 判断是否满足精度要求
        if (norm(r) < EPS) {
            break;
        }

        // 计算搜索方向p = (AtA)^(-1)r
        Ap = multiply(AtA, p);
        for (int i = 0; i < m; i++) {
            p[i] = r[i] - Ap[i];
        }

        // 计算步长alpha = (r^T)p / (p^T Ap)
        double alpha = dot(r, p) / dot(p, Ap);

        // 更新x = x + alpha * p
        for (int i = 0; i < n; i++) {
            x[i] += alpha * p[i];
        }
    }

    return x;
}

int main() {
    vector<vector<double>> A = {{1, 1}, {1, 2}, {1, 3}, {1, 4}, {1, 5}};
    vector<double> b = {2.1, 3.9, 6.0, 8.0, 9.8};
    vector<double> x = iterative_least_squares(A, b, 100);

    // 输出结果
    cout << "The solution is: ";
    for (int i = 0; i < x.size(); i++) {
        cout << x[i] << " ";
    }
    cout << endl;

    return 0;
}

以上代码实现了迭代最小二乘法的计算过程，其中 `iterative_least_squares` 函数接收一个系数矩阵 `A` 和一个常数向量 `b`，并返回最小二乘解向量 `x`。在实现中，使用了共轭梯度法来求解迭代方向。