尝试使用矩阵在 C++ 中实现 Durand-Kerner-Method

Question

我的 Durand-Kerner-Method (https://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method) 实施似乎不起作用。我相信（见以下代码）我没有在算法部分本身正确计算新的近似值。我似乎无法解决问题。非常感谢任何建议。

#include <complex>
#include <cmath>
#include <vector>
#include <iostream>
#include "DurandKernerWeierstrass.h"

using namespace std;
using Complex = complex<double>;
using vec = vector<Complex>;
using Matrix = vector<vector<Complex>>;


//PRE: Recieves input value of polynomial, degree and coefficients
//POST: Outputs y(x) value
Complex Polynomial(vec Z, int n, Complex x) {

    Complex y = pow(x, n);
    for (int i = 0; i < n; i++){
        y += Z[i] * pow(x, (n - i - 1));
    }
    return y;
}

/*PRE: Takes a test value, degree of polynomial, vector of coefficients and the desired
precision of polynomial roots to calculate the roots*/
//POST: Outputs the roots of Polynomial

Matrix roots(vec Z, int n, int iterations, const double precision) {
    Complex z = Complex(0.4, 0.9);
    Matrix P(iterations, vec(n, 0));
    Complex w;
    
    //Creating Matrix with initial starting values
    
    for (int i = 0; i < n; i++) {
        P[0][i] = pow(z, i);
    }

    //Durand Kerner Algorithm

    for (int col = 0; col < iterations; col++) {

        *//I believe this is the point where everything is going wrong*

        for (int row = 0; row < n; row++) {
            Complex g = Polynomial(Z, n, P[col][row]);
            for (int k = 0; k < n; k++) {
                if (k != row) {
                    g = g / (P[col][row] - P[col][k]);
                }

            }
                
            P[col][row] -= g;

        }
            
        return P;
    }   
    

}

以下代码是我用来测试功能的代码：

int main() {
    //Initializing section

    vec A = {1, -3, 3,-5 };
    int n = 3;
    int iterations = 10;
    const double precision = 1.0e-10;
    Matrix p = roots(A, n, iterations,precision);
    for (int i = 0; i < iterations; i++) {
        for (int j = 0; j < n; j++) {
            cout << "p[" << i << "][" << j << "] = " << p[i][j] << " ";
            
        }
        cout << endl;
    }
    return 0;

}

重要的是要注意 Durand-Kerner-Algorithm 连接到此代码中未包含的头文件。

Answer 1

您的问题是您没有将新值转录到索引为 col+1 的下一条数据记录中。因此，在下一个循环中，您将再次从零条目的数据集开始。改为

        P[col+1][row] = P[col][row] - g;

如果您想立即对所有后续近似使用新的改进近似，请使用

        P[col+1][row] = (P[col][row] -= g);

那么数据集都包含下一个近似值，尤其是第一个将不再包含初始设置的幂。

尝试使用矩阵在 C++ 中实现 Durand-Kerner-Method

Trying to implement Durand-Kerner-Method in C++ using Matrices

c++

math

matrix

numerical-methods

polynomials