C#，码海拾贝（55）——求“非线性方程组”“最小二乘解”的“广义逆法”之C#源代码

在数学建模和数据分析中，经常会遇到求解非线性方程组的最小二乘解的问题。其中，广义逆法是一种常用的求解方法。下面我们来看一下如何用C#实现广义逆法求解非线性方程组的最小二乘解。

首先，我们需要定义一个表示矩阵的类Matrix，其中包含矩阵的基本运算方法，如矩阵乘法、转置等。

csharppublic class Matrix{
 private double[,] data;
 public Matrix(int rows, int cols)
 {
 data = new double[rows, cols];
 }

 public double this[int i, int j]
 {
 get { return data[i, j]; }
 set { data[i, j] = value; }
 }

 public Matrix Transpose()
 {
 Matrix result = new Matrix(data.GetLength(1), data.GetLength(0));
 for (int i =0; i < data.GetLength(0); i++)
 {
 for (int j =0; j < data.GetLength(1); j++)
 {
 result[j, i] = data[i, j];
 }
 }
 return result;
 }

 public static Matrix operator *(Matrix a, Matrix b)
 {
 Matrix result = new Matrix(a.data.GetLength(0), b.data.GetLength(1));
 for (int i =0; i < a.data.GetLength(0); i++)
 {
 for (int j =0; j < b.data.GetLength(1); j++)
 {
 for (int k =0; k < a.data.GetLength(1); k++)
 {
 result[i, j] += a[i, k] * b[k, j];
 }
 }
 }
 return result;
 }
}

csharppublic class NonlinearEquationSystem{
 private Func<double[], double[]> equations;
 private Func<double[], Matrix> jacobian;

 public NonlinearEquationSystem(Func<double[], double[]> equations, Func<double[], Matrix> jacobian)
 {
 this.equations = equations;
 this.jacobian = jacobian;
 }

 public double[] Solve(double[] initialGuess)
 {
 double[] x = initialGuess;
 double[] f = equations(x);
 Matrix J = jacobian(x);
 Matrix Jt = J.Transpose();
 Matrix A = Jt * J;
 Matrix b = Jt * new Matrix(f.Length,1);
 Matrix xMatrix = A.Inverse() * b;
 return xMatrix.ToArray();
 }
}

csharpFunc<double[], double[]> equations = x => new double[] { x[0] * x[0] + x[1] * x[1] -1, x[0] + x[1] -1 };
Func<double[], Matrix> jacobian = x => new Matrix(2,2) { [0,0] =2 * x[0], [0,1] =2 * x[1], [1,0] =1, [1,1] =1 };

NonlinearEquationSystem system = new NonlinearEquationSystem(equations, jacobian);
double[] initialGuess = new double[] {0.5,0.5 };
double[] result = system.Solve(initialGuess);

Console.WriteLine($"x = {result[0]}, y = {result[1]}");