Fitting a polynomial to a series of points

In this article I will develop a function that takes a series of x and y points and fits a univariate polynomial to them.

A polynomial equation can written as shown below, where n is the number of points to be fitted.

y = c₀x⁰ + c₁x¹ + c₂x² + … + c_n-1x^n-1

For n points, this can be written

y₀ = c₀x₀⁰ + c₁x₀¹ + c₂x₀² + … + c_n-1x₀^n-1

y₁ = c₀x₁⁰ + c₁x₁¹ + c₂x₁² + … + c_n-1x₁^n-1

y₂ = c₀x₂⁰ + c₁x₂¹ + c₂x₂² + … + c_n-1x₂^n-1

+

…

+

y _n-1 = c₀x_n-1⁰ + c₁x _n-1¹ + c₂x _n-1² + … + c_n-1x _n-1^n-1

For these equations, the y’s and x’s are known and the c’s are the variables to be calculated.

Putting this in matrix form gives the following equation.

M C = Y

Where Y is

y₀

y₁

y₂

+

…

+

y_n-1

C is the coefficient vector that I want to solve for.

c₀

c₁

c₂

+

…

+

c_n-1

M is the matrix shown below.

x₀⁰ x₀¹ x₀² … x₀^n-1

x₁⁰ x₁¹ x₁² … x₁^n-1

x₂⁰ x₂¹ x₂² … x₂^n-1

+

…

+

x_n-1⁰ x _n-1¹ x _n-1² … x _n-1^n-1

To solve this equation, I will add a static function called SolvePolyCoefs to a class called Polynomial. It will create and fill in the M and Y matrix, solve for the C matrix and return it as the function result. The code to do this is shown below.

using System;
using System.Diagnostics;
using System.Threading.Tasks;
using NumericalMethods.FourthBlog;
 
namespace NumericalMethods
{
    public class Polynomial
    {
        public static double[] SolvePolyCoefs(double[] x, double[] y)
        {
            Debug.Assert(x.Length > 0);
            Debug.Assert(x.Length == y.Length);
 
            Matrix m = new Matrix(x.Length, x.Length);
 
            Parallel.For(0, x.Length, i =>
            {
                m[i, 0] = 1.0;
                for (int j = 1; j < x.Length; j++)
                {
                    m[i, j] = m[i, j - 1] * x[i];
                }
            }
            );
 
            return m.SolveFor(y);
        }
}