In this post 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 + … + cn-1xn-1
For n points, this can be written
y0 = c0x00 + c1x01 + c2x02 + … + cn-1x0n-1
y1 = c0x10 + c1x11 + c2x12 + … + cn-1x1n-1
y2 = c0x20 + c1x21 + c2x22 + … + cn-1x2n-1
+
…
+
y n-1 = c0xn-10 + c1x n-11 + c2x n-12 + … + cn-1x n-1n-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
+
…
+
yn-1
C is the coefficient vector that I want to solve for.
c0
c1
c2
+
…
+
cn-1
M is the matrix shown below.
x00 x01 x02 … x0n-1
x10 x11 x12 … x1n-1
x20 x21 x22 … x2n-1
+
…
+
xn-10 x n-11 x n-12 … x n-1n-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);
}
}
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