Advanced Polynomials in C# QuickStart Sample

Illustrates more advanced uses of the Polynomial class, including real and complex root finding, calculating least squares polynomials and polynomial arithmetic in C#.

This sample is also available in: Visual Basic, F#, IronPython.

Overview

This QuickStart sample demonstrates advanced capabilities of the Polynomial class in Numerics.NET, including working with complex roots, least squares polynomial fitting, and polynomial arithmetic operations.

The sample shows how to:

• Evaluate polynomials with complex arguments using ComplexValueAt and ComplexSlopeAt methods
• Find both real and complex roots of polynomials using FindRoots and FindComplexRoots
• Perform least squares polynomial fitting to approximate a set of data points
• Execute basic arithmetic operations on polynomials including addition, subtraction, multiplication, and division
• Use the Divide method to simultaneously obtain both quotient and remainder when dividing polynomials

Each operation is demonstrated with clear examples and the output is displayed to the console. The sample serves as a practical guide for working with polynomials in scientific and engineering applications where both real and complex analysis is required.

The code

using System;

// The Complex<T> structure resides in the Numerics.NET namespace.
using Numerics.NET;
// The Polynomial class resides in the Numerics.NET.Curves namespace.
using Numerics.NET.Curves;

// Illustrates the more advanced uses of the Polynomial class
// in the Numerics.NET.Curve namespace of Numerics.NET.

// The license is verified at runtime. We're using
// a 30 day trial key here. For more information, see
//     https://numerics.net/trial-key
Numerics.NET.License.Verify("your-trial-key-here");

// Basic operations on polynomials are covered in the
// BasicPolynomials QuickStart Sample. This QuickStart
// Sample focuses on more advanced topics, including
// finding complex roots, calculating least-squares
// polynomials, and polynomial arithmetic.

// Index variable.
int index;

//
// Complex numbers and polynomials
//

Polynomial polynomial = new Polynomial(new Double[] {-2, 0, 1, 1});

// The Polynomial class supports complex numbers
// as arguments for polynomials. It does not support
// polynomials with complex coefficients.
//
// For more about complex numbers, see the
// ComplexNumbers QuickStart Sample.
Complex<double> z1 = new Complex<double>(1, 2);

// Polynomial provides variants of ValueAt and
// SlopeAt for complex arguments:
Console.WriteLine("polynomial.ComplexValueAt({0}) = {1}",
    z1, polynomial.ComplexValueAt(z1));
Console.WriteLine("polynomial.ComplexSlopeAt({0}) = {1}",
    z1, polynomial.ComplexSlopeAt(z1));

//
// Real and complex roots
//
// Our polynomial has only one real root:
double[] roots = polynomial.FindRoots();
Console.WriteLine("Number of roots of polynomial1: {0}",
    roots.Length);
Console.WriteLine($"Value of root 1 = {roots[0]}");
// The FindComplexRoots method returns all three
// roots, two of which are complex:
Complex<double>[] complexRoots = polynomial.FindComplexRoots();
Console.WriteLine("Number of complex roots: {0}",
    complexRoots.Length);
Console.WriteLine("Value of root 1 = {0}",
    complexRoots[0]);
Console.WriteLine("Value of root 2 = {0}",
    complexRoots[1]);
Console.WriteLine("Value of root 3 = {0}",
    complexRoots[2]);

//
// Least squares polynomials
//

// Let's approximate 7 points on the unit circle
// by a fourth degree polynomial in the least squares
// sense.
// First, we create two arrays containing the x and
// y values of our data points:
double[] xValues = new double[7];
double[] yValues = new double[7];
double angle = 0;
for(index = 0; index < 7; index++)
{
    xValues[index] = Math.Cos(angle);
    yValues[index] = -Math.Sin(angle);
    angle = angle + Constants.Pi / 6;
}
// Now we can find the least squares polynomial
// by calling the ststic LeastSquaresFit method.
// The last parameter is the degree of the desired
// polynomial.
Polynomial lsqPolynomial =
    Polynomial.LeastSquaresFit(xValues, yValues, 4);
// Note that, as expected, the odd coefficients
// are close to zero.
Console.WriteLine("Least squares fit: {0}",
    lsqPolynomial.ToString());

//
// Polynomial arithmetic
//

// We can add, subtract, multiply and divide
// polynomials using overloaded operators:
Polynomial a = new Polynomial(new Double[] {4, -2, 4});
Polynomial b = new Polynomial(new Double[] {-3, 1});

Console.WriteLine($"a = {a.ToString()}");
Console.WriteLine($"b = {b.ToString()}");
Polynomial c = a + b;
Console.WriteLine($"a + b = {c.ToString()}");
c = a - b;
Console.WriteLine($"a - b = {c.ToString()}");
c = a * b;
Console.WriteLine($"a * b = {c.ToString()}");
c = a / b;
Console.WriteLine($"a / b = {c.ToString()}");
c = a % b;
Console.WriteLine($"a % b = {c.ToString()}");
// You can also calculate quotient and remainder
// at the same time by calling the overloaded Divide
// method:
Polynomial d;
c = Polynomial.Divide(a, b, out d);
Console.WriteLine("Using Divide method:");
Console.WriteLine($"  a / b = {c.ToString()}");
Console.WriteLine($"  a % b = {d.ToString()}");

Console.Write("Press Enter key to exit...");
Console.ReadLine();