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Roots of polynomials with Complex Coefficients #1143

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e58578b
Added function to FindRoots class for polynomials with complex coeffi…
bCosgrovePayne Dec 23, 2025
b3a9b16
Fixed error with degree and edited function to return empty array ins…
bCosgrovePayne Dec 24, 2025
e92fd86
Split up unit tests by degree through quartic, and included an edge c…
bCosgrovePayne Dec 25, 2025
0eec597
In FindRootsTest, changed test function names to omit the word test. …
bCosgrovePayne Dec 31, 2025
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39 changes: 38 additions & 1 deletion src/Numerics/FindRoots.cs
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Expand Up @@ -27,8 +27,12 @@
// OTHER DEALINGS IN THE SOFTWARE.
// </copyright>

using System;
using MathNet.Numerics.LinearAlgebra;
using MathNet.Numerics.LinearAlgebra.Complex;
using MathNet.Numerics.LinearAlgebra.Factorization;
using MathNet.Numerics.RootFinding;
using System;
using System.Linq;
using Complex = System.Numerics.Complex;

namespace MathNet.Numerics
Expand Down Expand Up @@ -118,6 +122,39 @@ public static Complex[] Polynomial(double[] coefficients)
return new Polynomial(coefficients).Roots();
}

/// <summary>
/// Find all roots of a polynomial with complex coefficients by calculating the characteristic polynomial of the companion matrix
/// </summary>
/// <param name="coefficients">The coefficients of the polynomial in ascending order, e.g. new Complex[] {new (5, 0), new (0, 2) new (2,3)} = "5 + 2i x^1 + (2 + 3i) x^2"</param>
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@jkalias jkalias Dec 24, 2025

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new (0, 2) new (2,3) -> new (0, 2), new (2,3) (a comma is missing)

/// <returns>The roots of the polynomial</returns>

public static Complex[] Roots(Complex[] coefficients)
{
int n = coefficients.Length;
if (n < 2)
{
return null;
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@jkalias jkalias Dec 24, 2025

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Wouldn't it be better to return an empty array/list instead of null?

}

// Negate, and normalize (scale such that the polynomial becomes monic)
Complex aN = coefficients[n];
Complex[] p = new Complex[n];
for (int i = n - 1; i >= 0; i--)
{
p[i] = -coefficients[i] / aN;
}

DenseMatrix A0 = DenseMatrix.CreateDiagonal(n - 1, n - 1, 1.0);
DenseMatrix A = new DenseMatrix(n);

A.SetSubMatrix(1, 0, A0);
A.SetRow(0, p.Reverse().ToArray());
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@jkalias jkalias Dec 25, 2025

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Shouldn't this be the last column, based on this wiki article?
https://en.wikipedia.org/wiki/Companion_matrix

Unless you decide to set the submatrix from the second column and first row. Or maybe I am confused :)

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@bCosgrovePayne bCosgrovePayne Dec 25, 2025

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So to be entirely honest, I don't know the theory. This is based entirely on the existing .Roots() property in the Polynomial class, which uses Evd to approximate the roots when the polynomial is higher than degree 3, if I'm not mistaken. I monkeyed around with the complex version of the DenseMatrix class and found out everything still works.

I'll triple check that this is producing accurate roots, but it seemed to pass the unit test w/o issue.


Evd<Complex> eigen = A.Evd(Symmetricity.Asymmetric);

return eigen.EigenValues.ToArray();
}

/// <summary>
/// Find all roots of a polynomial by calculating the characteristic polynomial of the companion matrix
/// </summary>
Expand Down