Polynomial Roots

Root Finding for Polynomials

Finding the roots of a polynomial—the values of \(x\) for which \(P(x) = 0\)—is a fundamental task in engineering. Whether you are finding the natural frequencies of a structure or the break-even point in a cost model, SepalSolver provides robust methods to extract these critical values from the coefficient array.

1. The Companion Matrix Method

For general polynomials, the most stable way to find all roots (including complex ones) is to construct a Companion Matrix. The eigenvalues of this matrix are exactly the roots of the polynomial. By transforming a polynomial problem into an eigenvalue problem, we leverage the power of our linear algebra engine.

// P(x) = x^2 - 5x + 6
double[] coeffs = [1.0, -5.0, 6.0];// Returns an array of Complex numbers: { (2,0), (3,0) }
Complex[] roots = Roots(coeffs);
Console.WriteLine("Roots = \n " + string.Join("\n ", roots));

Ouput

Roots =
   2.0000 + 0.0000i
   3.0000 + 0.0000i

Examples

Example 1 : Finding Natural Frequencies

In vibration analysis, the roots of a characteristic polynomial :math:`(s^2 + 4s + 13)`represent the squared natural frequencies of a system. By passing the coefficients of the system’s governing equation to Roots, we can identify the resonance points.

// Characteristic Equation: s^2 + 4s + 13 = 0
double[] characteristic = { 1.0, 4.0, 13.0 };
var frequencies = Roots(characteristic);

// Results will be complex: -2 + 3i and -2 - 3i
Console.WriteLine($"Principal Root = \n " + string.Join("\n ", frequencies));

Ouput

Principal Root =
  -2.0000 - 3.0000i
  -2.0000 + 3.0000i

Example 2 : Intersection of a Curve and an Axis

Imagine a robot arm path described by \(f(x) = x^3 - 2x^2 - 5x + 6\). We need to know exactly where the arm crosses the baseline (the x-axis). Using the Roots function allows us to find all three points of intersection simultaneously.

double[] path = [1.0, -2.0, -5.0, 6.0];
var intersections = Roots(path);

foreach (var r in intersections)
{
    if(Complex.IsReal(r))
        Console.WriteLine($"Crossing at: {r.Real}");
}

Ouput

Crossing at: 1
Crossing at: -2
Crossing at: 3

Example 3 :

Compute the roots of \(x^8 + 2x^7 + 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 2x + 3\)

// Roots of polynomials
double[] coeff = [1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 2, 3];
var roots = Roots(coeff);
Console.WriteLine("Roots = \n " + string.Join("\n ", roots));

Ouput

Roots =
   0.1514 - 0.7608i
   0.1514 + 0.7608i
  -0.1111 - 0.8529i
  -0.1111 + 0.8529i
   0.8943 - 0.4616i
   0.8943 + 0.4616i
  -0.9522 - 0.5038i
  -0.9522 + 0.5038i
   0.6864 - 1.1010i
   0.6864 + 1.1010i
  -1.2464 - 0.5377i
  -1.2464 + 0.5377i
  -0.4224 - 1.2924i
  -0.4224 + 1.2924i

Example 4 : Refining a Guess with Newton-Raphson

When high precision is required for a specific physical boundary, we take a rough estimate from a graph and refine it. By combining Evaluate and Polyder, we can implement a custom search that converges in just a few iterations.

double[] p = { 1.0, 0.0, -2.0 }; // x^2 - 2 (Roots are ±sqrt(2))
double x = 1.5; // Initial guess
var dp = Polyder(p, 1);
for (int i = 0; i < 5; i++)
{
    double f = Polyval(p, x);
    double df = Polyval(dp, x);
    x -= (f / df);
}
Console.WriteLine($"Refined sqrt(2): {x}");

Ouput

Refined sqrt(2): 1.414213562373095

Numerical Note: Complex Roots

Even if your input coefficients are real, the roots can be complex. SepalSolver always returns roots as an array of Complex numbers to prevent data loss. If you only care about real solutions, you can filter the results by checking if the imaginary part is below a small tolerance (e.g., Math.Abs(r.Imaginary) < 1e-12).

Polynomial Roots

Root Finding for Polynomials

1. The Companion Matrix Method

2. Iterative Refinement (Newton-Raphson)

Examples

Numerical Note: Complex Roots