A Twilight Zone: Where Do The Real and Imaginary Planes Blend?
By: Agni Kumar
Tags: Numerical Analysis, Analytics, Editors Pick, Data Visualization, Mathematics

Perturbed polynomials

Let’s observe what happens to the roots of a polynomial similar to Wilkinson’s when we randomly perturb its coefficients. We consider the polynomial

which has degree 10 and roots 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Expanding, we get

where C = {cᵢ}, where i ranges from 0 to 9 inclusive, is the set of coefficients of the non-leading terms. We randomly perturb the coefficients of p(x), replacing each element cᵢ ∈ C with cᵢ ⋅ (1 + aεᵢ), where each εᵢ is a normally distributed random variable with mean 0 and variance 1, and a is some small overall scaling factor. If we plug in each cᵢ, we get that the perturbed polynomial is

Below, p(x) is plotted in blue, while a randomly perturbed version of p(x) is given in pink. We observe that the number of real roots decreases after perturbation and that the roots change position, as evidenced by the differing positions of the blue dots and pink stars.

Roots values change significantly with small coefficient perturbations, as do the number of real roots.

Next, we investigate what happens to the roots of p(x) over multiple small values of a, confirming that the roots are indeed highly sensitive to perturbations applied to the coefficients.

Root distributions

We attempt to figure out whether there is a distinct point at which real roots begin transforming into complex ones.

Real

We observe distributions of real roots of a perturbed version of p(x), across various a values, for a single run per a.

Distributions of real roots of a perturbed version of our polynomial

The points above are randomly horizontally dodged for visibility purposes, and the leftmost a value indicates no perturbation. We see that some real roots change position or are even lost are lost, that is, turned into complex ones, as a increases. This leads to the question:

Is there a particular value of a at which it is almost certain that complex roots will appear? How can we characterize the spread of the real roots up until that point?

Let’s call the a value at which the roots of p(x), originally all real, begin to switch into the strictly complex plane with some probability pᵣ the critical point a*. This pᵣ, informed by computational experiments described in the sections below, is set somewhat arbitrarily but succeeds in capturing some of the relevant features of the problem.

Complex

Here we show a plot of the roots of 100 different perturbed versions of p(x), for each of 10 different a values. As previously noted, for greater a values, we see more and more roots entering the complex plane.

Perturbed polynomial roots over multiple scaling factors

Only larger a values seem to account for the spread in the imaginary direction. These observations motivate the exploration of perturbed real root spread as compared to the roots of p(x) over multiple values of a.

Critical scaling factor

We zoom in onto two particular a values, one below the critical point and one greater than the critical point a*. The plots depict eight different perturbed polynomials’ roots.