A Twilight Zone: Where Do The Real and Imaginary Planes Blend?

A Twilight Zone: Where Do The Real and Imaginary Planes Blend?

A Twilight Zone: Where Do The Real and Imaginary Planes Blend? Numerical analysis with perturbed polynomials

It is a natural question in numerical analysis to ask whether the problem of finding the roots of a polynomial is well-conditioned. That is, we’d hope that small changes in the coefficients of the polynomial would lead to small changes in the values of its roots. Interestingly, this is often not the case.

Wilkinson’s polynomial is a specific polynomial used by mathematician James Wilkinson in 1963 to illustrate the idea that the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial, even if it has well-separated zeros. He later described the personal impact of this discovery as:

“Speaking for myself I regard it as the most traumatic experience in my career as a numerical analyst.”

Wilkinson’s polynomial is often used to illustrate the undesirability of a common technique to compute the eigenvalues of a matrix, which involves deriving the coefficients of the matrix’s characteristic polynomial and then solving for its roots. It should be noted that using the coefficients as an intermediate step may introduce an extreme ill-conditioning even if the original problem was well-conditioned.

data-visualization mathematics python analytics numerical-analysis

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