Chebyshev Polynomials and Their Derivatives in C

Quick Summary

We use the C language to observe the incredible features of chebyshev polynomials: orthogonality. easy-to-find approximations, and even easier-to-find derivatives.

Advanced math concepts turned into code for programmers. Subscribe.

Aggrandizing quotes about Chebyshev Polynomials. Taken from (Bray, 2016)

We provide Python starter code and complete C code (15 June update) at these links.

1.0 Chebyshev Polynomials for Programmers

Numerical analysis is the design of computer algorithms to solve math and physics problems (Shi, 2024) 1.

Chebyshev polynomials are like ferraris among numerical analysts because they are fast, precise and the math is beautiful.

This article serves as a friendly introduction to chebyshev polynomials for programmers.

2.1 Introduction to Chebyshev Polynomials

In approximation theory, Chebyshev Polynomials are sequences of polynomials defined on the interval [-1, 1] that satisfy the recurrence (Sidharth et al., 2024):

Chebyshev polynomials of the second kind satisfy the recurrence:

Chebyshev polynomials possess these properties:

Orthogonality:

The chebyshev polynomials of the first kind form a set of orthogonal polynomials over the real interval [-1, 1] (Mason & Handscomb, 2023) 2.

Being orthogonal ensures the polynomial coefficients are uncorrelated for convergence and numerical stability (Sidharth et al., 2024).

Recursive definition:

This we observed in the definition of chebyshev polynomials.

Inversion symmetry (Deus Ex Machina, 2021) 3:

This means that T n (x) = T n (-x)

Rapid Convergence and Uniform Approximation (Sidharth et al., 2024):

The approximation error of chebyshev polynomials decreases faster than other polynomial bases as degree increases.

Compared to same-degree polynomials, chebyshev polynomials minimize better the maximum approximation error.

Easy to map:

Chebyshev polynomials on the general interval [a,b] can be mapped to the real interval [-1,1] either:

non-linearly using the hyperbolic…