Numerical Linear Algebra with Applications

Appendix C

Chebyshev Polynomials

Chebyshev polynomials have important uses in developing convergence results for algorithms; for instance, the analysis for the convergence of the conjugate gradient method in Chapter 21 can be done using these polynomials (see Ref. [1, pp. 312-316]). They also play a large role in convergence results for methods that compute eigenvalues of large sparse matrices (see Ref. [3, pp. 151-159]). There are additional applications to least-squares and interpolation. The book does not actually use Chebyshev polynomials in proofs, but the statement of certain theorems involve Chebyshev polynomials, and so we will give a brief overview of their definition and properties.

C.1 Definition

The definition begins with the form ...