Effective solution of a linear system with Chebyshev coefficients

TitleEffective solution of a linear system with Chebyshev coefficients
Publication TypeJournal Article
Year of Publication2009
AuthorsKujan, Petr, Martin Hromčík, and Michael Šebek
JournalIntegral Transforms and Special Functions

This paper presents an efficient algorithm for a special triangular linear system with Chebyshev coefficients. We present two methods of derivations, the first is based on formulae where the nth power of x is solved as the sum of Chebyshev polynomials and modified for a linear system. The second deduction is more complex and is based on the Gauss-Banachiewicz decomposition for orthogonal polynomials and the theory of hypergeometric functions which are well known in the context of orthogonal polynomials. The proposed procedure involves O(nm) operations only, where n is matrix size of the triangular linear system L and m is number of the nonzero elements of vector b. Memory requirements are O(m), and no recursion formula is needed. The linear system is closely related to the optimal pulse-wide modulation problem.