Effective solution of a linear system with Chebyshev coefficients
|Title||Effective solution of a linear system with Chebyshev coefficients|
|Publication Type||Journal Article|
|Year of Publication||2009|
|Authors||Kujan, Petr, Martin Hromčík, and Michael Šebek|
|Journal||Integral 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.