Abstract
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.
Highlights
Orthogonal polynomials are very strong tools in approximation theory and play a fundamental role in finding the least-squares approximation in explicit form
Orthogonal polynomials over triangular domains have to be determined in a different way, see [4]
The Legendre weight function W (u, v, w) = 1 is used; these Legendre-weighted orthogonal polynomials are given in the Bernstein basis form, in order to take full advantage of the numerical stability property of the later basis
Summary
Orthogonal polynomials are very strong tools in approximation theory and play a fundamental role in finding the least-squares approximation in explicit form. They are well-studied, and a lot of research has been done for the univariate case, see [1]. In [5], to find these bivariate orthogonal polynomials on triangular domains, the sums have to be expanded, and many multiplications have to be performed. This leads to a time-consuming and costly method of construction.
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