Hypergeometric Summation Algorithms for High-order Finite Elements

Hypergeometric Summation Techniques for High Order Finite Elements

The goal of this paper is to discuss the application of computer algebra methods in the design of a high order finite element solver. The finite element method is nowadays the most popular method for the computer simulation of partial differential equations. The performance of iterative solution methods depends on the condition number of the system matrix, which itself depends on the chosen bas...

Algorithms for m-Fold Hypergeometric Summation

Algorithms for q-Hypergeometric Summation in Computer Algebra

together with their Maple implementations, which is relevant both to people being interested in symbolic computation and in q-series. For all these algorithms, the theoretical background is already known and has been described, so we give only short descriptions, and concentrate ourselves on introducing our corresponding Maple implementations by examples. Each section is closed with a descripti...

Pyramid Algorithms for Bernstein-Bézier Finite Elements of High, Nonuniform Order in Any Dimension

The archetypal pyramid algorithm is the de Casteljau algorithm, which is a standard tool for the evaluation of Bézier curves and surfaces. Pyramid algorithms replace an operation on single high order polynomial by a recursive sequence of self-similar affine combinations, and are ubiquitous in CAGD for computations involving high order curves and surfaces. Pyramid algorithms have received no att...

Hypergeometric summation revisited

We consider hypergeometric sequences i e the sequences which satisfy linear rst order homogeneous recurrence equations with rela tively prime polynomial coe cients Some results related to necessary and su cient conditions are discussed for validity of discrete Newton Leibniz formula P w k v t k u w u v when u k R k t k and R k is a rational solution of Gosper s equation