Abstract
Deployment of the recurrence relation or difference equation to generate discrete classical orthogonal polynomials is vulnerable to error propagation. This issue is addressed for the case of Krawtchouk functions, i.e., the orthonormal basis derived from the Krawtchouk polynomials. An algorithm is proposed for stable determination of these functions. This is achieved by defining proper initial points for the start of the recursions, balancing the order of the direction in which recursions are executed and adaptively restricting the range over which equations are applied. The adaptation is controlled by a user-specified deviation from unit norm. The theoretical background is given, the algorithmic concept is explained and the effect of controlled accuracy is demonstrated by examples.
Highlights
This paper describes a method for generating the Krawtchouk functions where the deviation of generated functions from unit norm is user-controlled
Calculation of the Krawtchouk functions using the recurrence relation and/or the difference equation runs into numerical problems when executed outside of the effective width
We will first look into the generation of the zeroth-order Krawtchouk function considering the control and number of samples that need to be calculated
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. For the normalized discrete Chebyshev polynomials several mitigation schemes have been proposed; the latest one a computation scheme characterized by truncation of the iterations under the control of a user-defined deviation from unit norm [8]. The solutions defined for the discrete Chebyshev polynomials do not carry over since they make use of the (anti-)symmetry of the polynomials with respect to the mid of the domain, i.e., a property that does not generally hold for Krawtchouk polynomials. It introduces the well-known concepts of center of energy and effective width, notions underlying the proposed scheme.
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