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Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials.

Khaled A Al-Utaibi1, Sadiq H Abdulhussain2, Basheera M Mahmmod2

  • 1Department of Computer Engineering, University of Ha'il, Ha'il 682507, Saudi Arabia.

Entropy (Basel, Switzerland)
|September 28, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces a novel recurrence algorithm for Krawtchouk polynomials (KPs) to overcome numerical errors in high-order coefficient computation. The new method significantly reduces computation cost and improves polynomial size, enhancing applications in information and coding theory.

Keywords:
Krawtchouk momentscomputation costdiscrete Krawtchouk polynomialsenergy compactionpropagation error

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Area of Science:

  • Applied Mathematics
  • Information Theory
  • Coding Theory
  • Signal Processing

Background:

  • Krawtchouk polynomials (KPs) are valuable for feature extraction and classification.
  • Existing KP recurrence algorithms suffer from numerical errors, especially for large polynomial sizes and parameter values near 0 or 1.

Purpose of the Study:

  • To develop a new recurrence relation and algorithm for computing Krawtchouk polynomial coefficients with improved accuracy and efficiency.
  • To address the limitations of existing methods in handling large polynomial sizes and extreme parameter values.

Main Methods:

  • A novel recurrence relation for computing KP coefficients in high orders.
  • A new mathematical model for the initial value of the KP parameter.
  • A diagonal recurrence relation derived from existing n-direction and x-direction algorithms.
  • Exploitation of symmetry relations for efficient computation across the KP plane.

Main Results:

  • The proposed algorithm reliably computes KP coefficients with reduced errors and lower computational cost.
  • It achieves a significant improvement ratio (18.64% to 81.55%) in computed coefficients compared to existing methods.
  • The algorithm can generate polynomials of an order approximately 8.5 times larger than state-of-the-art algorithms.

Conclusions:

  • The new recurrence algorithm offers a reliable and efficient solution for computing Krawtchouk polynomial coefficients.
  • It overcomes numerical challenges associated with large polynomial sizes and extreme parameter values.
  • This advancement enhances the applicability of KPs in information theory, coding theory, and signal processing.