# IJPAM: Volume 106, No. 5 (2016)

NUMERICAL INTEGRATION USING HYBRID OF
BLOCK-PULSE FUNCTIONS AND LAGRANGE POLYNOMIAL

G. Uma, V. Prabhakar, S. Hariharan
School of Advanced Sciences
VIT University Chennai, INDIA
S.A. Engineering College
Chennai, INDIA

Abstract. This paper attempts to solve the definite integrals numerically using Gaussian quadrature rule with variable spacing. In this paper, the orthogonal polynomial is chosen to be the hybrid function formed from the block-pulse function of order N and Lagrange basis polynomial of order M. Here, the domain of the integral is partitioned into N sub-intervals, and the M roots of the Legendre polynomial of order M, are chosen as the nodes in each sub-interval. Having identified M nodes in each sub-interval, the Lagrange polynomial basis of order M is constructed. The total number of abscissas (or quadrature points) in the domain of the definite integral is . The weights can be easily obtained from the Lagrange polynomial. A significant advantage of this method is that the Hybrid matrix turns out to be an Identity matrix of order and the hybrid coefficients are simply the value of the integrand at the nodal points. The method is explained for single definite integrals, and this rule is extended to cover definite double and triple integrals with constant or variable limits. A comparative study of this method for both definite single and double integrals with Haar Wavelet and Hybrid function [2] reveals that better accuracy can be achieved with this quadrature rule with less number of points. It can be considered to cover a broad class of integrands.

Received: January 23, 2016

AMS Subject Classification: 65D30, 34K28

Key Words and Phrases: block-pulse function, Lagrange basis polynomial, Legendre polynomial, hybrid function, numerical integration

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DOI: 10.12732/ijpam.v106i5.4 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 106
Issue: 5
Pages: 33 - 44