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December 2017, 7(4): 457-464. doi: 10.3934/naco.2017028

## A study of numerical integration based on Legendre polynomial and RLS algorithm

 1 Changsha University of Science and Technology, Changsha 410114, P. R. China 2 Measurement and Testing Research Institute of Hunan Province, Changsha 410014, P. R. China

* Corresponding author: Wen Tan

The reviewing process of the paper was handled by Nanjing Huang as Guest Editors

Received  March 2016 Revised  July 2017 Published  October 2017

Fund Project: The first author is supported by The National Natural Science Foundation of China (41201468) and The National Nonprofit Industry Research (201510003-5)

A quadrature rule based on Legendre polynomial functions is proposed to find approximate values of definite integrals in this paper. This method uses recursive least squares (RLS) algorithm to compute coefficients of Legendre polynomial fitting functions, and then approximately computes values of definite integrals by using obtained the coefficients. The main advantage of this approach is its efficiency and simple applicability. Finally some examples are given to test the convergence and accuracy of the method.

Citation: Hongguang Xiao, Wen Tan, Dehua Xiang, Lifu Chen, Ning Li. A study of numerical integration based on Legendre polynomial and RLS algorithm. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 457-464. doi: 10.3934/naco.2017028
##### References:
 [1] S. Arora and S. Khot, Fitting algebraic curves to noisy data, Journal of Computer and System Sciences, 67 (2003), 325-340. doi: 10.1016/S0022-0000(03)00012-6. [2] K. Atkinson, An Introduction to Numerical Analysis, second ed., Wiley, 1989. [3] E. Babolian, M. MasjedJamei and M. R. Eslahchi, On numerical improvement of Gauss-Legendre quadrature rule, Applied Mathematics and Computation, 160 (2005), 779-789. doi: 10.1016/j.amc.2003.11.031. [4] R. L. Burden and J. Douglas Faires, Numerical Analysis, Seventh ed., Thomson Learning, 2001. [5] F. Cazals and M. Pouget, Estimating differential quantities using polynomial fitting of osculating jets, Computer Aided Geometric Design, 22 (2005), 121-146. doi: 10.1016/j.cagd.2004.09.004. [6] Hamza Chaggara and Wolfram Koepf, On linearization and connection coefficients for generalized Hermite polynomials, Journal of Computational and Applied Mathematics, 236 (2011), 65-73. doi: 10.1016/j.cam.2011.03.010. [7] W. N. Everitta, K. H. Kwonb, L. L. Littlejohnc and R. Wellman, Orthogonal polynomial solutions of linear ordinary differential equations, Journal of Computational and Applied Mathematics, 133 (2001), 85-109. doi: 10.1016/S0377-0427(00)00636-1. [8] G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), 206-216. doi: 10.1007/BF01436075. [9] G. H. Golub, Matrix decompositions and statistical calculations, in Statistical Computations (eds. R. C. Milton, J. A. Nedler), Academic Press, New York, (1969), 365–397. [10] S. M. Hashemiparast, Numerical integration using local Taylor expansions in nodes, Applied Mathematics and Computation, 192 (2007), 332-336. doi: 10.1016/j.amc.2007.03.009. [11] Siraj-ul-Islam, Imran Aziz and Fazal Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Computers and Mathematics with Applications, 59 (2010), 2026-2036. doi: 10.1016/j.camwa.2009.12.005. [12] Ana Marco and José-Javier Martínez, Polynomial least squares fitting in the Bernstein basis, Linear Algebra and its Applications, 433 (2010), 1254-1264. doi: 10.1016/j.laa.2010.06.031. [13] Luis J. Morales-Mendoza, Hamurabi Gamboa-Rosales and Yuriy S. Shmaliy, A new class of discrete orthogonal polynomials for blind fitting of finite data, Signal Processing, 93 (2013), 1785-1793. [14] Tomasz Pander, New polynomial approach to myriad filter computation, Signal Processing, 90 (2010), 1991-2001. [15] C. F. So, S. C. Ng and S. H. Leung, Gradient based variable forgetting factor RLS algorithm, Signal Processing, 83 (2003), 1163-1175. doi: 10.1109/TSP.2005.851110. [16] Peter Strobach, Solving cubics by polynomial fitting, Journal of Computational and Applied Mathematics, 235 (2011), 3033-3052. doi: 10.1016/j.cam.2010.12.025. [17] Peter Strobach, A fitting algorithm for real coefficient polynomial rooting, Journal of Computational and Applied Mathematics, 236 (2012), 3238-3255. doi: 10.1016/j.cam.2012.02.027. [18] Li-yun Su, Prediction of multivariate chaotic time series with local polynomial fitting, Computers and Mathematics with Applications, 59 (2010), 737-744. doi: 10.1016/j.camwa.2009.10.019. [19] Yegui Xiao, Liying Ma and Rabab Kreidieh Ward, Fast RLS Fourier analyzers capable of accommodating frequency mismatch, Signal Processing, 87 (2007), 2197-2212. [20] Zhe-zhao Zeng and Xu Zhou, A neural-network method based on RLS algorithm for solving special linear systems of equations, Journal of Computational Information Systems, 8 (2012), 2915-2920.

show all references

##### References:
 [1] S. Arora and S. Khot, Fitting algebraic curves to noisy data, Journal of Computer and System Sciences, 67 (2003), 325-340. doi: 10.1016/S0022-0000(03)00012-6. [2] K. Atkinson, An Introduction to Numerical Analysis, second ed., Wiley, 1989. [3] E. Babolian, M. MasjedJamei and M. R. Eslahchi, On numerical improvement of Gauss-Legendre quadrature rule, Applied Mathematics and Computation, 160 (2005), 779-789. doi: 10.1016/j.amc.2003.11.031. [4] R. L. Burden and J. Douglas Faires, Numerical Analysis, Seventh ed., Thomson Learning, 2001. [5] F. Cazals and M. Pouget, Estimating differential quantities using polynomial fitting of osculating jets, Computer Aided Geometric Design, 22 (2005), 121-146. doi: 10.1016/j.cagd.2004.09.004. [6] Hamza Chaggara and Wolfram Koepf, On linearization and connection coefficients for generalized Hermite polynomials, Journal of Computational and Applied Mathematics, 236 (2011), 65-73. doi: 10.1016/j.cam.2011.03.010. [7] W. N. Everitta, K. H. Kwonb, L. L. Littlejohnc and R. Wellman, Orthogonal polynomial solutions of linear ordinary differential equations, Journal of Computational and Applied Mathematics, 133 (2001), 85-109. doi: 10.1016/S0377-0427(00)00636-1. [8] G. H. Golub, Numerical methods for solving linear least squares problems, Numer. Math., 7 (1965), 206-216. doi: 10.1007/BF01436075. [9] G. H. Golub, Matrix decompositions and statistical calculations, in Statistical Computations (eds. R. C. Milton, J. A. Nedler), Academic Press, New York, (1969), 365–397. [10] S. M. Hashemiparast, Numerical integration using local Taylor expansions in nodes, Applied Mathematics and Computation, 192 (2007), 332-336. doi: 10.1016/j.amc.2007.03.009. [11] Siraj-ul-Islam, Imran Aziz and Fazal Haq, A comparative study of numerical integration based on Haar wavelets and hybrid functions, Computers and Mathematics with Applications, 59 (2010), 2026-2036. doi: 10.1016/j.camwa.2009.12.005. [12] Ana Marco and José-Javier Martínez, Polynomial least squares fitting in the Bernstein basis, Linear Algebra and its Applications, 433 (2010), 1254-1264. doi: 10.1016/j.laa.2010.06.031. [13] Luis J. Morales-Mendoza, Hamurabi Gamboa-Rosales and Yuriy S. Shmaliy, A new class of discrete orthogonal polynomials for blind fitting of finite data, Signal Processing, 93 (2013), 1785-1793. [14] Tomasz Pander, New polynomial approach to myriad filter computation, Signal Processing, 90 (2010), 1991-2001. [15] C. F. So, S. C. Ng and S. H. Leung, Gradient based variable forgetting factor RLS algorithm, Signal Processing, 83 (2003), 1163-1175. doi: 10.1109/TSP.2005.851110. [16] Peter Strobach, Solving cubics by polynomial fitting, Journal of Computational and Applied Mathematics, 235 (2011), 3033-3052. doi: 10.1016/j.cam.2010.12.025. [17] Peter Strobach, A fitting algorithm for real coefficient polynomial rooting, Journal of Computational and Applied Mathematics, 236 (2012), 3238-3255. doi: 10.1016/j.cam.2012.02.027. [18] Li-yun Su, Prediction of multivariate chaotic time series with local polynomial fitting, Computers and Mathematics with Applications, 59 (2010), 737-744. doi: 10.1016/j.camwa.2009.10.019. [19] Yegui Xiao, Liying Ma and Rabab Kreidieh Ward, Fast RLS Fourier analyzers capable of accommodating frequency mismatch, Signal Processing, 87 (2007), 2197-2212. [20] Zhe-zhao Zeng and Xu Zhou, A neural-network method based on RLS algorithm for solving special linear systems of equations, Journal of Computational Information Systems, 8 (2012), 2915-2920.
The calculation results
 examples This proposed method rer of Hybrid[20] 1 rer Example 1 0.32197059919282 3.7271e-17 9.6947e-13 Example 2 26.08328771414714 1.4155e-16 3.7148e-14 Example 3 0.34196249133027 1.8001e-11 3.7947e-8
 examples This proposed method rer of Hybrid[20] 1 rer Example 1 0.32197059919282 3.7271e-17 9.6947e-13 Example 2 26.08328771414714 1.4155e-16 3.7148e-14 Example 3 0.34196249133027 1.8001e-11 3.7947e-8
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