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Computation of nodes and weights of Gaussian Quadrature rule by using Jacobi method

Iqbal, Raja Zafar (2010)
M.Phil. thesis, University of Birmingham.

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Abstract

Numerical analysis has become important in solving large eigenvalue problems in science and engineering due to the increasing spread of quantitative analysis. These problems occur in a wide variety of applications. Eigenvalues are very useful in engineering for the dynamic analysis of large-scale structures such as aerospace. There is also a useful connection between nodes and weights of Gaussian quadrature and eigenvalues and eigenvectors. Thus the need for faster methods to solve these larger eigenvalue problems has become very important. A standard textbook method for finding the eigenvalues of a matrix A is to solve for the roots of \lambda It is quite easy to solve analytically when matrix is small but it is more difficult for large matrices. For such problems numerical methods are used to find eigenvalues and corresponding eigenvectors. There are numerous numerical methods for the solution of the symmetric eigenvalue problems. Of these the QR algorithm, Cholesky iteration and Jacobi rotational methods are commonly used. In this project we checked rate the of convergence and accuracy of the Cholesky-iterative method and the Jacobi's method for finding eigenvalues and eigen vectors and found that the Jacobi's method converges faster than the Cholesky method. Then by using "three-term recurrence relation" we calculated nodes and weights of Gaussian quadrature by eigenvalues and eigenvectors. The nodes and weights computed were found to be highly accurate, so this method allows one to perform Gaussian Quadrature without using standard tables of nodes and weights, saving time and avoiding the risk of errors in entering the nodes and weights from tables.

Type of Work:M.Phil. thesis.
Supervisor(s):Mathias, Roy
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:1352
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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