Iqbal, Raja Zafar
(2010).
Computation of nodes and weights of Gaussian Quadrature rule by using Jacobi method.
University of Birmingham.
M.Phil.
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.
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