Approximation of the Scattering Amplitude using Nonsymmetric Saddle Point Matrices

In this thesis we look at iterative methods for solving the primal (Ax = b) and dual (AT y = g) systems of linear equations to approximate the scattering amplitude defined by gTx =yTb. We use a conjugate gradient-like iteration for a unsymmetric saddle point matrix that is contructed so as to have a real positive spectrum. We find that this method is more consistent than known methods for computing the scattering amplitude such as GLSQR or QMR. Then, we use techniques from matrices, moments, and quadrature to compute the scattering amplitude without solving the system directly

Chebyshev Polynomial Approximation to Solutions of Ordinary Differential Equations

In this thesis, we develop a method for finding approximate particular solutions for second order ordinary differential equations. We use Chebyshev polynomials to approximate the source function and the particular solution of an ordinary differential equation. The derivatives of each Chebyshev polynomial will be represented by linear combinations of Chebyshev polynomials, and hence the derivatives will be reduced and differential equations will become algebraic equations. Another advantage of the method is that it does not need the expansion of Chebyshev polynomials. This method is also compared with an alternative approach for particular solutions. Examples including approximation, particular solution, a class of variable coefficient equation, and initial value problem are given to demonstrate the use and effectiveness of these methods