Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2004

Problem 1 To compute , we consider the following Eudoxos iterations: starting with , we set followed by . Then Problem 1a Explain the Eudoxos method in terms of the power method. Solution 1a The iteration can be represented in matrix formulation as follows: which can be written as Thus the iteration is just the power method since each step is represented by a multiplication by the matrix . The power method converges to the eigenvector of the largest eigenvalue. The eigenvalues of are computed to be . Hence the largest eigenvalue is The corresponding eigenvector is then Then as desired. Problem 1b How many iterations are required for an error Solution 1b Since convergence is linear, 7 steps is required to achieve the error bound. Problem 2 Let be a sequence of monic polynomials orthogonal on with respect to the positive weight function ( has degree ). Show that satisfy a three term recursion formula of the form Give expressions for the coefficients and Solution 2a First notice that and therefore we can express it as a linear combination of the monic polynomials of degree or less i.e. Taking the inner product of both side of with yields from the orthogonality of the polynomials: Rearranging terms then yields Similarly, taking the inner product of both side of with yields from the orthogonality of the polynomials: Notice that Therefore, Finally, taking inner product of both side of with yields, Notice that which implies for Problem 3a Find such that is a polynomial of degree and this set is orthogonal on with respect to the weight function Solution 3a Using Gram Schmidt with inner product defined as and power basis as starting vectors, we get Problem 3b Find the weights and nodes of the 2 point Gaussian formula Note: Solution 3b Using test functions and and using the roots of as nodes we find