Elementary boundary value problems are used as a vehicle to introduce upper level undergraduate students or first year graduate students to descent algorithms. The paper is expository in nature and includes references for Euclidean and Sobolev steepest descent while serving as an introduction to optimization techniques such as variable metric and conjugate gradient methods. Numerical algorithms for solving boundary value problems with both Euclidean and Sobolev descent are developed. Results for constrained, unconstrained, and singular problems are displayed and properties of descent algorithms are outlined.

In this paper we present a method, parallel in nature, for finding all eigenvalues of a symmetric generalized tridiagonal matrix. Our method employs the determinant evaluation and the Durand-Kerner root-finding scheme. It will be shown that the method converges quadratically and is reliable, efficient, and easy to implement in practice.