CHAPTER 6 - EIGENVALUE PROBLEMS

Abstract

This chapter discusses eigenvalue problems. A common problem involves a differential system that has solutions only for certain values of a parameter occurring in the coefficients of the system. The most suitable method for matrices obtained from differential systems can depend both on the system concerned and on the requirements of the problem. The matrix can generally be sparse, with nonzero elements only in the region of the diagonal. The negative eigenvalues correspond to eigenfunctions of decreasing exponential form, while the positive eigenvalues correspond to functions with bounded oscillation. The differential system usually has infinity of solutions, whereas the algebraic problem has only a finite number equal to the number of pivotal points in the range of integration. The choice between an initial-value technique and a matrix technique depends on many diverse factors, including the size of the high-speed store of the machine and the extent of a priori knowledge of the eigenvalue and eigenfunction required.