Abstract : The purpose of this paper was to illustrate how the techniques of the theory of dynamic programming may be used to convert a number of eigenvalue problems, where one is interested only in maximum or minimum values, into problems involving recurrence relations. Treated Jacobi matrices, some special types of quadratic forms possessing certain features of regularity, and finally Sturm-Liouville problems. The method discussed is not only useful for computational purposes, but provides a method for studying the analytic dependence of the maximum and minimum eigenvalues upon the analytic structure of the matrix.