International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 4, Pages 401-405

The Rayleigh quotient and dynamic programming

Richard Bellman

Department of Mathematics, Electrical Engineering, and Medicine, University of Southern California, Los Angeles 90007, California, USA

Received 8 August 1978

Copyright © 1978 Richard Bellman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The purpose of this paper is to derive a nonlinear partial differential equation for which λ given by (1.3), is one value of the solution. In Section 2, we derive this equation using a straightforward dynamic programming approach. In Section 3, we discuss some computational aspects of derermining the solution of this equation. In Section 4, we show that the same method may be applied to the nonlinear characteristic value problem. In Section 5, we discuss how the method may by applied to find the higher characteristic values. In Section 5, we discuss how the same method may be applied to some matrix problems. Finally, in Section 7, we discuss selective computation.