Reduction of a generalized matrix of polynomial elements to triangular form on the IBM 704

Abstract

A new method [1], developed by the author, for reducing matrices with polynomial elements (hereafter referred to as lambda-matrices) to triangular form is presented. The method presented has been programmed on the IBM 704. The process may be used for various applications such as obtaining the characteristic equation of a constant matrix A,| λI-A| = 0, determining the characteristic equation for generalized lambda-matrices such as those which arise in systems of linear differential equations with constant coefficients, and providing a mathematical solution in analytical form of such systems.Existing computer programming methods for obtaining the characteristic equation of lambda-matrices employ some variation of the method of determinants. It is well known that the computing time for such methods increases rapidly as the order of the matrix increases. The method presented requires far fewer operations than determinant methods and although it is equivalent to the division algorithm for polynomials discussed in [2], it has an advantage over this method in that it does not require polynomial division and replaces general polynomial multiplication with a trivial form of this operation.The paper is presented in three sections; (1) the mathematical method employed in the reduction, (2) some special programming techniques, and (3) a brief description of the scaling techniques used.