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

Abstract

i o Introdu,~tion new method [!] ~ developed by the author s for reducing A matrices with polynomial elements (hereafter referred to as lambda~matrices) to triangular form is presented° The method presented has been programmed on the IBM 704o The prc~ess may be used for various applications such as obtaining the characteristic equation of a constant matrix A~I ~I-A I = O~ 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 so_~ation in analytical form of such systems° Existing comp~ter programming methods for obtaining the characteristic equation of !ambda~matrices employ some variation of the method of determinants° It i~ well k~o~a~ 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 ~lynomia!s discussed in [21~ it has an advantage over this method ~ that it does not require polynomial division and replaces general polynomial ~ltiplication with a trivial form of this operation° The paper is presented in three sections~ (1) the mathematical method employed in the reductions (2) some special programming techniques, and (3) a brief description of the scaling techniques used° 29 -I