Finding Zeros of Arbitrary Functions

A method for finding real and complex roots of polynomial equations, due to D. Muller, is applied to finding'roots of general equations of the form F(z) = O, , where F(z) is analytic in the neighborhood of the roots. The procedure does not depend on any prior knowledge of the location of the roots nor on any special starting process. All that is required is the ability to evaluate F(z) for any desired value of z. Multiple roots can also be obtained. A general purpose program, prepared for the UNIVAC Scientific 1103 and 1103A, is described and numerical results are presented for the following applications: finding eigenvalues of differential operators; finding eigenvalues of arbitrary matrices; finding zeros of the generalized eigenvalue problem; finding roots of a number of transcendental equations.