The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of J₀(z) - iJ₁(z) and of Bessel functions J_m(z) of any real order m

Consider computing simple eigenvalues of a given compact infinite matrix re- garded as operating in the complex Hilbert space l² by computing the eigenvalues of the truncated finite matrices and taking an obvious limit. In this paper we deal with a special case where the given matrix is compact, complex, and symmetric (but not necessarily Hermitian). Two examples of application are studied. The first is con- cerned with the equation J₀(z) - iJ₁(z)=0 appearing in the analysis of the solitary-wave runup on a sloping beach, and the second with the zeros of the Bessel function J_m(z) of any real order m. In each case, the problem is reformulated as an eigenvalue problem for a compact complex symmetric tridiagonal matrix operator in l² whose eigenvalues are all simple. A complete error analysis for the numerical solution by truncation is given, based on the general theorems proved in this paper, where the usefulness of the seldom used generalized Rayleigh quotient is demonstrated.