THE EIGENVALUE PROBLEM FOR INFINITE COMPACT COMPLEX SYMMETRICAL MATRICES WITH APPLICATION TO THE NUMERICAL COMPUTATION OF COMPLEX ZEROS OF J(0)(Z)-IJ(1)(Z) AND OF BESSEL-FUNCTIONS J(M)(Z) OF ANY REAL ORDER-M

Consider computing simple eigenvalues of a given compact infinite matr ix regarded 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 matr ix is compact, complex, and symmetric (but not necessarily Hermitian). Two examples of application are studied. The first is concerned with the equation J0(z) - iJ1(z) = 0 appearing in the analysis of the solit ary-wave runup on a sloping beach, and the second with the zeros of th e Bessel function J(m)(z) of any real order m. In each case, the probl em is reformulated as an eigenvalue problem for a compact complex symm etric tridiagonal matrix operator in l2 whose eigenvalues are all simp le. A complete error analysis for the numerical solution by truncation is given, based on the general theorems proved in this paper, where t he usefulness of the seldom used generalized Rayleigh quotient is demo nstrated.