Consider computing zeros z of Whittaker function Mκ,μ (z) for given κ and μ, and eigenvalues of Lamé equation for given ν and 0 < k¹ < 1, by reformulating them as eigenvalue problems of infinite symmetric tridiagonal matrices. Miyazaki et al. have demonstrated a matrix method to compute zeros of Whittaker function of the first kind Mκ,μ (z), using the three-term recurrence relations, as a difference equation regarding μ, introduced by Boersma. This method, in which the coefficient matrix is considered as a compact operator from Hilbert space ℓ² to itself, also gives an asymptotically accurate error analysis. Erdéryi et al. and Jansen have investigated matrix methods for computing eigenvalues of Lamé equation. Eigenvalue problems of infinite symmetric tridiagonal matrices are considered where the matrix itself is NOT compact and its inverse is compact in Hilbert space ℓ², with their diagonal and super- and sub- diagonal elements diverging. In this paper, we attempted to rewrite the problem of computing zeros of Mκ,μ (z) by another three-term recurrence relations, as a difference equation regarding κ. This approach fails to compute the zeros of Mκ,μ (z) although the elements of the matrix behave asymptotically as in Lamé's case. Our objective is to point out the critical difference between the two cases of Lamé equation and Whittaker function, and to supply a condition under which the matrix method works. [ABSTRACT FROM AUTHOR]