Remarks on the Continued Fraction Calculation of Eigenvalues and Eigenvectors

Abstract

For eigenvalue problems in which the secular determinant has tridiagonal form, e.g., the rigid asymmetric rotor; the secular equation may be written in the form f(λ′)=0, where f(λ′) is a continued fraction and λ′ an eigenvalue. Furthermore, if the secular problem is of nth order, then the continued fraction f(λ′) may be developed in n different ways. Since the eigenvalues are roots of a function f(λ), it is convenient to find the eigenvalues by means of the Newton-Raphson iterative procedure. This requires that the derivative of f(λ) with respect to λ(f′(λ)) be determined. An exact expression for f′(λ) is derived and it is shown that f′(λ′) is in fact the norm of the eigenvector belonging to the eigenvalue λ′. A simple recursion formula, in continued fraction form, for the eigenvector elements is also derived. The Newton-Raphson procedure is further shown to be equivalent to the variational method for iterative calculation of eigenvalues. The former procedure has, however, the advantage of bypassing the necessity of solving a set of simultaneous equations. Advantage is taken of the relation between f′(λ′) and the eigenvector of λ′ to formulate a reasonable criterion for choosing the best possible development of f(λ) is order to avoid convergence to an undesired root of f(λ).