Accurate Approximation of Eigenvalues and Zeros of Selected Eigenf unctions of Regular Sturm-Liouville Problems

A method for simultaneously approximating to high accuracy the corresponding eigenvalue and zeros of the (n + l)st eigenfunction of a regular Sturm-Liouville eigenvalue problem is presented. It is based upon equilibrating the minimum eigenvalues of several problems on subintervals that form a partition of the orginal interval. The method is easily derived from classical mini-max variational principles. The equilibration is accomplished iteratively using an approximate Newton Method. Numerical results are given. Introduction. The problem of approximating the eigenvalues of regular SturmLiouville equations has attracted the attention of analysts for a long time. An annoying aspect of most numerical schemes is that the accuracy of the approximation of the nth eigenvalue decreases as n increases. This is due to the fact that the higher eigenfunctions are more oscillatory and therefore more difficult to approximate accurately. In this paper is presented a simple method by which any eigenvalue can be approximated as accurately as the first, which accuracy will depend of course on the particular scheme utilized for this basic (minimum eigenvalue) calculation. The method is based upon approximating the minimum eigenvalues of several problems on subintervals that form a partition of the original interval. These "subeigenvalues" are then equilibrated by iteratively selecting appropriate breakpoints for the subintervals. A related question, for Sturm-Liouville problems, is that of calculating the n distinct zeros of the (« 4l)st eigenfunction. These points are of interest in some applications. One approach to approximating the zeros of such a "special function" would be to use a standard root finder together with an analytic approximation (such as a truncated series or continued fraction) to the eigenfunction. The accuracy of the computed zeros then would depend on the accuracy of the approximation to the eigenfunction. Here we will see that these zeros are precisely the equilibration points mentioned previously. Thus they can be approximated, simultaneously with the eigenvalue, in a general way, which requires no specific information about the eigenfunction; and the accuracy of the approximation will depend only on the accuracy to which one can approximate the minimum eigenvalues on the subproblems. Received August 30, 1982. 1980 Mathematics Subject Classification. Primary 65L15.