ABSTRACT
In the past few years several high quality Sturm-Liouville software packages have been written (the NAG routine SL02FM, SLEDGE, SLEIGN). All excel at providing eigenfunction estimates at a finite set of points but only the NAG routine provides interpolation capability, and this by a fairly crude method (especially when compared with the sophistication of its eigenvalue solver). We seek efficient, accurate methods of approximating Sturm-Liouville eigenfunctions based on a small sample of given accurate values. Since the author is most familiar with SLEDGE, it will get most of the attention. The proposed methods are closely tied to SLEDGE’s basic approximations, its mesh selection, and its classification scheme, so these are briefly reviewed. The known theory on SLEDGE’s approximate eigenfunction errors are also discussed. Several alternatives are studied, both numerically and analytically. The first, coefficient approximation, is just SLEDGE’s underlying algorithm. The second is based on residual correction, and the other is a piecewise L-spline of sorts. While we concentrate on regular problems for which the mathematics is easier, we mention extensions to the more interesting singular case.
