Qualitatively Correct Sturm-Liouville Eigenfunction Approximation

In recent years several high quality Sturm-Liouville software packages have been written (e.g., SLEDGE, SLEIGN, the NAG routine SL02FM). 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 describe an accurate, efficient method of approximating Sturm-Liouville eigenfunctions based on a small sample of given accurate values. Even though the theory is developed only for regular problems, the algorithm can be extended to many singular cases. This research was partially supported by National Science Foundation grant DMS-9322615 to the Colorado School of Mines. 1 Introduction In this paper the Sturm-Liouville problem \Gamma(p(x)u 0 ) 0 + q(x)u = r(x)u (1.1) is studied for a ! x ! b; we are especially interested in approximating the eigenfunctions associated with (1.1). The package SLEDGE of ..