A finite-difference algorithm for an inverse Sturm-Liouville problem

We study a method for approximating a potential q(x) in y(0)=y()=0 from finite spectral data. When the potential is symmetric,the data are the first M Dirichlet eigenvalues. In the generalcase, the first M terminal velocities are also specified. Acentred finite-difference scheme reduces the inverse Sturm-Liouvilleproblem to a matrix inverse eigenvalue problem. Our approachis motivated by the work of Paine, de Hoog and Anderssen, whoinvestigated the discrepancy between continuous and matrix eigenvaluesunder finite differences. Our modified Newton scheme is basedon choosing the number of interior mesh points in the discretizationto be 2M. The modified Newton scheme is shown to be convergentfor both the case of a symmetric and general potential. Somenumerical experiments are given. Supported in part by Institute for Scientific Computation,Texas A&M University.