E cient approximation of Sturm { Liouville problems using Lie - group methods

We present a new approach to the numerical solution of Sturm{Liouville eigenvalue problems based on Magnus expansions. Our algorithms are closely related to Pruess' methods [Pre73], but provide for high order approximations at nearly the same cost as the second-order Pruess method. By using Newton iteration to solve for the eigenvalues, we are able to present an e cient algorithm for computing a range of eigenvalues and eigenfunctions. Numerical experiments display promising results, and asymptotic corrections are made to improve further the accuracy of the schemes. Introduction In this paper we are concerned with the numerical approximation of parameter estimation problems of the form y0 = A(x; )y; (1) where A 2 R , y 2 R. The parameter 2 K (where K = R or C ) is unknown, and is to be found subject to some kind of boundary conditions. Of particular interest are conditions of the form Bay(a; ) +Bby(b; ) = 0; where Ba; Bb 2 R n are possibly dependent on the parameter . Among the problems tting this category are Sturm{Liouville and Schrodinger eigenvalue problems. Other interesting problems are AKNS eigenvalue problems obtained when generalising the inverse scattering transform for non-linear partial di erential equations. A fourth example is provided by multi-point boundary-value problems. These are obtained e.g. when modeling wave propagation in range-independent multi-region uid-solid media. Mathematically these problems are formulated as follows: Given a set of matrices Gj 2 Rj n together with corresponding points xj 2 R, j = 1; : : : ; k; the problem is to determine a parameter value such that Gjy(xj) = 0; j = 1; 2; : : : ; k