Comments on the Comparison of Global Methods for Linear Two-Point Boundary Value Problems

A more careful count of the operations involved in solving the linear system associated with collocation of a two-point boundary value problem using rough splines reverses results recently reported by others in this journal. In addition, it is observed that the use of the technique of "condensation of parameters" can decrease the computer storage required. Furthermore, the use of a particular highly localized basis can also reduce the setup time when the mesh is irregular. Finally, operation counts are roughly estimated for the solution of certain linear systems associated with two competing collocation methods; namely, collocation with smooth splines and collocation of the equivalent first order system with continuous piecewise polynomials. In a recent paper [1] in this journal, R. D. Russell and J. M. Varah carry out a comparison of various global methods for the numerical solution of the (2m)th order linear two-point boundary value problem (1) Lu(x) := Z (-DW^Ef^x)) = fix), a<x< b, i=0 (2) flW) = LVuQ)) = 0, 0<i<m. We wish to take exception to their account of the computational effort required to solve (l)-(2) approximately by collocation at Gauss points with c2m~x piecewise polynomials of degree less than 2«.. Suppose we collocate at