Packages for solving almost block diagonal linear systems arising in spline collocation at Gaussian points with monomial basis functions

Abstract

Numerical techniques based on spline collocation at Gaussian points have proved to be exceedingly effective for the approximate solution of boundary value problems for ordinary differential equations and in the method of lines solution of parabolic and hyperbolic initial-boundary value problems. In this paper, we describe a linear equation software package designed to solve the special almost block diagonal systems of linear algebraic equations which arise in spline collocation at Gaussian points when certain monomial basis functions are employed. The new package is an improvement over existing ones from the points of view of speed and storage requirements. We also describe a package for solving the almost block diagonal systems which arise when condensation is applied to the collocation equations. This package can also be used to solve similar almost block diagonal systems arising in other methods for solving two-point boundary value problems, such as implicit Runge-Kutta methods and multiple shooting.