A DERIVATIVE–FREE BLOCK HYBRID METHOD FOR NUMERICALQUADRATURE

Abstract

We derive a new ninth-order block hybrid method for the numerical solution of systems of differential equations and we compare results
of numerical experiments with an already existing method in the literature. Both methods are bye-products of linear multistep methods using the interpolation and collocation approach. We show computationally that in the absence of round-off errors, the solution obtained by solving systems of differential equations by the existing block hybrid method derived by differentiating
the continuous scheme at a particular off–grid point is the same as those obtained in the new method which is derivative free. Besides, the new block hybrid method which is derivative free results in a well conditioned system as opposed to the ill–conditioned one in the literature. Therefore, providing an answer to Shampine’s claim that matrices arising from the numerical approximation of stiff initial value problems using Linear Multistep Methods are mostly ill–conditioned. Finally, we showed computationally how an LUtype preconditioned Quasi Minimal Residual with a fixed default tolerance reduced the condition number of the old and new methods, with the latter
resulting in the smallest minimum norm of residual.