Computational Complexity Comparison of a New Linear Block Approach and Modified Taylor Series Approach for Developing k-Step Third Derivative Block Methods

This article introduces two approaches to develop block methods for solving second order ordinary differential equations directly. Both approaches, namely a new linear block approach and the modified Taylor series approach are capable of producing a family of methods that will simultaneously approximate the solutions of any ordinary differential equation at the respective grid points of the block method. The computational complexities of both approaches are examined, and the results show the new linear block approach require less computations compared to the modified Taylor series approach.