Runge-Kutta pairs of order 5(4) satisfying only the first column simplifying assumption

Among the most popular methods for the solution of the Initial Value Problem are the Runge-Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. To achieve this, we usually admit various simplifying assumptions. The most common of them are the so-called row simplifying assumptions. Here we neglect them and present an algorithm for the construction of Runge-Kutta pairs of orders 5 and 4 based only in the first column simplifying assumption. The result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of DETEST. A cost free fourth order formula is also derived for handling dense output. (C) 2011 Elsevier Ltd. All rights reserved.