Hermite-Birkhoff-Obrechkoff 3-stage 4-step ODE solver of order14 with quantized stepsize

A 3-stage Hermite-Birkhoff-Obrechkoff method of order 14 with 4 quantized variable steps, denoted by HBOQ(14)3, is constructed for solving non-stiff systems of firstorder differential equations of the form y′ = f(x, y) with initial conditions y(x0) = y0. Its formula uses y′, y′′ and y′′′ as in Obrechkoff method. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear Vandermonde-type systems. To reduce overhead, simple formulae are derived once for all to obtain the values of Hermite-Birkhoff interpolation polynomials in terms of Lagrange basis functions for 16 quantized stepsize ratios. The stepsize is controlled by a local error estimator. When programmed in C++, HBOQ(14)3 is superior to the Dormand-Prince Runge-Kutta pair DP(8,7)13M of order 8 in solving several problems often used to test higher order ODE solvers at stringent tolerances, When programmed in Matlab, it is superior to ode113 in solving costly problems, on the basis of number of steps, CPU time, and maximum global error.