# Some Efficient Schemes with Improving Rate of Convergence for Two-stage Gauss Method

### Abstract

Various iteration schemes have been proposed to solve the non-linear equations arising in the implementation of implicit Runge-Kutta methods. A modified Newton scheme is typically used to solve these equations. As an alternative to this scheme, iteration schemes, which sacrifice super-linear convergence for

reduced linear algebra costs, have been proposed. A scheme of this type proposed in [11] avoids expensive vector transformations and is computationally more

efficient. The rate of convergence of this scheme is examined in [11] when it is applied to the scalar test differential equation ′ and the convergence rate

depends on the spectral radius of the iteration matrix , a function of , where is the step-length. In this scheme, some conditions are imposed on the spectral radius of its iteration matrix in order to get super-linear convergence at two points in the -plane. Then the supremum of the spectral radius of each

scheme is minimized over the left-half -plane in order to improve the convergence rate of the scheme. Two new schemes with optimal parameters are obtained for the two-stage Gauss method and some numerical experiments are reported.

**GSTF Journal of Mathematics, Statistics and Operations Research (JMSOR)**, [S.l.], v. 2, n. 1, may 2018. ISSN 2251-3396. Available at: <http://dl6.globalstf.org/index.php/jmsor/article/view/1519>. Date accessed: 16 feb. 2019.