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A contribution to Runge-Kutta formulas of the 7th order with rational coefficients for systems of differential equations of the first order
Applications of Mathematics, Tome 29 (1984) no. 6, pp. 411-422
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The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development $\sum^\infty_{i=1} \frac {h^i} {i!} \frac {d^{i-1}} {dx^{i-1}} \bold f\left[x,\bold y(x)\right]$ up to $h^7$ with the development given by successive insertion into the formula $h.f_i(k_0,k_1,\ldots, k_{i-1})$ for $i=1,2,\ldots, 10$ and $k=\sum^{10}_{i=0} p_i, k_i$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers.
The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development $\sum^\infty_{i=1} \frac {h^i} {i!} \frac {d^{i-1}} {dx^{i-1}} \bold f\left[x,\bold y(x)\right]$ up to $h^7$ with the development given by successive insertion into the formula $h.f_i(k_0,k_1,\ldots, k_{i-1})$ for $i=1,2,\ldots, 10$ and $k=\sum^{10}_{i=0} p_i, k_i$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers.
DOI : 10.21136/AM.1984.104115
Classification : 34A34, 65L05
Keywords: Runge-Kutta formulas; rational coefficients; systems; 7th order formulas 
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Huťa, Anton; Penjak, Vladimír. A contribution to Runge-Kutta formulas of the 7th order with rational coefficients for systems of differential equations of the first order. Applications of Mathematics, Tome 29 (1984) no. 6, pp. 411-422. doi: 10.21136/AM.1984.104115

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[6] A. Huťa: The Algorithm for Computation of the n-order Formula for Numerical Solution of Initial Value Problem of Differential Equations. 5th Symposium on Algorithms - Algorithms' 79. Proceedings of Lectures (1979), 53 - 61.

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