We consider the solution of an autonomous system of ordinary differential equations y′(x) = f(y(x)) by a step-by-step method with step size h. In the general linear method, N approximations \(y_1^{\left( n \right)},y_2^{\left( n \right)},...,y_N^{\left( n \right)}\) are computed during step number n (n = 1,2, ..) by the formula
$$y_i^{\left( n \right)} = \sum\limits_{j = 1}^N {{a_{ij}}y_j^{\left( {n - 1} \right)}}+ h\sum\limits_{j = 1}^N {{b_{ij}}f\left( {y_j^{\left( n \right)}} \right) + h\sum\limits_{j = 1}^N {{c_{ij}}f\left( {y_j^{\left( {n - 1} \right)}} \right),\;\left( {i = 1,2,...,N} \right)} } $$
where the matrices A,B,C with components a ij , b ij , c ij characterize the method. Without loss of generality we can assume C = o and we denote the method by (A,B).


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  1. 1.
    Butcher, J. C.: On the convergence of numerical solutions to ordinary differential equations. Math. Comp. 20 (1966), 1–10.CrossRefGoogle Scholar
  2. 2.
    Butcher, J. C.: An algebraic theory of integration methods. Math. Comp. 26 (1972), 79–106.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • J. C. Butcher
    • 1
  1. 1.AucklandNew Zealand

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