Abstract Application of an idea originally due to Ch. Hermite allows the derivation of an approximate formula for expressing the integral ∫ x i x i −1 y ( x ) dx as a linear combination of y ( x i −1), y ( x i ), and their derivatives y ( v )( x i −1) up to order v = α and y ( v ) ( x i ) up to order v = β . In addition to this integro-differential form a purely differential form of the 2-point Hermite approximation will be derived. Both types will be denoted by H αβ -approximation. It will be shown that the well-known Obreschkoff-formulas contain no new elements compared to the much older H αβ -method. The H αβ -approximation will be applied to the solution of systems of ordinary differential equations of the type y '( x ) = M ( x ) y ( x ) + q ( x ), and both initial value and boundary value problems will be treated. Function values at intermediate points x ϵ ( x i −1 , x i ) are obtained by the use of an interpolation formula given in this paper. An advantage of the H αβ -method is the fact that high orders of approximation (α, β) allow an increase in step size h i . This will be demonstrated by the results of several test calculations.
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