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The characteristics and capabilities of the best codes for solving the initial value problem for ordinary differential equations are studied. Only codes which are readily available, portable, and very efficient are examined. Their sources, distinctive features, documentation, and ease of use are described. Their efficiency is compared with respect to storage, overhead, and derivative evaluations. Their behavior when faced with difficult problems is studied. Some guidelines are developed as to when a particular code is to be preferred.

Abstract. A class of one-step methods which obtain a block of r new values at each step are studied. The asymptotic behavior of both implicit and predictor-corrector procedures is examined.

Abstract.A class of methods for solving the initial value problem for ordinary differential equations is studied. We develop r-block implicit one-step methods which compute a block of r new values simultaneously with each step of application. These methods are examined for the property of A-stability. A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizes r = 1, 2 ..... 8 these methods are A-stable while those for r = 9,10 are not. We construct A-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly) A-stable formulas.

The user of a code for solving the initial value problem for ordinary differential equations is normally interested in the global error, i.e. the difference between the solution of the problem posed and the numerical result returned by the code. This paper describes a way of estimating the global error reliably while still solving the problem with acceptable efficiency. Global extrapolation procedures are applied to parallel solutions obtained by a Runge-Kutta-Fehlberg method. These ideas are implemented in a Fortran program called GERK, which is ACM Algomthm 504.The Algorithm: Algorithm 504, GERK: Global Error Estimation for Ordinary Differential Equations.