Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here we give a brief overview of differential equations that can now be solved by R.

Following the Test Set for Initial Value Problem Solvers available at http://www.dm.uniba.it/~testset a new Fortran Test Set for Boundary Value Problem (BVP) Solvers has been developed and it is now available at http: //www.dm.uniba.it/~bvpsolve. The BVPTestSet includes documentation of the test problems, experimental results from a number of proven solvers, and Fortran subroutines providing a common interface to the defining problem functions.

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In this paper, starting from a sequence of results which can be traced back to I. J. Schoenberg, we analyze a class of spline collocation methods for the numerical solution of ordinary differential equations (ODEs) with collocation points coinciding with the knots. Such collocation methods are naturally associated to a special class of linear multistep methods, here called B-spline (BS) methods, which are able to generate the spline values at the knots. We prove that, provided the additional conditions are appropriately chosen, such methods are all convergent and A-stable. The convergence property of the BS methods is naturally inherited by the related spline extensions, which, by the way, are easily and safely computable using their B-spline representation.