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New algorithms for the numerical solution of Ordinary Differential Equations (ODEs) with initial condition are proposed. They are designed for work on a new kind of a supercomputer - the Infinity Computer. - that is able to deal numerically with finite, infinite and infinitesimal numbers. Due to this fact, the Infinity Computer allows one to calculate the exact derivatives of functions using infinitesimal values of the stepsize. As a consequence, the new methods described in this paper are able to work with the exact values of the derivatives, instead of their approximations. [ABSTRACT FROM AUTHOR] more...

In this article we describe the code bvptwp.m, a MATLAB code for the solution of two point boundary value problems. This code is based on the well-known Fortran codes, twpbvp.f, twpbvpl.f and acdc.f, that employ a mesh selection strategy based on the estimation of the local error, and on revisions of these codes, called twpbvpc.f, twpbvplc.f and acdcc.f, that employ a mesh selection strategy based on the estimation of the local error and the estimation of two parameters which characterize the conditioning of the problem. The codes twpbvp.f/tpbvpc.f use a deferred correction scheme based on Mono-Implicit Runge-Kutta methods (MIRK); the other codes use a deferred correction scheme based on Lobatto formulas. The acdc.f/acdcc.f codes implement an automatic continuation strategy. The performance and features of the new solver are checked by performing some numerical tests to show that the new code is robust and able to solve very difficult singularly perturbed problems. The results obtained show that bvptwp.m is often able to solve problems requiring stringent accuracies and problems with very sharp changes in the solution. This code, coupled with the existing boundary value codes such as bvp4c.m, makes the MATLAB BVP section an extremely powerful one for a very wide range of problems. [ABSTRACT FROM AUTHOR] more...

Abstract: Codes for the numerical solution of two-point boundary value problems can now handle quite general problems in a fairly routine and reliable manner. When faced with particularly challenging equations, such as singular perturbation problems, the most efficient codes use a highly non-uniform grid in order to resolve the non-smooth parts of the solution trajectory. This grid is usually constructed using either a pointwise local error estimate defined at the grid points or else by using a local residual control. Similar error estimates are used to decide whether or not to accept a solution. Such an approach is very effective in general providing that the problem to be solved is well conditioned. However, if the problem is ill conditioned then such grid refinement algorithms may be inefficient because many iterations may be required to reach a suitable mesh on which to compute the solution. Even worse, for ill conditioned problems an inaccurate solution may be accepted even though the local error estimates may be perfectly satisfactory in that they are less than a prescribed tolerance. The primary reason for this is, of course, that for ill conditioned problems a small local error at each grid point may not produce a correspondingly small global error in the solution. In view of this it could be argued that, when solving a two-point boundary value problem in cases where we have no idea of its conditioning, we should provide an estimate of the condition number of the problem as well as the numerical solution. In this paper we consider some algorithms for estimating the condition number of boundary value problems and show how this estimate can be used in the grid refinement algorithm. [Copyright &y& Elsevier] more...

Abstract: We present a hybrid mesh selection strategy for use in codes for the numerical solution of two-point boundary value problems. This new mesh strategy is based on the estimation of two parameters which characterise the conditioning of the continuous problem as well as on a standard estimate of the local discretisation error. We have implemented this algorithm in the well known code TWPBVP and have found that the modified code is often considerably more efficient than the original. Another strong advantage of using the new mesh selection algorithm is that it automatically provides an estimate of the conditioning of the discrete problem. This is very valuable (arguably indispensible) for use either in an a posteriori error estimate or, in situations where the conditioning constants are large, as a warning that the accuracy obtained in the solution may be worse than anticipated. [Copyright &y& Elsevier] more...

Block-boundary value methods are considered and applied to solve initial value problems. Results on their convergence and stability properties are presented and their intermediate position between the class of multistep and Runge{Kutta methods is established. [ABSTRACT FROM AUTHOR] more...

A new backward error analysis of LU factorization is presented. It allows to obtain a sharper upper bound for the forward error and a new definition of the growth factor that we compare with the well known Wilkinson growth factor for some classes of matrices. Numerical experiments show that the new growth factor is often of order approximately log 2n whereas Wilkinson's growth factor is of order n or √n. [ABSTRACT FROM AUTHOR] more...

Concerns with the unique solvability of the equations that arise when a Boundary Value Method (BVM) is applied to solve a nonlinear initial value problem satisfying Lipschitz condition. Formulation of a BVM and related algebraic properties; Equivalent conditions for the uniqueness; Topological degree for the existence of the unique solution; Estimation of eigenvalue for some classes of BVM. more...

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the Infinity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique. [ABSTRACT FROM AUTHOR] more...