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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]

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]

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]

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]

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.