A problem-solving environment for the numerical solution of boundary value problems

Boundary value problems (BVPs) are systems of ordinary differential equations (ODEs) with boundary conditions imposed at two or more distinct points. Such problems arise within mathematical models in a wide variety of applications. Numerically solving BVPs for ODEs generally requires the use of a series of complex numerical algorithms. Fortunately, when users are required to solve a BVP, they have a variety of BVP software packages from which to choose. However, all BVP software packages currently available implement a specific set of numerical algorithms and therefore function quite differently from each other. Users must often try multiple software packages on a BVP to find the one that solves their problem most effectively. This creates two problems for users. First, they must learn how to specify the BVP for each software package. Second, because each package solves a BVP with specific numerical algorithms, it becomes difficult to determine why one BVP package outperforms another. With that in mind, this thesis offers two contributions. First, this thesis describes the development of the BVP component to the fully featured problem-solving environment (PSE) for the numerical solution of ODEs called pythODE. This software allows users to select between multiple numerical algorithms to solve BVPs. As a consequence, they are able to determine the numerical algorithms that are effective at each step of the solution process. Users are also able to easily add new numerical algorithms to the PSE. The effect of adding a new algorithm can be measured by making use of an automated test suite. Second, the BVP component of pythODE is used to perform two research studies. In the first study, four known global-error estimation algorithms are compared in pythODE. These algorithms are based on the use of Richardson extrapolation, higher-order formulas, deferred corrections, and a conditioning constant. Through numerical experimentation, the algorithms based on higher-order formulas and deferred corrections are shown to be computationally faster than Richardson extrapolation while having similar accuracy. In the second study, pythODE is used to solve a newly developed one-dimensional model of the agglomerate in the catalyst layer of a proton exchange membrane fuel cell

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nag_zhpcon (f07puc) nag_zhpcon (f07puc) estimates the condition number of a complex Hermitian indefinite matrix A, where A has been factorized by nag_zhptrf (f07prc), using packed storage

Fast Taylor polynomial evaluation for the computation of the matrix cosine

[EN] In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and a mixed forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the matrix cosine function more efficiently than using Paterson-Stockmeyer method. A sequential Matlab implementation of the new algorithm is provided, giving better efficiency and accuracy than state-of-the-art algorithms. Moreover, we provide an implementation in Matlab that can use NVIDIA CPUs easily and efficiently. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by Spanish Ministerio de Economía y Competitividad and European Regional Development Fund (ERDF) grants TIN2014-59294-P, and TIN2017-89314-P.Sastre, J.; Ibáñez González, JJ.; Alonso-Jordá, P.; Peinado Pinilla, J.; Defez Candel, E. (2019). Fast Taylor polynomial evaluation for the computation of the matrix cosine. Journal of Computational and Applied Mathematics. 354:641-650. https://doi.org/10.1016/j.cam.2018.12.041S64165035