# Defect estimation in Adams PECE codes

Higham, D.J.
(1989)
*Defect estimation in Adams PECE codes.*
SIAM Journal on Scientific Computing, 10 (5).
pp. 964-976.
ISSN 1064-8275

## Abstract

Many modern codes for solving the nonstiff initial value problem $y'(x) - f(x,y(x)) = 0,y(a)$ given, $a leqq x leqq b$, produce, in addition to a discretised solution, a function $p(x)$ that approximates $y(x)$ over $[a,b]$. The associated defect $delta (x): = p'(x) - f(x,p(x))$ is a natural measure of the error. In this paper the problem of reliably estimating the defect in Adams PECE methods is considered. Attention is focused on the widely used Shampine-Gordon variable order, variable step code fitted with a continuously differentiable interpolant $p(x)$ due to Watts and Shampine [SIAM .J. Sci. Statist. Comput, 7 (1986), pp. 334-345]. It is shown that over each step an asymptotically correct estimate of the defect can be obtained by sampling at a single, suitably chosen point. It is also shown that a valid "free" estimate can be formed without recourse to sampling. Numerical results are given to support the theory.

Item type: | Article |
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ID code: | 212 |

Keywords: | Adams PECE method, interpolant, defect, numerical mathematics, Probabilities. Mathematical statistics, Computational Mathematics, Applied Mathematics |

Subjects: | Science > Mathematics > Probabilities. Mathematical statistics |

Department: | Faculty of Science > Mathematics and Statistics |

Depositing user: | Ms Sarah Scott |

Date Deposited: | 08 Mar 2006 |

Last modified: | 20 Apr 2017 00:02 |

URI: | http://strathprints.strath.ac.uk/id/eprint/212 |

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