### 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.

Original language | English |
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Pages (from-to) | 964-976 |

Number of pages | 12 |

Journal | SIAM Journal on Scientific Computing |

Volume | 10 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1989 |

### Keywords

- Adams PECE method
- interpolant
- defect
- numerical mathematics

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## Cite this

Higham, D. J. (1989). Defect estimation in Adams PECE codes.

*SIAM Journal on Scientific Computing*,*10*(5), 964-976. https://doi.org/10.1137/0910056