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Two variable-order, varlable-step size methods for the numerical solution of the initial value problem for ordinary differential equations are presented. These methods share a common philosophy and have been combined in a single program. The two integrators are for stiff and nonstiff ordinary differential equations, respectively. The former integrator is… (More)

- N. M. Steen, G. D. Byrne, E. M. Gelbard, E. M. GELBAKD
- 2010

Gaussian quadratures are developed for the evaluation of the integrals given in the title. The weights and abscissae for the semi-infinite integral are given for two through fifteen points with fifteen places. For 6 = 1, the weights and abscissae are given for two through ten points with fifteen places. 1. Introduction. In nuclear reactor design… (More)

- George D. Andria, George D. Byrne, David R. Hill, GEORGE D. ANDRIA, GEORGE D. BRYNE
- 2010

Numerical integration formulas of interpolatory type are generated by the integration of g-splines. These formulas, which are best in the sense of Sard, are used to construct predictor-corrector and block implicit schemes. The schemes are then compared with Adams-Bashforth-Adams-Moulton and Rosser schemes for a particular set of prototype problems.… (More)

The idea of rank-one updates for the inverse of the Newton iteration matrix is considered in the context of solving stiff systems of ordinary differential equations. A specific and simple problem (linear, with a constant, diagonal Jacobian) and a specific and simple method (backward Euler, with constant step) are studied. A Newton iteration matrix which is… (More)

- G. D. Byrne, S. Thompson
- 2013

Sometimes an ordinary differential equation (ODE) solver gives the results it should, even if they are unexpected or undesirable. To obtain correct, expected numerical results, both relative and absolute error control tolerances must be chosen judiciously. Here we attempt to clear up some possible misunderstandings regarding the performance of a popular ODE… (More)

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