George D. Byrne

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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)
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)
If u and P are 2V-dhuensiomfl vectors a:ud cx, ~, % (3 are matrices of order N, the atmlogue of eq, (:12) is dA'/dT = ~ + c~,9 + Ra + lgvtf, as shown i~ [6]. The correspondiug equatiol~ for ,u(t) = Z(7'~c is 2 *:) Qe~) dZ, It. has been shown in several ex:m~ples that, iu some cases, one can convert unstable boundary-value problems into stable initial-value(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)