On the use of rank-one updates in the solution of stiff systems of ordinary differential equations

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Abstract
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 in error because of a change in step size is then considered. The various resulting Newton matrices (original and updated) and the spectral radii of the corresponding error reduction matrices are computed. Two formulations of the Newton process are considered. On the whole the results are rather discouraging. The observed reductions in spectral radii as a result of the use of updates were usually only slight, unless sufficiently large numbers of updates were performed.

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