A generalization of singly-implicit methods
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Singly-implicit Runge-Kutta methods are characterized by a one-point spectrum property of the coefficient matrix. If a method of this type is also a collocation method, then its abscissae are proportional to the zeros of a Laguerre polynomial. The generalization introduced here is a multistep method in the style of Nordsieck and also a multistage method under the one-point spectrum constraint. It is found that much of the theory of singly-implicit methods carries over but with Laguerre polynomials replaced by their usual generalizations. Amongst the formal properties of the new method which are studied is a derivation of the similarity transformations which allow their efficient implementation. A preliminary investigation is made of the stability of the new methods.
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