# Convergence behaviour of the Newton iteration for first order differential equations

5. Differential Equations

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## Abstract

In a typical application of the Newton iteration in a power series domain (e.g. to compute an algebraic function), the number of correct power series coefficients in the k-th iterate is exactly double the number of correct coefficients in the preceding iterate. This paper considers the application of the Newton iteration to compute the power series solution of a first-order nonlinear differential equation. It is proved that in one iteration the number of correct coefficients is more than doubled in the case of an explicit differential equation, and is less than doubled in the most general case.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1979