First-principles Derivation of a Third-order Method for Solving a Two-dimensional Nonlinear System

J.S.C. Prentice


In this paper, we adopt a `first-principles' approach to deriving a cubically convergent unipoint iterative method for a two-dimensional system of nonlinear equations. We demand that the Jacobian and Hessian of an iteration function be identically zero at the fixed point, and these conditions allow us to determine various terms in the iteration function. We present analytical expressions for the inverses of three matrices appearing in the algorithm, which allows the iteration function to be written explicitly. We demonstrate the cubic convergence rate by means of a few numerical examples, and we determine the asymptotic error constant for these examples.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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