Simple geometry facilitates iterative solution of a nonlinear equation via a special transformation to accelerate convergence to third order

✍ Scribed by M. Çetin Koçak

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 202 KB
• Volume: 218
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2007.02.036

✦ Synopsis

Direct substitution x k+1 = g(x k ) generally represents iterative techniques for locating a root z of a nonlinear equation f (x). At the solution, f (z) = 0 and g(z) = z. Efforts continue worldwide both to improve old iterators and create new ones. This is a study of convergence acceleration by generating secondary solvers through the transformation

is the point of intersection of a linearised g with the g = x line. Aitken's and Wegstein's accelerators are special cases of g m . Simple geometry suggests that m(x) = (g (x) + g (z))/2 is a good approximation for the ideal slope of the linearised g. Indeed, this renders a third-order g m . The pertinent asymptotic error constant has been determined. The theoretical background covers a critical review of several partial substitution variants of the well-known Newton's method, including third-order Halley's and Chebyshev's solvers. The new technique is illustrated using first-, second-, and third-order primaries. A flexible algorithm is added to facilitate applications to any solver. The transformed Newton's method is identical to Halley's. The use of m(x) = (g (x) + g (z))/2 thus obviates the requirement for the second derivative of f (x). Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.