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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol.218, no.2, pp.350-363, 2008 (SCI-Expanded)
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 g(m)(x) = (g(x) - m(x)x)/(1 - m(x)) or, equivalently, through partial substitution g(mps)(x) = x + G(x)(g - x), G(x) = 1/(1 - m(x)). As a matter of fact, g(m)(x) equivalent to g(mps)(x) is the point of intersection of a linearised g with the g = x line. Aitken's and Wegstein's accelerators are special cases of gm. 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))12 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. (C) 2007 Elsevier B.V. All rights reserved.