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

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 m ps ( x ) = x + G ( x ) ( g - x ) , G ( x ) = 1 / ( 1 - m ( x ) ) . As a matter of fact, g m ( x ) ≡ g m ps ( x ) is the point of intersection of a linearised g with the g = x line. Aitkenʹs and Wegsteinʹs 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 Newtonʹs method, including third-order Halleyʹs and Chebyshevʹs 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 Newtonʹs method is identical to Halleyʹs. 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 Halleyʹs and Chebyshevʹs solvers are provided. Numerical results are from the square root and cube root examples.