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

Direct substitution _xk+1=g(_xk)$x_{k+1}=g(x_k)$ generally represents iterative techniques for locating a root z   of a nonlinear equation f(x)$f(x)$. At the solution, f(z)=0$f(z)=0$ and g(z)=z$g(z)=z$. 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 _gm(x)=(g(x)-m(x)x)/(1-m(x))$g_m(x)=(g(x)-m(x)x)/(1-m(x))$ or, equivalently, through partial substitution _gmps(x)=x+G(x)(g-x)$g_{m\mathrm{ps}}(x)=x+G(x)(g-x)$, G(x)=1/(1-m(x))$G(x)=1/(1-m(x))$. As a matter of fact, _gm(x)≡_gmps(x)$g_m(x)\equiv g_{m\mathrm{ps}}(x)$ is the point of intersection of a linearised g   with the g=x$g=x$ line. Aitken's and Wegstein's accelerators are special cases of _gm$g_m$. Simple geometry suggests that m(x)=(g^′(x)+g^′(z))/2$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ is a good approximation for the ideal slope of the linearised g  . Indeed, this renders a third-order _gm$g_m$. 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$m(x)=(g^{\prime }(x)+g^{\prime }(z))/2$ thus obviates the requirement for the second derivative of f(x)$f(x)$. Comparison and combination with Halley's and Chebyshev's solvers are provided. Numerical results are from the square root and cube root examples.