  Iteration methods based on barycentric rational interpolation are derived
that exhibit accelerating orders of convergence. For univariate root search,
the derivative-free methods approach quadratic convergence and the
first-derivative methods approach cubic convergence. For univariate
optimisation, the convergence order of the derivative-free methods approaches
1.62 and the order of the first-derivative methods approaches 2.42. Generally,
performance advantages are found with respect to low-memory iteration methods.
In optimisation problems where the objective function and gradient is
calculated at each step, the full-memory iteration methods converge
asymptotically 1.8 times faster than the secant method. Frameworks for
multivariate root search and optimisation are also proposed, though without
discovery of practical parameter choices.
