Householder's method

Householder's methods are a class of iterative root-finding algorithms for functions of one real variable that have continuous derivatives up to some order d + 1. Each method is characterised by its order d and converges with order d + 1. The methods are named after the American mathematician Alston Scott Householder. The case d = 1 is Newton's method and the case d = 2 is Halley's method. Orders of 3 and above are not widely used, because the gain in precision does not match the rise in effort. Judged by error reduction per function evaluation, the Ostrowski index, the second-order case, Halley's method, is the optimal choice for polynomials. Whether the iterates approach the root from above, from below or alternately from both sides depends on whether d is odd or even and on the sign of an error constant.