The condition of Steffensenʹs acceleration in several variables

To improve the efficiency of the straightforward algorithm for general secant methods in several variables, Wolfe initiated a fast secant algorithm in 1959, which became very popular in the 1980s and 1990s, notably with Brezinski, Sadok, and others. However, the condition number of such a fast algorithm increases as fast as the iterations converge, which guarantees at most one-half as many accurate digits as used in the computation. In contrast, the condition number of the straightforward algorithm may remain bounded, for example, in certain instances of the method—suggested by Henrici in 1964 and 1982—to compute Steffensenʹs acceleration by means of Aitkenʹs acceleration. Specifically, the present work shows that if the ambient space is cyclic with respect to the Jacobian matrix, then every neighbourhood of the fixed point contains initial estimates from which the iteration matrix for the next step remains uniformly nonsingular. In particular, for maps of the real plane with no real eigenvalues at the fixed point, Henriciʹs straightforward algorithm converges stably from all sufficiently close initial estimates. Numerical examples confirm that the straightforward algorithm converges faster than the fast secant algorithm.