Errata to “Elimination and fixed point iterations”

Fixed Point Iterations

Recall that a vector norm on R is a mapping ‖·‖ : R → R satisfying the following conditions: • ‖x‖ > 0 for x 6= 0. • ‖λx‖ = |λ|‖x‖ for x ∈ R and λ ∈ R. • ‖x+ y‖ ≤ ‖x‖+ ‖y‖ for all x, y ∈ R. Since the space Rn×n of all matrices is also a vector space, it is also possible to consider norms there. In contrast to usual vectors, it is, however, also possible to multiply matrices (that is, the matric...

Anderson Acceleration for Fixed-Point Iterations

This paper concerns an acceleration method for fixed-point iterations that originated in work of D. G. Anderson [Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Machinery, 12 (1965), 547-560], which we accordingly call Anderson acceleration here. This method has enjoyed considerable success and wide usage in electronic structure computations, where it is known as Anders...

Fixed Point Iterations for Real Functions

Fixed Point Iterations Using Infinite Matrices

Let £ be a closed, bounded, convex subset of a Banach space X, /: E —»E. Consider the iteration scheme defined by x"« = xQ e E, x , = ñx ), x = 2" na ,x., nal, where A is a regular weighted mean n + l ' n n * = 0 nk k o er matrix. For particular spaces X and functions /we show that this iterative scheme converges to a fixed point of /. Let X be a normed linear space, E a nonempty closed bounded...

Compositional Analysis of Boolean Networks Using Local Fixed-Point Iterations

We present a compositional method which allows to overapproximate the set of attractors and under-approximate the set of basins of attraction of a Boolean network (BN). This merely consists in replacing a global fixed-point computation by a composition of local fixedpoint computations. Once these approximations have been computed, it becomes much more tractable to generate the exact sets of att...