Fixed point iterations for real functions

Fixed Point Iterations for Real Functions

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 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...

Implicit Mann fixed point iterations for pseudo-contractive mappings

Let K be a compact convex subset of a real Hilbert space H and T:K→K a continuous hemicontractive map. Let {an},{bn} and {cn} be real sequences in [0, 1] such that an+bn+cn=1, and {un} and {vn} be sequences in K. In this paper we prove that, if {bn}, {cn} and {vn} satisfy some appropriate conditions, then for arbitrary x0K, the sequence {xn} defined iteratively by xn=anxn−1+bnTvn+cnun;n≥1, conv...