Interval methods for fixed-point problems
L. B. Rail · Numerical Functional Analysis and Optimization · 1987
Interval analysis is applied to the fixed-point problem x=ϕ(x) for continuous ϕ:S→S, where the space S is constructed from Cartesian products of the set R of real numbers, with componentwise definitions of arithmetic operations, ordering, and the product topology. With the aid of an interval inclusion φ:IS → IS in the interval space IS corresponding to S, interval iteration is used to establish the existence or nonexistence of a fixed point x∗ of ϕ in the initial interval X0. Each step of the interval iteration provides lower and upper bounds for fixed points of ϕ in the initial interval, from which approximate values and guaranteed error bounds can be obtained directly. In addition to interval iteration, operator equation and dissection methods are considered briefly. The theory of interval iteration applies directly when only finite subsets of S, IS are used, so this method is adaptable immediately to actual computation. A numerical example is given of the use of interval iteration for the computational solution of a nonlinear integral equation of radiative transfer. It is shown that numerical results with acceptable, guaranteed accuracy can be obtained with a modest amount of computation for an extended range of the parameter involved.