alternatingiteration

Alternatingiteration is a term used in numerical analysis to describe a class of iterative algorithms that progress toward a solution by alternately applying two (or more) distinct iterative procedures. In its simplest form, the method alternates between two maps or operators, often chosen to address different aspects of a problem, such as stability of updates and accuracy of refinement.

Formally, let X be a complete metric or normed space and let A and B be two operators X → X. Starting from an initial guess x0, an alternatingiteration scheme applies A and B in sequence, for example by setting x1 = A(B(x0)) or by choosing a fixed alternation pattern like x(2k+1) = A(x(2k)), x(2k+2) = B(x(2k+1)).

If A and B are contractions (or satisfy appropriate nonexpansive properties) and the composite AB (or the chosen pattern) is a contraction, the sequence (xn) converges to a fixed point x*, which is a solution to the problem (e.g., a fixed point of AB or a common fixed point of A and B). Convergence analysis typically relies on the contraction mapping principle or on operator-splitting theory.

Alternatingiteration is related to, and sometimes overlaps with, operator splitting methods, cyclic coordinate updates, and alternating projection methods used for finding a point in the intersection of sets. It can be viewed as a way to combine complementary strengths of the constituent steps, such as fast preliminary progress followed by refinement, or handling different constraints separately.

Applications of alternatingiteration appear in numerical linear algebra, convex optimization, image processing, and machine learning. Its effectiveness depends on the choice of A and B and on problem structure; without suitable conditions, convergence is not guaranteed.