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In numerical analysis, an iterative method is called locally convergent if the successive approximations produced by the method are guaranteed to converge to a solution when the initial approximation is already close enough to the solution.
4. every point of X has a local base of closed compact neighbourhoods. 5. ... local uniform convergence is the same as compact convergence. The point at infinity
Uniform convergence implies both local uniform convergence and compact convergence, since both are local notions while uniform convergence is global. If X is locally compact (even in the weakest sense: every point has compact neighborhood), then local uniform convergence is equivalent to compact (uniform) convergence. Roughly speaking, this is ...
Uniform convergence both local uniform convergence and compact (uniform) convergence. - "Local" modes of convergence tend to imply "compact" modes of convergence. E.g., Local uniform convergence compact (uniform) convergence. - If is locally compact, the converses to such tend to hold: Local uniform convergence compact (uniform) convergence.
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.. More specifically, given a function defined on the real numbers with real values and given a point in the domain of , the fixed-point iteration is + = (), =,,, … which gives rise to the sequence,,, … of iterated function applications , (), (()), … which is hoped to converge to a point .
Matetski-Quastel-Remenik constructed the KPZ fixed point for the (+)-dimensional KPZ universality class (i.e. one space and one time dimension) on the polish space of upper semicontinous functions (UC) with the topology of local UC convergence.
A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. [7] The term locally convex topological vector space is sometimes shortened to locally convex space or LCTVS.
The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. [1] [2] It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point. [3]