Iterationsmatrix

Iterationsmatrix, often referred to as the iteration matrix, is a concept in numerical analysis and linear algebra that describes how errors propagate under a fixed-point iteration. For a vector-valued function g, a fixed point x* satisfies g(x*) = x*. Starting from an initial guess x0 and iterating x_{k+1} = g(x_k), the behavior of the error e_k = x_k − x* is governed, near x*, by the derivative (Jacobian) of g: e_{k+1} ≈ J_g(x*) e_k, where J_g(x*) is the iteration matrix. In linear settings where the iteration has the form x_{k+1} = B x_k + c, the matrix B itself serves as the iteration matrix, and the fixed point condition becomes x* = B x* + c.

A common source of iteration matrices is the decomposition of a linear system A x = b used in iterative solvers. For example, in Jacobi iteration with A = D + L + U, the iteration is x^{k+1} = D^{-1}(b − (L + U)x^k), and the iteration matrix is B_J = −D^{-1}(L + U). In Gauss-Seidel iteration, B_GS = −(D + L)^{-1} U. Convergence of the iteration is determined by the spectral radius ρ(B): the method converges if ρ(B) < 1. Sufficient conditions for convergence include diagonal dominance of A for Jacobi, or symmetry and positive definiteness for certain Gauss-Seidel variants.

In nonlinear problems, the iteration matrix is the Jacobian of g at the fixed point, and convergence is typically local, governed by the spectral radius of this Jacobian. The rate of convergence is approximately proportional to ρ(B)^k, where k is the iteration count.