JacobiMethode

JacobiMethode, also known as the Jacobi method, is an iterative algorithm for solving a system of linear equations Ax = b. It is named after Carl Gustav Jacob Jacobi. The method updates all components of the vector x simultaneously, using values from the previous iteration. Writing A as D + R where D is the diagonal part of A and R contains the off-diagonal elements, the scheme solves D x^(k+1) = b - R x^(k). Equivalently, x_i^(k+1) = (b_i - sum_{j≠i} a_ij x_j^(k)) / a_ii for i = 1,...,n.

Convergence: The Jacobi iteration can be written as x^(k+1) = J x^(k) + c with J = -D^{-1} R and c = D^{-1} b. The method converges if the spectral radius of J, ρ(J), is less than 1. A standard sufficient condition is that A is strictly diagonally dominant by rows. For symmetric positive definite A, Gauss-Seidel is guaranteed to converge; Jacobi convergence is not guaranteed solely by positive definiteness, though it holds for many SPD matrices in practice.

Complexity: Each iteration requires solving diagonally simple equations, so the per-iteration cost is O(n^2) for dense A and O(nnz) for sparse A. Memory usage is O(n^2) to store A and O(n) for the vectors.

Variants and relations: A common variant is weighted Jacobi, x^(k+1) = x^(k) + ω D^{-1}(b - A x^(k)) with relaxation parameter ω; this leads to the relaxed Jacobi method. The Jacobi method is often used as a simple baseline solver for large sparse systems or as a component within more advanced iterative schemes.

---