Jacobimenetelmää

Jacobimenetelmää, often referred to as the Jacobi method, is an iterative technique used to solve a system of linear equations. It is a member of the stationary iterative methods, meaning that the update rule does not change from one iteration to the next. The method is particularly suited for solving large, sparse systems of linear equations that arise in various scientific and engineering applications, such as finite difference approximations of partial differential equations.

The core idea behind the Jacobi method is to iteratively refine an initial guess for the solution vector. In each step, a new approximation of each variable is calculated by using the values of the other variables from the *previous* iteration. This is in contrast to the Gauss-Seidel method, which uses the most recently updated values within the same iteration. For a system of linear equations Ax = b, where A is a square matrix, x is the vector of unknowns, and b is a known vector, the Jacobi method updates each component xi^(k+1) of the solution vector x at iteration k+1 as follows: xi^(k+1) = (bi - sum(Aij * xj^(k) for j != i)) / Aii. This process is repeated until the solution converges to a desired level of accuracy, typically measured by the difference between successive approximations. Convergence is guaranteed for diagonally dominant matrices.