In numerical linear algebra, the Jacobi method (a.k.a.
the Jacobi iteration method) is an iterative algorithm for determining the solutions of a strictly diagonally dominant system of linear equations.
Each diagonal element is solved for, and an approximate value is plugged in.
The process is then iterated until it converges.
This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization.
The method is named after Carl Gustav Jacob Jacobi.
be a square system of n linear equations, where:
is unknown, we can use the Jacobi method to approximate
denotes our initial guess for
as the k-th approximation or iteration of
Then A can be decomposed into a diagonal component D, a lower triangular part L and an upper triangular part U:
The solution is then obtained iteratively via The element-based formula for each row
requires each element in
Unlike the Gauss–Seidel method, we cannot overwrite
, as that value will be needed by the rest of the computation.
The minimum amount of storage is two vectors of size n. The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: A sufficient (but not necessary) condition for the method to converge is that the matrix A is strictly or irreducibly diagonally dominant.
Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms: The Jacobi method sometimes converges even if these conditions are not satisfied.
Note that the Jacobi method does not converge for every symmetric positive-definite matrix.
A linear system of the form
First, we rewrite the equation in a more convenient form
This process is repeated until convergence (i.e., until
The solution after 25 iterations is Suppose we are given the following linear system: If we choose (0, 0, 0, 0) as the initial approximation, then the first approximate solution is given by
Using the approximations obtained, the iterative procedure is repeated until the desired accuracy has been reached.
The following are the approximated solutions after five iterations.
The exact solution of the system is (1, 2, −1, 1).
The weighted Jacobi iteration uses a parameter
, this may also be expressed as In case that the system matrix
is of symmetric positive-definite type one can show convergence.
The spectral radius can be minimized for a particular choice of
is the matrix condition number.