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Wikipedia / Jacobi method

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.

Description

be a square system of n

as the k- is the next (or k.

Matrix-based formula

Then A can be decomposed into a diagonal component D, a lower triangular part L and an upper triangular part UThe solution is then obtained iteratively via

Element-based formula

except itself. Unlike the Gauss–Seidel method, as that value will be needed by the rest of the computation. The minimum amount of storage is two vectors of size n.

Algorithm

Input: initial guess x(0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion
Output: solution when convergence is reached
Comments: pseudocode based on the element-based formula above

k = 0
while convergence not reached do
    for i := 1 step until n do
        σ = 0
        for j := 1 step until n do
            if j ≠ i then
                σ = σ + aij xj(k)
            end
        end
        xi(k+1) = (bi − σ) / aii
    end
    increment k
end

Convergence

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

Examples

Example question

is given by

is small). The solution after 25 iterations is

Example question 2

Suppose we are given the following linear system:

If we choose 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 .

Python example

Weighted Jacobi method

to compute the iteration as

being the usual choice., this may also be expressed as

.

Convergence in the symmetric positive definite case

is symmetric positive-definite, one can show convergence.

be the iteration matrix. Then, convergence is guaranteed for

is the maximal eigenvalue.

is the matrix condition number.

See also

References