Category:Jacobi's Iterative Method

This category contains results about Jacobi's Iterative Method.
Definitions specific to this category can be found in Definitions/Jacobi's Iterative Method.

Jacobi's iterative method is an iterative method for solving a system $S$ of simultaneous linear equations.

Let $\mathbf A \mathbf x = \mathbf b$ be a system of simultaneous linear equations expressed in matrix form.

Let $\mathbf A$ be reduced to the form:

$\mathbf A = \mathbf D + \mathbf B$

where $\mathbf D$ denotes the diagonal matrix:

$D_{i i} : = A_{i i}$

where:

$A_{i i}$ and $D_{i i}$ are the elements at the $i$th row and $i$th column of $\mathbf A$ and $\mathbf D$ respectively.

$A_{i i} \ne 0$ for all $i$

One would arrange for $S$ to be expressed in a form as to make the latter statement so.

Let $\mathbf x_0$ be a first approximation to the vector $\mathbf x$.

Jacobi's iterative method generates a sequence of vectors $\mathbf x_1, \mathbf x_2, \ldots$ from the formula:

$\mathbf x_{n + 1} = \mathbf D^{-1} \paren {\mathbf b - \mathbf B \mathbf x_n}$

Suppose $\sequence {x_n}$ converges to the limit $\mathbf x$.

Then $\mathbf x$ is the solution to $\mathbf A \mathbf x = \mathbf b$.