Category:Definitions/Jacobi's Iterative Method
This category contains definitions related to Jacobi's Iterative Method.
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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 $\mathbf A \mathbf x = \mathbf b$.
Pages in category "Definitions/Jacobi's Iterative Method"
The following 4 pages are in this category, out of 4 total.