Jacobi method

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In numerical linear algebra, the Jacobi method is an algorithm for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value 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 Jakob Jacobi.

Description

Given a square system of n linear equations:

$A{\mathbf x}={\mathbf b}$

where:

$A={\begin{bmatrix}a_{{11}}&a_{{12}}&\cdots &a_{{1n}}\\a_{{21}}&a_{{22}}&\cdots &a_{{2n}}\\\vdots &\vdots &\ddots &\vdots \\a_{{n1}}&a_{{n2}}&\cdots &a_{{nn}}\end{bmatrix}},\qquad {\mathbf {x}}={\begin{bmatrix}x_{{1}}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\qquad {\mathbf {b}}={\begin{bmatrix}b_{{1}}\\b_{2}\\\vdots \\b_{n}\end{bmatrix}}.$

Then A can be decomposed into a diagonal component D, and the remainder R:

$A=D+R\qquad {\text{where}}\qquad D={\begin{bmatrix}a_{{11}}&0&\cdots &0\\0&a_{{22}}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &a_{{nn}}\end{bmatrix}}{\text{ and }}R={\begin{bmatrix}0&a_{{12}}&\cdots &a_{{1n}}\\a_{{21}}&0&\cdots &a_{{2n}}\\\vdots &\vdots &\ddots &\vdots \\a_{{n1}}&a_{{n2}}&\cdots &0\end{bmatrix}}.$

The solution is then obtained iteratively via

${\mathbf {x}}^{{(k+1)}}=D^{{-1}}({\mathbf {b}}-R{\mathbf {x}}^{{(k)}}).$

The element-based formula is thus:

$x_{i}^{{(k+1)}}={\frac {1}{a_{{ii}}}}\left(b_{i}-\sum _{{j\neq i}}a_{{ij}}x_{j}^{{(k)}}\right),\quad i=1,2,\ldots ,n.$

Note that the computation of x_i^(k+1) requires each element in x^(k) except itself. Unlike the Gauss–Seidel method, we can't overwrite x_i^(k) with x_i^(k+1), as that value will be needed by the rest of the computation. The minimum amount of storage is two vectors of size n.

Algorithm

Choose an initial guess $x^{{0}}$ to the solution

$k=0$

check if convergence is reached

while convergence not reached do

for i := 1 step until n do

$\sigma =0$

for j := 1 step until n do

if j ≠ i then

$\sigma =\sigma +a_{{ij}}x_{j}^{{(k)}}$

end if

end (j-loop)

$x_{i}^{{(k+1)}}={{\left({b_{i}-\sigma }\right)} \over {a_{{ii}}}}$

end (i-loop)

check if convergence is reached

$k=k+1$

loop (while convergence condition not reached)

Convergence

The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1:

$\rho (D^{{-1}}R)<1.$

The method is guaranteed to converge if 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:

$\left|a_{{ii}}\right|>\sum _{{j\neq i}}{\left|a_{{ij}}\right|}.$

The Jacobi method sometimes converges even if these conditions are not satisfied.

Example

A linear system of the form $Ax=b$ with initial estimate $x^{{(0)}}$ is given by

$A={\begin{bmatrix}2&1\\5&7\\\end{bmatrix}},\ b={\begin{bmatrix}11\\13\\\end{bmatrix}}\quad {\text{and}}\quad x^{{(0)}}={\begin{bmatrix}1\\1\\\end{bmatrix}}.$

We use the equation $x^{{(k+1)}}=D^{{-1}}(b-Rx^{{(k)}})$ , described above, to estimate $x$ . First, we rewrite the equation in a more convenient form $D^{{-1}}(b-Rx^{{(k)}})=Tx^{{(k)}}+C$ , where $T=-D^{{-1}}R$ and $C=D^{{-1}}b$ . Note that $R=L+U$ where $L$ and $U$ are the strictly lower and upper parts of $A$ . From the known values

$D^{{-1}}={\begin{bmatrix}1/2&0\\0&1/7\\\end{bmatrix}},\ L={\begin{bmatrix}0&0\\5&0\\\end{bmatrix}}\quad {\text{and}}\quad U={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}.$

we determine $T=-D^{{-1}}(L+U)$ as

$T={\begin{bmatrix}1/2&0\\0&1/7\\\end{bmatrix}}\left\{{\begin{bmatrix}0&0\\-5&0\\\end{bmatrix}}+{\begin{bmatrix}0&-1\\0&0\\\end{bmatrix}}\right\}={\begin{bmatrix}0&-1/2\\-5/7&0\\\end{bmatrix}}.$

Further, C is found as

$C={\begin{bmatrix}1/2&0\\0&1/7\\\end{bmatrix}}{\begin{bmatrix}11\\13\\\end{bmatrix}}={\begin{bmatrix}11/2\\13/7\\\end{bmatrix}}.$

With T and C calculated, we estimate $x$ as $x^{{(1)}}=Tx^{{(0)}}+C$ :

$x^{{(1)}}={\begin{bmatrix}0&-1/2\\-5/7&0\\\end{bmatrix}}{\begin{bmatrix}1\\1\\\end{bmatrix}}+{\begin{bmatrix}11/2\\13/7\\\end{bmatrix}}={\begin{bmatrix}5.0\\8/7\\\end{bmatrix}}\approx {\begin{bmatrix}5\\1.143\\\end{bmatrix}}.$

The next iteration yields

$x^{{(2)}}={\begin{bmatrix}0&-1/2\\-5/7&0\\\end{bmatrix}}{\begin{bmatrix}5.0\\8/7\\\end{bmatrix}}+{\begin{bmatrix}11/2\\13/7\\\end{bmatrix}}={\begin{bmatrix}69/14\\-12/7\\\end{bmatrix}}\approx {\begin{bmatrix}4.929\\-1.713\\\end{bmatrix}}.$

This process is repeated until convergence (i.e., until $\|Ax^{{(n)}}-b\|$ is small). The solution after 25 iterations is

$x={\begin{bmatrix}7.111\\-3.222\end{bmatrix}}.$

Weighted Jacobi method

The weighted Jacobi iteration uses a parameter $\omega$ to compute the iteration as

${\mathbf {x}}^{{(k+1)}}=\omega D^{{-1}}({\mathbf {b}}-R{\mathbf {x}}^{{(k)}})+\left(1-\omega \right){\mathbf {x}}^{{(k)}}$

with $\omega =2/3$ being the usual choice.^[1]

References

↑ Saad, Yousef (2003). Iterative Methods for Sparse Linear Systems (2 ed.). SIAM. p. 414. ISBN 0898715342.

External links

This article incorporates text from the article Jacobi_method on CFD-Wiki that is under the GFDL license.

Black, Noel; Moore, Shirley; and Weisstein, Eric W., "Jacobi method", MathWorld.
Jacobi Method from www.math-linux.com
Module for Jacobi and Gauss–Seidel Iteration
Numerical matrix inversion

v t e Numerical linear algebra

Key concepts	Floating point Numerical stability BLAS

Problems	Matrix multiplication Matrix decompositions Linear equations Sparse problems

Hardware	CPU cache TLB Cache-oblivious algorithm SIMD Multiprocessing

Software	Specialized libraries General purpose software

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