Jacobi method

Iterative method

Jacobi method is an iterative method for solving linear systems such as

For this, we use a sequence $x^{(k)}$ which converges to the fixed point(solution)

. For $x^{(0)}$ given, we build a sequence $x^{(k)}$ such $x^{(k+1)}=F(x^{(k)})$ with $k \in \mathbf{N}$ .

where is an invertible matrix.

$\begin{array}{cccc} Ax=b \Leftrightarrow Mx=Nx+b \Leftrightarrow & x &=& M^{-1}Nx+M^{-1}b \\ & &=& F(x) \end{array}$

where

is an affine function.

Algorithm

$\left\{ \begin{array}{cc} x^{(0)} \textrm{ given}& ,\\ x^{(k+1)} = M^{-1}Nx^{(k)}+M^{-1}b& \textrm{else}. \end{array} \right.$

If is solution of then $x = M^{-1}Nx+M^{-1}b$

Error

Let $e^{(k)}$ be the error vector

$e^{(k+1)}=x^{(k+1)}-x^{(k)}=M^{-1}N(x^{(k)}-x^{(k-1)})=M^{-1}Ne^{(k)}$
We put $B = M^{-1}N$ , which gives

$e^{(k+1)}=Be^{(k)}=B^{(k+1)}e^{(0)}.$

Convergence

The algorithm converges if $\lim_{k \to \infty} \| e^{(k)} \| = 0 \Leftrightarrow \lim_{k \to \infty} \| B^k \| = 0$ (null matrix).

Theorem: $\lim_{k \to \infty} \| B^k \| = 0$ if and only if the spectral radius of the matrix checks:

$\rho(B)<1,$

we remind that $\rho(B) = \max_{i = 1,\ldots,n} |\lambda_i|$ where $\lambda_1,\ldots,\lambda_n$ represent the eigenvalues of

Theorem: If A is strictly diagonally dominant,

$\left | a_{ii} \right | > \sum_{i \ne j} {\left | a_{ij} \right |},\forall i=1,\ldots,n$

then for all

the Jacobi algorithm will converge to the solution

of the system

We decompose in the following way :

with

the diagonal

the strictly lower triangular part of

the strictly upper triangular part of

In the Jacobi’s method, we choose and (in the Gauss-Seidel Method, and ).

$x^{(k+1)}=D^{-1}(E+F) x^{(k)}+D^{-1}b$

The -th line of $D^{-1}(E+F)$ is : $-(\frac{a_{i,1}}{a_{i,i}},\cdots, \frac{a_{i,i-1}}{a_{i,i}},0,\frac{a_{i,i+1}}{a_{i,i}},\cdots, \frac{a_{i,n}}{a_{i,i}})$

We obtain :

$x^{(k+1)}_i= -\frac{1}{a_{ii}} \sum_{j=1,j \ne i}^n a_{ij}x^{(k)}_j + \frac{b_i}{a_{ii}}$

Residual vector

Let $r^{(k)}=b-Ax^{(k)}$ be the residual vector. We can write $x_i^{(k+1)}=\frac{r_i^{(k)}}{a_{ii}} + x_i^{(k)}$ with $r_i^{(k)}$ calculated like follows

$r_i^{(k+1)}=-\sum_{j=1,j \ne i}^n a_{ij} \frac{r_i^{(k)}}{a_{jj}}$

Stop criteria

For the stop criteria , we can use the residual vector, wich gives for a given precision $\epsilon$ :

$\frac{\|r^{(k)} \|}{\|b\|}=\frac{\|b-Ax^{(k)} \|}{\|b\|} < \epsilon$