Eigenvalues of Jacobi matrix and convergence of Jacobi method.

numerical methods

Let $J$ be the Jacobian iteration matrix of the linear system $\begin{bmatrix}
1&2&1\\ 2&1&2\\ -4&2&1
\end{bmatrix}\begin{bmatrix}
x\\y\\z
\end{bmatrix} = \begin{bmatrix}
1\\2\\3
\end{bmatrix}$. Consider the following statements:

(P): one of the eigen values of J lies in $[2, 3]$.

(Q): The Jacobi iteration matrix converges for the above system.

Which of the above statements are true?

My attempt:

First we write given matrix as $D+R$. Then, $x_{k+1}= D^{-1}(b-Rx_k)$. The Jacobi matrix is $D^{-1}R$. Am i correct? How to proceed further. Thanks in advance.

Best Answer

Here's a really big theorem:

A consistent iteration scheme $x^{(k+1)} = Bx^{(k)} + f$ (where $B \in \mathbb{R}^{n \times n}$ is the iteration matrix and $f \in \mathbb{R}^n$) converges iff $\rho(B) < 1$ where we define $$\rho(B) = \max_{i} |\lambda_i(B)|$$ is the spectral radius of the matrix (the maximum of the absolute values of the eigenvalues of the matrix).

This theorem will be of great use to us. The iteration matrix for Jacobi method is $-D^{-1}R$ as you said. So in this case, we have $B = -D^{-1}R$ so all we have to do is check if $\rho(-D^{-1}R) < 1$. If this is the case, we have convergence.

Related Question