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Matrix

a)I need to find the spectral radius of the Jacobi and Gauss-Seidel iteration matrices from the given matrix.

I know that the spectral radius is the maximum eigenvalue however I am still confused by the question.

Any help would be very appreciated

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  • $\begingroup$ Check wikipedia. $\endgroup$
    – u_sre
    Mar 3 '20 at 15:44
  • $\begingroup$ @Moo Yes thank you. I was able to find the spectral radius of the Gauss-Seidel Method however, i was not able to do it for the relaxation (SOR) method. Is there any chance you can help me with this? $\endgroup$
    – Jared
    Mar 4 '20 at 20:02
  • $\begingroup$ Wouldn't you have to find the parameter $\omega$. Then using this result, find the spectral radius using the preconditioning matrix (where $D$ is the diagonal and $E$ is the lower triangular like you used in the GS-Method) $$\dfrac{1}{\omega}\left(D - \omega E\right)$$ $\endgroup$
    – Moo
    Mar 4 '20 at 20:30
  • $\begingroup$ @Moo I tried doing that however i did not get the correct answer. I worked out w to be 1.1716 however then when working out p I get 0.3719 which is incorrect. I know it is incorrect because the answer does not satisfy this equation: p = w - 1 $\endgroup$
    – Jared
    Mar 4 '20 at 20:55
  • $\begingroup$ $\omega$ looks correct, but the spectral radius does not. Let me take a quick look. $\endgroup$
    – Moo
    Mar 4 '20 at 21:04
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We are given

$$A = \begin{pmatrix} 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 4 & -1 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 4 & 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 4 & -1 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & -1 & 4 & -1 & 0 & -1 & 0 \\ 0 & 0 & -1 & 0 & -1 & 4 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 0 & 0 & 4 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 & -1 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 & 4 \\ \end{pmatrix}$$

For the Jacobi Method, we have

$$mjac = \begin{pmatrix} 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 \\ \end{pmatrix}$$

$$tjac = I - mjac^{-1} A = \begin{pmatrix} 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 & 0 \\ \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 & 0 & 0 \\ \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 \\ 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \\ 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} \\ 0 & 0 & 0 & \frac{1}{4} & 0 & 0 & 0 & \frac{1}{4} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 & \frac{1}{4} \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{4} & 0 & \frac{1}{4} & 0 \\ \end{pmatrix}$$

The eigenvalues of $tjac$ are

$$-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},-\frac{1}{2 \sqrt{2}},-\frac{1}{2 \sqrt{2}},\frac{1}{2 \sqrt{2}},\frac{1}{2 \sqrt{2}},0,0,0$$

The largest in absolute value makes the spectral radius of the Jacobi Method

$$\rho_j = \frac{1}{\sqrt{2}}$$

Can you do this for the Gauss-Seidel Method?

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