How do I find optimal ω for SOR method?

Question

Following is the example from this book. My question is, what value of λ did he put in? He did not explain that, can anybody explain how did he get 1.24?

Answer 1

For the matrix $T_j$, he calculates the eigenvalues. We have the characteristic polynomial:

$$\tag 1 \det(T_j - \lambda I) = -\lambda(\lambda^2 -0.625) \implies \lambda_{1,2,3} = 0, \pm \sqrt{0.625}$$

The spectral radius is given by the maximum magnitude eigenvalue:

$$\rho(A) = \displaystyle \max ~ |\lambda_i| $$

From $(1)$, we have:

$$\rho(T_j) = \displaystyle \max ~ |\lambda_i| = \sqrt{0.625}$$

Now, when $A$ is positive definite and tridiagonal, we have:

$$\omega = \dfrac{2}{1 + \sqrt{1 - [\rho(T_j)]^2}}$$

The calculation follows.

There is a write-up of the theory of this choice in Section 3.