Computational Physics Lectures: Eigenvalue problems

Discussion of Jacobi's method for eigenvalues

Choose a tolerance $\epsilon$ , making it a small number, typically $10^{-8}$ or smaller.
Setup a while test where one compares the norm of the newly computed off-diagonal matrix elements $\mathrm{off}(\mathbf{A}) = \sqrt{\sum_{i=1}^n\sum_{j=1,j\ne i}^n a_{ij}^2} > \epsilon.$
Now choose the matrix elements $a_{kl}$ so that we have those with largest value, that is $|a_{kl}|=\mathrm{max}_{i\ne j} |a_{ij}|$ .
Compute thereafter $\tau = (a_{ll}-a_{kk})/2a_{kl}$ , $\tan\theta$ , $\cos\theta$ and $\sin\theta$ .
Compute thereafter the similarity transformation for this set of values $(k,l)$ , obtaining the new matrix $\mathbf{B}= \mathbf{S}(k,l,\theta)^T \mathbf{A}\mathbf{S}(k,l,\theta)$ .
Compute the new norm of the off-diagonal matrix elements and continue till you have satisfied $\mathrm{off}(\mathbf{B}) \le \epsilon$