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 \( \mathrm{off}(\mathbf{B}) \le \epsilon \)