# Error propagation on GSL eigenvalues computation

The problem comes from the need to estimate the error propagation of the spectral norm computation of a square matrix $A$ of which I know the components' $A_{ij}$ absolute error.

The fundamental step of the computation of the spectral norm $||A||_2$ is the computation of the largest eigenvalue of the matrix $A^TA$. In fact the spectral norm is defined as $$||A||_2 = \sqrt{\lambda_\mathrm{max}(A^TA)}$$ where $\lambda_\mathrm{max}(A^TA)$ denotes the maximum eigenvalue of $A^TA$.

In my algorithm I'm using the GSL library to compute this $\lambda_\mathrm{max}$. On it's manual the GSL states that the algorithm used is a "symmetric bidiagonalization followed by QR reduction".

My question is: given the absolute error of the components $A_{ij}$ how can I estimate the error on the largest eigenvalue of $A^TA$? Can you point me on the direction to further investigate the problem?