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?


I am not sure how deep you want to get into it, but you may want to check out Z. Bai's "Error Analysis of the Lanczos algorithm for the non-symmetric eigenvalue problem", Math. Comp. 62 (1994), 209-226. It is open access, so you can access it freely.

C.C. Paige's "Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem", Linear Algebra and its Applications, Volume 34, (1980), Pages 235-258, may be more appropriate, as it deals with symmetric problems.


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