Eigenvalues and eigenvectors play important role in many real-world applications, including control systems modeling, partial differential equations, data mining and clusterization, chemistry, vibration analysis, to name a few examples.
The main practical difficulty is that eigenproblem might be ill-conditioned and hard to compute even when matrix itself is well-conditioned.
This comes from the fact that computing eigenvalues is equivalent to finding roots of a polynomial, which is proven to be unavoidably ill-conditioned problem (even when roots are well-separated, as shown by J.H. Wilkinson [1-3]) .
More precisely, conditioning of an eigenproblem depends on eigenvalues clustering, magnitude and angles between the eigenvectors. All the properties are inherent to the problem and ill-conditioning, if it is present, cannot be easily alleviated or even detected beforehand. The only case when it is considered stable is computing eigenvalues of a symmetric matrix.
At the moment the only reliable way of dealing with ill-conditioned eigenproblems is to use the extended-precision computations. In fact, there is no other alternative in contrast to solving systems of linear equations where various pre-conditioners can be used to improve the conditioning.
Here we want to show two examples of such problems and how toolbox solves them in comparison to MATLAB.
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. Naturally this limits applicability of the methods for large matrices, as complexity might be prohibitive.
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