The real polynomial eigenvalue problem is well conditioned on the average (Q2309519)

The paper deals with polynomial eigenvalue problem, namely its condition number is studied. First, the solution variety \(\mathcal{S}\) is introduced, which turns out to be real algebraic or semialgebraic subset of \(\mathbb{R}^m\setminus\{0\}\times S^1\), the product of the variety of inputs and the variety of outputs endowed with Finsler structures. The condition number \(\mu(a)\) of a given input \(a\) is the sum of the local condition numbers \(\mu(a,x)\) for all solutions for the input. If the solution variety \(\mathcal{S}\) is co-called nondegenerate, the formula for the squared condition number is proven. Afterwards the formula is used for the case of polynomial eigenvalue problem and so a new proof of the latter is obtained.