Alexei Yu. Uteshev

We treat the problem of the Frobenius distance evaluation from a given matrix $ A \in \mathbb R^{n\times n} $ with distinct eigenvalues to the manifold of matrices with multiple eigenvalues. On restricting considerations to the rank $ 1 $ real perturbation matrices, we prove that the distance in question equals $ \sqrt{z_{\ast}} $ where $ z_{\ast} $ is a positive (generically, the least positive) zero of the algebraic equation $$ \mathcal F(z) = 0, \ \mbox{where} \ \mathcal F(z):= \mathcal D_λ \left( \det \left[ (λI - A)(λI - A^{\top})-z I_n \right] \right)/z^n $$ and $ \mathcal D_λ $ stands for the discriminant of the polynomial treated with respect to $λ$. In the framework of this approach we also provide the procedure for finding the nearest to $ A $ matrix with multiple eigenvalue. Generalization of the problem to the case of complex perturbations is also discussed. Several examples are presented clarifying the computational aspects of the approach.

We treat the interpolation problem ${\{f\left(x_j\right)=y_j\}}_{j=1}^N$ for polynomial and rational functions. Developing the approach originated by C. Jacobi, we represent the interpolants by virtue of the Hankel polynomials generated by the sequences of special symmetric functions of the data set like ${\{{\sum }_{j=1}^N{x}_j^k{y}_j/W^{\prime }\left(x_j\right)\}}_{k\in \mathbb{N}}$ and ${\{{\sum }_{j=1}^N{x}_j^k/\left(y_j{W}^{\prime }\left(x_j\right)\right)\}}_{k\in \mathbb{N}}$; here, $W\left(x\right)={\prod }_{j=1}^N(x-x_j)$. We also review the results by Jacobi, Joachimsthal, Kronecker and Frobenius on the recursive procedure for computation of the sequence of Hankel polynomials. The problem of evaluation of the resultant of polynomials $p\left(x\right)$ and $q\left(x\right)$ given a set of values ${\{p\left(x_j\right)/q\left(x_j\right)\}}_{j=1}^N$ is also tackled within the framework of this approach. An effective procedure is suggested for recomputation of rational interpolants in case of extension of the data set by an extra point.