Fast Reduction of Generalized Companion Matrix Pairs for Barycentric Lagrange Interpolants

For a barycentric Lagrange interpolant $p(z)$, the roots of $p(z)$ are exactly the eigenvalues of a generalized companion matrix pair $(\mathbf{A},\mathbf{B})$. For real interpolation nodes, the matrix pair $(\mathbf{A},\mathbf{B})$ can be reduced to a pair $(\mathbf{H},\mathbf{B})$, where $\mathbf{H}$ has tridiagonal plus rank-one structure. In this paper we propose two fast algorithms for reducing the pair $(\mathbf{A},\mathbf{B})$ to Hessenberg-triangular form. The matrix pair $(\mathbf{A},\mathbf{B})$ has two spurious infinite eigenvalues, and if the leading coefficients of the interpolant are zero, there will also be other infinite eigenvalues. We propose tools for detecting when the leading coefficients of $p(z)$ are zero, and describe a procedure to deflate all of the infinite eigenvalues from the reduced matrix pair $(\mathbf{H},\mathbf{B})$, while still maintaining the tridiagonal plus rank-one structure of the resulting standard eigenvalue problem. Since fast $QR$ algorithms exist for such struc...