Fast and reliable algorithms for computing the zeros of Althammer polynomials

In this manuscript, we propose a stable algorithm for computing the zeros of Althammer polynomials. These polynomials are orthogonal with respect to a Sobolev inner product, and are even if their degree is even, odd otherwise. Furthermore, their zeros are real, distinct, and located inside the interval $(-1,1)$. The Althammer polynomial $p_n\left(x\right)$ of degree n satisfies a long recurrence relation, whose coefficients can be arranged into a Hessenberg matrix of order n, with eigenvalues equal to the zeros of the considered polynomial.

Unfortunately, the eigenvalues of this Hessenberg matrix are very ill–conditioned, and standard balancing procedures do not improve their condition numbers. Here, we introduce a novel algorithm for computing the zeros of $p_n\left(x\right)$, which first transforms the Hessenberg matrix into a similar symmetric tridiagonal one, i.e., a matrix whose eigenvalues are perfectly conditioned, and then computes the zeros of $p_n\left(x\right)$ as the eigenvalues of the latter tridiagonal matrix. Moreover, we propose a second algorithm, faster but less accurate than the former one, which computes the zeros of $p_n\left(x\right)$ as the eigenvalues of a truncated Hessenberg matrix, obtained by properly neglecting some diagonals in the upper part of the original matrix. The computational complexity of the proposed algorithms are, respectively, $O\left(\frac{n^3}{6}\right)$, and $O\left({\ell }^{2}n\right)$, with $\ell \ll n$ in general.