Spectral Chebyshev Collocation for the Poisson and Biharmonic Equations
SIAM Journal on Scientific Computing
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An algorithm based on the Ehrlich--Aberth iteration is presented for the computation of the zeros of $p(\lambda)=\det(T-\lambda I)$, where $T$ is a real irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of $p(\lambda)/p'(\lambda)=-1/\mathrm{trace}(T-\lambda I)^{-1}$, which is done by exploiting the QR factorization of $T-\lambda I$ and the semiseparable structure of $(T-\lambda I)^{-1}$. The choice of initial approximations relies on a divide-and-conquer strategy, and some results motivating this strategy are given. Guaranteed a posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the effectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine \texttt{dhseqr} show that our algorithm is faster for large dimensions.