Rank-structured QR for Chebyshev rootfinding
Angelo Casulli, Leonardo Robol

TL;DR
This paper introduces a rank-structured QR algorithm for Chebyshev polynomial rootfinding that is efficient, parallelizable, and provides improved backward stability guarantees compared to traditional methods.
Contribution
It extends the QR iteration with an aggressive deflation strategy exploiting rank structure, achieving quadratic complexity and better backward stability for Chebyshev rootfinding.
Findings
Achieves quadratic complexity and linear storage requirements.
Guarantees small backward error up to an explicit amplification factor.
Demonstrates improved accuracy in numerical tests.
Abstract
We consider the computation of roots of polynomials expressed in the Chebyshev basis. We extend the QR iteration presented in [Eidelman, Y., Gemignani, L., and Gohberg, I., Numer. Algorithms, 47.3 (2008): pp. 253-273] introducing an aggressive early deflation strategy, and showing that the rank-structure allows to parallelize the algorithm avoiding data dependencies which would be present in the unstructured QR. We exploit the particular structure of the colleague linearization to achieve quadratic complexity and linear storage requirements. The (unbalanced) QR iteration used for Chebyshev rootfinding does not guarantee backward stability on the polynomial coefficients, unless the vector of coefficients satisfy , an hypothesis which is almost never verified for polynomials approximating smooth functions. Even though the presented method is mathematically equivalent to…
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