2020/11/25 by Evan S. Gawlik, Gawlik, Evan S.
Computer Science · Mathematics · #15A23 #30E10 #41A20 #41A50 #65F15 #65F60 #Advanced Optimization Algorithms Research #Algebraic and Geometric Analysis #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2011.12449
openalex publication_date 2020/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct fast, structure-preserving iterations for computing the sign decomposition of a unitary matrix A with no eigenvalues equal to ± i. This decomposition factorizes A as the product of an involutory matrix S = sign(A) = A(A2)-1/2 times a matrix N = (A2)1/2 with spectrum contained in the open right half of the complex plane. Our iterations rely on a recently discovered formula for the best (in the minimax sense) unimodular rational approximant of the scalar function sign(z) = z/√(z2) on subsets of the unit circle. When A has eigenvalues near ± i, the iterations converge significantly faster than Padé iterations. Numerical evidence indicates that the iterations are backward stable, with backward errors often smaller than those obtained with direct methods. This contrasts with other iterations like the scaled Newton iteration, which suffers from numerical instabilities if A has eigenvalues near ± i. As an application, we use our iterations to construct a stable spectral divide-and-conquer algorithm for the unitary eigendecomposition.