Iterations for the Unitary Sign Decomposition and the Unitary Eigendecomposition

11/25/2020 ∙ by Evan S. Gawlik, et al. ∙ 0

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(A^2)^-1/2 times a matrix N = (A^2)^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/√(z^2) 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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.