Zolotarev Iterations for the Matrix Square Root
Provides a new iterative method for the matrix square root problem, particularly beneficial for matrices with ill-conditioned eigenvalue distributions.
This work introduces a family of iterations for computing the principal square root of a matrix using Zolotarev's rational minimax approximants, achieving faster convergence for matrices with eigenvalues of widely varying magnitudes. Numerical experiments show improved performance over existing methods.
We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these rational functions obey a recursion, allowing one to iteratively generate optimal rational approximants of $\sqrt{z}$ of high degree using compositions and products of low-degree rational functions. The corresponding iterations for the matrix square root converge to $A^{1/2}$ for any input matrix $A$ having no nonpositive real eigenvalues. In special limiting cases, these iterations reduce to known iterations for the matrix square root: the lowest-order version is an optimally scaled Newton iteration, and for certain parameter choices, the principal family of Padé iterations is recovered. Theoretical results and numerical experiments indicate that the iterations perform especially well on matrices having eigenvalues with widely varying magnitudes.