Approximating the pth Root by Composite Rational Functions

Abstract

A landmark result from rational approximation theory states that x 1/p on [0, 1] can be approximated by a type-(n, n) rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating x 1/p by composite rational functions of the form. While this class of rational functions ceases to contain the minimax (best) approximant for p ≥ 3, we show that it achieves approximately pth-root exponential convergence with respect to the degree. Moreover, crucially, the convergence is doubly exponential with respect to the number of degrees of freedom, suggesting that composite rational functions are able to approximate x 1/p and related functions (such as |x| and the sector function) with exceptional efficiency.

Iterations for the Unitary Sign Decomposition and the Unitary Eigendecomposition

“…A related composition law for rational minimax approximants of √ z has been used to construct iterations for the matrix square root [4]. These iterations were generalized to the matrix pth root in [5] and used to derive approximation theoretic results in [6]. An even more recent advancement-a composition law for rational minimax approximants of sign(z) on subsets of the unit circle [7]-is what inspired the present paper.…”