Bounding Zolotarev Numbers Using Faber Rational Functions

Abstract

By closely following a construction by Ganelius, we use Faber rational functions to derive tight explicit bounds on Zolotarev numbers. We use our results to bound the singular values of matrices, including complex-valued Cauchy matrices and Vandermonde matrices with nodes inside the unit disk. We construct Faber rational functions using doubly connected conformal maps and use their zeros and poles to supply shift parameters in the alternating direction implicit method.

Appendix A: Ensuring \({\mathbf {n}}\) is Large Enough for a Valid Bound

For the bound in (1.4) to be effective, n must be taken large enough so that both $h^n > C_n$, and the lower bound in (4.8) must also be greater than 0, or

$$\begin{aligned} \frac{1-h^{-2n}}{M_n(E,F)} - \frac{M_n(F,E)}{1-h^{-2n}}h^{-n} - \frac{1}{h^n-C_n} \ge 0 \end{aligned}$$

(A.1)

so that the denominator in (1.4) is positive. This latter condition subsumes the first, and so a minimal effective value of n may be given as follows.

Lemma A.1

For any integer $n>\log (x_1)/\log (h)$, we have that the expression in (4.8) is greater than 0, where $x_1$ is the largest positive real root of the seventh-degree polynomial

$$\begin{aligned}&x^7 - (4ef+8e+4f+9)x^6 + (8e^2f+4e^2+20ef+12e-4f^2-3)x^5\\&+(8e^3+28e^2+16ef^2+32ef+56e+20f^2+40f+43)x^4\\&+ (16e^2f+20e^2+32ef+40e+8f^3+28f^2+56f+43)x^3\\&+ (-4e^2+8ef^2+20ef+4f^2+12f-3)x^2\\&-(4ef+4e+8f+9)x+1, \end{aligned}$$

where we have abbreviated $\mathrm{Rot}(E)$ and $\mathrm{Rot}(F)$ as e and f, respectively.

In the case where E and F are convex, the polynomial above reduces to

$$\begin{aligned} x^7 -25x^6+37x^5+243x^4+243x^3+37x^2-25x+1, \end{aligned}$$

(A.2)

whose rightmost root satisfies $29.9< x_1 < 29.901$.