Minimal condition number for positive definite Hankel matrices using semidefinite programming (Q989049)

A Hankel matrix is a square matrix having constant anti-diagonal elements. For \(h=(h_1,h_2,\dots,h_{2n-1})^T\in{\mathbb R}^{2n-1}\), denote the Hankel matrix \[ H(h):=\begin{bmatrix} h_1 &h_2 &\cdots &h_{n-1} &h_n\\ h_2 &h_3 &\cdots &h_n &h_{n+1}\\ \vdots &\vdots &\ddots &\vdots &\vdots\\ h_{n} &h_{n+1} &\cdots &h_{2n-2} &h_{2n-1} \end{bmatrix}. \] The paper gives a semidefinite programming approach to compute a positive definite real Hankel matrix \(\widehat H_n=H(h)\) that minimizes the condition number \(\kappa(\widehat H_n):=\lambda_{\max}(\widehat H_n)/\lambda_{\min}(\widehat H_n)\). The approach is guaranteed to find such an \(\widehat H_n\) within any desired tolerance. Some semidefinite programming solvers are used to compute \(\widehat H_n\) up to \(n=100\).