The infinity norm bounds and characteristic polynomial for high order RK matrices

Abstract

This paper shows that $t_m \leq \|\mathbf{A}\|_\infty \leq \sqrt{t_m}$ holds, when $\mathbf{A} \in \mathbb{R}^{m \times m}$ is a Runge-Kutta matrix which nodes originating from the Gaussian quadrature that integrates polynomials of degree $2m-2$ exactly. It can be shown that this is also true for the Gauss-Lobatto quadrature. Additionally, the characteristic polynomial of $\mathbf{A}$, when the matrix is nonsingular, is $p_A(\lambda) = m!t^m + (m-1)!a_{m-1}t^{m-1} + \dots + a_0$, where the coefficients $a_i$ are the coefficients of the polynomial of nodes $\omega(t) = (t - t_1) \dots (t - t_m) = t^m + a_{m-1}t^{m-1} + \dots + a_0$.