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

Abstract

This paper shows that $t_m\le \Vert \mathbf{A}{\Vert }_{\infty }\le \sqrt{t_m}$ 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 $p_A(\lambda )=m!t^m+(m-1)!a_{m-1}{t}^{m-1}+\cdots +a_0$, where the coefficients $a_i$ are the coefficients of the polynomial of nodes $\omega (t)=(t-t_1)\dots (t-t_m)=t^m+a_{m-1}{t}^{m-1}+\cdots +a_0$.