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

03/08/2022

∙

by   Gayatri Čaklović, et al.

∙

0

∙

share

This paper shows that t_m ≤𝐀_∞≤√(t_m) holds, when 𝐀∈ℝ^m × 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 𝐀, when the matrix is nonsingular, is p_A(λ) = m!t^m + (m-1)!a_m-1t^m-1 + … + a_0, where the coefficients a_i are the coefficients of the polynomial of nodes ω(t) = (t - t_1) … (t - t_m) = t^m + a_m-1t^m-1 + … + a_0.