论文标题
高级RK矩阵的无限规范界限和特征多项式
The infinity norm bounds and characteristic polynomial for high order RK matrices
论文作者
论文摘要
本文表明，$ t_m \ leq \ | \ mathbf {a} \ | _ \ infty \ leq \ sqrt {t_m} $保持，完全集成了$ 2M-2 $的多项式。可以证明，高斯 - 洛巴托正交也是如此。此外，当矩阵非发音时，$ \ mathbf {a} $的特征多项式为$ p_a（λ）= m！ （t -t_1）\ dots（t -t_m）= t^m + a_ {m -1} t^{m -1} + \ dots + a_0 $。
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 $p_A(λ) = m!t^m + (m-1)!a_{m-1}t^{m-1} + \dots + a_0$, where the coefficients $a_i$ are the coefficients of the polynomial of nodes $ω(t) = (t - t_1) \dots (t - t_m) = t^m + a_{m-1}t^{m-1} + \dots + a_0$.
