高阶RK矩阵的无穷范数界和特征多项式
The infinity norm bounds and characteristic polynomial for high order RK matrices
摘要：
本文证明了当$\mathbf{A} \in \mathbb{R}^{m \times m}$是一个Runge-Kutta矩阵时，$t_m \leq \|\mathbf{A}\|_\infty \leq \sqrt{t_m}$成立，其中该矩阵的节点来自于对$2m-2$次多项式精确积分的高斯求积.可以证明，对于高斯-洛巴托求积也是如此。另外，当矩阵非奇异时，$\mathbf{A}$的特征多项式是$p_A(λ) = m!t^m + (m-1)!a_{m-1}t^{m-1} + \dots + a_0$，其中系数$a_i$是节点$ω(t) = (t - t_1) \dots (t - t_m) = t^m + a_{m-1}t^{m-1} + \dots + a_0$的多项式的系数。
英文摘要：
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 $ω(t) = (t - t_1) \dots (t - t_m) = t^m + a_{m-1}t^{m-1} + \dots + a_0$.
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