论文标题
矩形多参数特征值问题的数值方法，并应用于找到最佳ARMA和LTI模型
Numerical methods for rectangular multiparameter eigenvalue problems, with applications to finding optimal ARMA and LTI models
论文作者
论文摘要
标准多参数特征值问题（MEP）是$ k \ ge 2 $ linear $ k $ -parameter Square矩阵铅笔的系统。最近，出现了一种新形式的多参数特征值问题：一种只有一种多元矩形矩阵铅笔的矩形MEP（RMEP），我们正在寻找铅笔等级不完整的参数的组合。应用包括找到最佳最小二乘自动移动平均值（ARMA）模型以及最佳最小二乘实现自主线性时间流动（LTI）动力学系统的实现。对于线性和多项式RMEP，我们给出了解决方案的数量，并显示如何通过转换为标准MEP来数值解决这些问题。对于转换，我们为具有特定单体结构的二次多元基质多项式提供了新的线性化，并考虑了矩形和方形多元基质多项式的混合系统。这种数值方法在计算上似乎比Block Macaulay方法更具吸引力，这是唯一用于多项式RMEPS的其他可用数值方法。
Standard multiparameter eigenvalue problems (MEPs) are systems of $k\ge 2$ linear $k$-parameter square matrix pencils. Recently, a new form of multiparameter eigenvalue problems has emerged: a rectangular MEP (RMEP) with only one multivariate rectangular matrix pencil, where we are looking for combinations of the parameters for which the rank of the pencil is not full. Applications include finding the optimal least squares autoregressive moving average (ARMA) model and the optimal least squares realization of autonomous linear time-invariant (LTI) dynamical system. For linear and polynomial RMEPs, we give the number of solutions and show how these problems can be solved numerically by a transformation into a standard MEP. For the transformation we provide new linearizations for quadratic multivariate matrix polynomials with a specific structure of monomials and consider mixed systems of rectangular and square multivariate matrix polynomials. This numerical approach seems computationally considerably more attractive than the block Macaulay method, the only other currently available numerical method for polynomial RMEPs.
