Numerical Algorithms for the Polynomial Eigenvalue Problem

多项式特征值问题（polynomials Eigenvalue Problem，PEP）是求一个方阵的特征值和特征向量的问题，其中特征值是λ的一个值，且λ的值是矩阵奇异的。对于多项式次数为1的情况，有很好的数值方法。二次或更高次的矩阵多项式通常出现在结构力学、声学系统和电路模拟等领域。极端设计（如微机电（MEMS）设备和超大型喷气机）的趋势意味着这些特征问题往往条件不佳（因此难以精确求解），同时也具有代数结构，应该在数值方法中加以利用。作为一个具体的例子，在柏林工业大学的一个项目中，对欧洲高速列车的声音和振动水平进行建模，发现标准有限元软件包在计算解中没有提供正确的图形，直到在该提案中使用的线性代数技术在基础二次方程中发挥作用。（度2）特征值问题（参见SIAM News，Nov. 2004的封面文章）。解决PEP的标准方法是将问题转化为更大维度的度1特征值问题--线性化过程。这几乎总是使用具有公知的伴随矩阵形式的线性化来完成，但这只是许多可能的线性化之一。在最近的工作中，三个向量空间的线性化已被研究，推广的同伴形式，并提供了一个系统的方式产生广泛的线性化。这些空间使得有可能确定具有特定属性的线性化，例如最佳条件，最佳向后误差界和保留结构，例如对称性。我们最近的工作已经表明，这些新的线性化可以产生显着更好的质量比同伴form.This项目的目的是开发新的算法求解PEP线性化，有更好的精度和稳定性比目前在实践中使用的，并充分利用结构特性，如对称性和明确性。这项工作将涉及新的理论的发展，包括向后误差界的新空间的线性化，研究一个新的特征向量恢复公式，并研究双曲多项式和他们的明确的线性化。(The双曲多项式是对称的、具有真实的特征值的多项式的子集，并且它们在工程应用中是常见的，包括在过阻尼机械系统中。这项工作的一个重要成果是可以作为图书馆软件基础的算法，因为目前还没有解决PEP的标准图书馆软件。

The polynomial eigenvalue problem (PEP) is to find the eigenvalues and eigenvectors of a square matrix whose elements are polynomials in a variable lambda, where an eigenvalue is a value of lambda for which the matrix is singular.Excellent numerical methods exist for the case where the polynomial degree is 1. Matrix polynomials of degree 2 or higher arise commonly in areas such as structural mechanics, acoustic systems and electrical circuit simulation. The trend to towards extreme designs (such as in micro-electromechanical (MEMS) devices and superjumbo jets) means that these eigenproblems are often poorly conditioned (hence difficult to solve accurately) while also having algebraic structure that should be exploited in a numerical method. As a specific example, in a project at TU Berlin modelling the sound and vibration levels in European high-speed trains it was found that standard finite element packages provided no correct figures in the computed solutions until linear algebra techniques of the type to be used in this proposal were brought into play in the underlying quadratic (degree 2) eigenvalue problem (see the cover article in SIAM News, Nov. 2004).The standard way of solving the PEP is by converting the problem to a degree 1 eigenvalue problem of larger dimension---the process of linearization. This is almost invariably done using a linearization having the well known companion matrix form, but this is just one of many possible linearizations. In very recent work three vector spaces of linearizations have been studied that generalize the companion form and which provide a systematic way of generating a wide class of linearizations. These spaces make it possible to identify linearizations having specific properties such as optimal conditioning, optimal backward error bounds and preservation of structure such as symmetry. Our recent work has already shown that these new linearizations can produce numerical solutions of significantly better quality than the companion form.This project aims to develop new algorithms for solving the PEP by linearization that have substantially better accuracy and stability properties than those currently used in practice, and which take full advantage of structural properties such as symmetry and definiteness. The work will involve the development of new theory, including backward error bounds for the new spaces of linearizations, the study of a new eigenvector recovery formula, and the study of hyperbolic polynomials and their definite linearizations. (The hyperbolic polynomials are a subset of those that are symmetric and have real eigenvalues, and they are common in engineering applications, including in overdamped mechanical sytems.) An important output of the work will be algorithms that can serve as the basis for library software, since at present there is no standard library software for solving the PEP.