论文标题
非线性特征值问题的导数插值子空间框架
Derivative Interpolating Subspace Frameworks for Nonlinear Eigenvalue Problems
论文作者
论文摘要
我们首先考虑近似于最接近规定目标的有理矩阵值函数的一些特征值的问题。假定有理矩阵值值函数的适当合理部分在传输函数表格$ h（s）= c（si-a）^{ - 1} b $中表示，其中中间因子很大，而$ c $的行数和$ b $的列数和$ b $相等和小。我们提出了一个子空间框架，该框架对$ h（\ cdot）$的状态空间表示进行双面或单面投影，该框架通常用于模型降低并引起降低的传输功能。在每次迭代中，投影子空间都会扩展，以在最接近目标的降低传递函数的特征值下达到Hermite插值条件，进而导致新的降低传递函数。从理论上讲，我们证明，当降低的传递函数的特征值序列收敛到整个问题的特征值时，它至少以二次速率收敛。在第二部分中，我们扩展了提出的框架，以找到一般广场的大尺度非线性meromormormorormormormormorix-balued函数$ t（\ cdot）$的特征值，我们在其中利用表示$ \ nathcal {r}（r}（r}（r}（r}（s）= c（s）= c（s）a（s）a（s）a（s）a（s）^{s） $ t（\ cdot）$。数值实验表明，所提出的框架可靠地定位最接近目标点的特征值，并且对于运行时，它与非线性特征值问题的已建立方法具有竞争力。
We first consider the problem of approximating a few eigenvalues of a rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational part of the rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided or one-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a reduced transfer function. At every iteration, the projection subspaces are expanded to attain Hermite interpolation conditions at the eigenvalues of the reduced transfer function closest to the target, which in turn leads to a new reduced transfer function. We prove in theory that, when a sequence of eigenvalues of the reduced transfer functions converges to an eigenvalue of the full problem, it converges at least at a quadratic rate. In the second part, we extend the proposed framework to locate the eigenvalues of a general square large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we exploit a representation $\mathcal{R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined in terms of the block components of $T(\cdot)$. The numerical experiments illustrate that the proposed framework is reliable in locating a few eigenvalues closest to the target point, and that, with respect to runtime, it is competitive to established methods for nonlinear eigenvalue problems.
