Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient.

中文翻译：

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Cayley变换方法的扩展，用于参数化广义特征值逆问题

由于最近的研究表明，Cayley变换方法可以作为解决特征值反问题的有效迭代方法，因此在这项工作中，我们考虑将其扩展用于解决一类参数化的广义特征值反问题，并证明其局部二次收敛性。这种类型的反特征值问题，包括乘法和加法反特征值问题，出现在许多应用中。另外，我们考虑给定特征值是多个的情况。在这种情况下，我们描述了一个未确定的修正问题，并讨论了针对该修正问题的Cayley变换方法的扩展。最后，为了证明这些算法的有效性，我们提供了一些数值算例，表明所提出的方法是切实有效的。