This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu (SIAM J Numer Anal 33:2417–2430, 1996). We provide a geometric Gauss–Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is established under some assumptions. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the geometric Gauss–Newton equation. Finally, some numerical tests, including an application in the inverse Sturm–Liouville problem, are reported to illustrate the efficiency of the proposed method.

中文翻译：

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最小二乘特征值反问题的几何高斯-牛顿方法

本文涉及从最小二乘意义上的指定部分特征值重建线性参数化实对称矩阵的最小二乘逆特征值问题，该问题最初由 Chen 和 Chu 提出（SIAM J Numer Anal 33:2417–2430， 1996）。我们提供了一种几何高斯-牛顿方法来解决最小二乘特征值逆问题。所提出方法的全局和局部收敛分析是在一些假设下建立的。此外，提出了一种具有有效预处理器的预处理共轭梯度方法来求解几何高斯-牛顿方程。最后，报告了一些数值测试，包括在逆 Sturm-Liouville 问题中的应用，以说明所提出方法的效率。