We consider an inverse eigenvalue problem for constructing an $n\times n$ Jacobi matrix $J_n$ under the circumstance that its all eigenvalues, except for one and a part of the matrix $J_n$ are given. To be precise, the known partial data of $J_n$ means either its leading principal submatrix $J_{[(n+1)/2]}$ when $n$ is odd, or the submatrix $J_{[(n+1)/2]}$ together with the $[(n+1)/2]\times (n/2+1)$ codiagonal element when $n$ is even. The necessary and sufficient conditions for the solvability of the problem is derived, also the numerical algorithm and a numerical example are provided.

中文翻译：

---

具有缺失特征值的 Jacobi 矩阵的逆特征值问题

我们考虑一个反特征值问题来构造一个$n\times n$雅可比矩阵${Ĵ}_n$在它的所有特征值，除了一个和一部分矩阵的情况下${Ĵ}_n$给出。准确地说，已知的部分数据${Ĵ}_n$表示其领先的主子矩阵${Ĵ}_{[(n+1)/2]}$什么时候$n$是奇数，或子矩阵${Ĵ}_{[(n+1)/2]}$连同$[(n+1)/2]\times (n/2+1)$对角元素当$n$甚至。推导了该问题可解性的充要条件，给出了数值算法和数值算例。