Mathematical Sciences: Numerical Solution of Spectral Problems for Efficiently Structured Hermitian Matrices

构造了一类Hermitian矩阵族H 如果它包含每阶n的成员， 指定任意n x n所需的独立参数 H的成员是O（n）。 Hermitian Toeplitz矩阵与真实的 Hankel矩阵，其中第i行和第j行中的元素 列分别仅依赖于i-j或i+j， 结构有效。 矩阵的形式A+B，其中A是一个 B是真实的Hankel矩阵， 也有效地构造，作为一个厄米特矩阵， 给定位移秩，即厄米特矩阵，它可以是 写为上和下的固定数量的乘积的和， 下三角Toeplitz矩阵 有两个有趣的 有效结构矩阵的谱问题：（a） 普通特征值问题;即，找到特征值， 给定有效结构矩阵的特征向量， 利用简单结构的特定优点的方法， 和（B）逆特征值问题;在a中找到矩阵 给定有效结构的家庭， 特征值是一组给定的真实的数字。 的目标 拟议的研究将考虑这两个问题， 都处于相对原始的发展阶段。

A family H of Hermitian matrices is efficiently structured if it contains members of every order n and the number of independent parameters required to specify an arbitrary n x n member of H is O(n). Hermitian Toeplitz matrices and real Hankel matrices, for which the element in the i-th row and j-th column depends only on i-j or i+j, respectively, are efficiently structured. Matrices of the form A+B, where A is a Hermitian Toeplitz matrix and B is a real Hankel matrix, are also efficiently structured, as are Hermitian matrices of a given displacement rank, i.e. Hermitian matrices which can be written as the sum of a fixed number of products of upper and lower triangular Toeplitz matrices. There are two interesting spectral problems for efficiently structured matrices: (a) the ordinary eigenvalue problem; i.e., finding the eigenvalues and eigenvectors of a given efficiently structured matrix by methods which take specific advantage of the simple structure, and (b) the inverse eigenvalue problem; finding a matrix in a given efficiently structured family which has as its eigenvalues a given set of real numbers. The objectives of the proposed research are to consider these two problems, which are both in relatively primitive stages of development.