对称矩阵特征值反问题的最佳逼近解的一种数值解法

A Numerical Algorithm for the Optimal Approximation Solution to Inverse Eigenvalue Problem for Symmetric Matrices

利用复合最速下降法,给出了对称矩阵特征值反问题AX=XΛ有解和无解两种情况下最佳逼近解的通用数值算法,对任意给定的初始矩阵A0,经过有限步迭代可以得到对称矩阵特征值反问题的最佳逼近解,并分别给出有解和无解两种情况下的数值实例,证明了此算法的可行性.另外,结合投影算法,可以用此算法来求解其它凸约束下矩阵特征值反问题的最佳逼近解,从而扩大了此算法的求解范围.

By applying the hybrid steepest descent method,this paper gives a general numerical algorithm to find the optimal approximation solution to inverse eigenvalue problem,AX=XΛ,for symmetric matrices.For any given initial matrix,the optimal approximation can be derived by finite iteration steps.Some numerical examples are provided to illustrate the feasibility of the algorithm.Moreover,combined with projection algorithm,the numerical algorithm can also be used to calculate the optimal approximation solution to other convex constrained inverse eigenvalue problem,thus extending the applicable scope of this algorithm.