Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure
The authors develop a numerical isospectral method including convergence results and error estimates for the solution of the following generalized inverse eigenvalue problem: Given a set of complex numbers \(\lambda = \{\lambda_j\} \in {\mathbb{C}, \, j = 1(1)n}\), a subset of index subscript pairs \({(i_{\nu}, j_{\nu})}_{\nu = 1}^l, \, 1 \leqq i_{\nu}, j_{\nu} \leqq n\), and a subset of prescribed values \(a_\nu \in \mathbb{R}, \, a_\nu \geqq 0, \, \nu = 1(1)l\). Find a nonnegative matrix \(X = \{x_{i,j}\} \in {\mathbb{R}}^{n \times n}\) such that \(\lambda (X) = \lambda\) and \(x_{i_\nu, j_\nu} = a_{\nu}, \, \nu = 1(1)l\). A corresponding problem is treated for symmetric nonnegative matrices. The described problems reduce to the common nonnegative inverse eigenvalue problems if the subset of prescribed entries \(\{a_\nu\}\) is empty. The here investigated problems have multiple solutions as the most inverse eigenvalue problems. Roughly speaking, the isospectral method for the reconstruction of the matrix \(X\) consists of the finding the intersection of a set determinated by the spectral constraints \(\{\lambda_j\}\) and a set depending on the structural constraints \(\{a_\nu\}\). The intersection is formulated as the distance between the two sets which is minimized using the steepest descent method. The first set is built of the isospectral surface \(V \Lambda V^{-1}, \, V \in {\mathbb{R}}^{n \times n}\), where \(V\) is nonsingular and \(\Lambda \in {\mathbb{R}}^{n \times n}\) denotes a diagonal block matrix (with blocks of order 1 or 2) with the given spectrum. The second set is constructed by the Hadamard product \(A \circ A\) of the matrix \(A\) built by the prescribed entries. The ill-posedness of nonnegative inverse eigenvalue problems (NIEP) caused by nonunique solutions is transferred to the dependence of solutions of ordinary differential equations on initial values, i.e., different initial values correspond with different solutions of the NIEP. The results are validated by numerical tests, particulary a nonnegative symmetric tridiagonal matrix is reconstructed from the prescribed spectrum and with positive codiagonal entries. For important applications the authors refer to other publications.
nonnegative symmetric inverse eigenvalue problem
scientific article; zbMATH DE number 5914067
scientific article