On the construction of real non-selfadjoint tridiagonal matrices with prescribed three spectra

Non-selfadjoint tridiagonal matrices play a role in the discretization and truncation of the Schrodinger equation in some extensions of quantum mechanics, a research field particularly active in the last two decades. In this article, we consider an inverse eigenvalue problem that consists of the reconstruction of such a real non-selfadjoint matrix from its prescribed eigenvalues and those of two complementary principal submatrices. Necessary and sufficient conditions under which the problem has a solution are presented, and uniqueness is discussed. The reconstruction is performed by using a modified unsymmetric Lanczos algorithm, designed to solve the proposed inverse eigenvalue problem. Some illustrative numerical examples are given to test the efficiency and feasibility of our reconstruction algorithm. Key words. inverse eigenvalue problem, non-selfadjoint tridiagonal matrix, modified unsymmetric Lanczos algorithm, spectral data AMS subject classifications. 65F18, 65F15, 15A18, 15A29