Application of the Faber polynomials in calculation of eigenvalues
Let \(A\) be a complex matrix of order \(n\times n\) with eigenvalues \(\lambda_ 1,\dots,\lambda_ n\); \(K\) be a bounded continuum on the plane \(\overline{\mathbb{C}}\). Let the domain \(D= \overline{\mathbb{C}}\backslash K\) be simply connected and let there be known that \(\lambda_ 1\in D\), \(\lambda_ j\in K\) \((j=2,\dots,n)\). The construction of such a continuum \(K\) for a given matrix \(A\) can be done, in number of cases, by means of the well-known technique of the localization of eigenvalues. If \(K\) is a disk, then, in order to calculate \(\lambda_ 1\) and the corresponding eigenvector \(\bar g_ 1\), \(A\bar g_ 1= \lambda_ 1\bar g_ 1\), one can use the power method. But if \(K\) is a segment, then \(\lambda_ 1\) and \(\bar g_ 1\) can be calculated by the Chebyshev method. In this article we show that if there exists a technique for the calculation of the Faber polynomials for the continuum \(K\), then the mentioned problem can be solved by Faber's integrative method. This method generalizes the classical power method and the Chebyshev method, having higher rate of convergence in certain cases.
scientific article; zbMATH DE number 640170