An inverse problem for Toeplitz matrices
Let \(T_ m\) be an \(m\times m\) positive definite symmetric Toeplitz matrix, and let \(T_ n\) be the \(n\times n\) matrix consisting of the first n rows and columns of \(T_ m(1\leq n\leq m)\). The central mass sequence of \(T_ m\) is \(\{\rho_ 1,...,\rho_ m\}\) defined by \(\rho_ n=\sup\{\rho:T_ n-\rho\Pi_ n\) is positive definit\(e\}\) where \(\Pi_ n\) is the \(n\times n\) matrix consisting of all 1's. The problem referred to in the title is the problem of computing \(T_ m\) from a knowledge of its central mass sequence, and it has applications in the design of discrete transmission lines. This problem was solved by \textit{R. E. Caflisch} [ibid. 38, 207-225 (1981; Zbl 0468.35080)]. In the present paper the author gives a simpler solution based on a Cholesky factorization of the inverse of \(T_ m\) (referred to by the author as the Levinson-Durbin algorithm).
positive definite symmetric Toeplitz matrix
scientific article; zbMATH DE number 3873513