William F. Trench

On the eigenvalue problem for Toeplitz band matrices

Abstract A formula is given for the characteristic polynomial of an n th order Toeplitz band matrix, with bandwidth k n , in terms of the zeros of a k th degree polynomial with coefficients independent of n . The complexity of the formula depends on the bandwidth k , and not on the order n . Also given is a formula for eigenvectors, in terms of the same zeros and k coefficients which can be obtained by solving a k × k homogeneous system.

Numerical Solution of the Inverse Eigenvalue Problem for Real Symmetric Toeplitz Matrices

H. J. Landau has recently given a nonconstructive proof of an existence theorem for the inverse eigenvalue problem for real symmetric Toeplitz (RST) matrices. This paper presents a procedure for the numerical solution of this problem. The procedure is based on an implementation of Newtons method that exploits Landaus theorem and other special spectral properties of RST matrices. With this version of Newtons method, together with the strategy proposed for applying it, the method appears to be globally convergent; however, this is not proved.

Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations

SummaryConditions are given for the nonlinear differential equation (1)Lny+f(t, y, ..., ...,y(n−1)=0to have solutions which exist on a given interval [t0, ∞)and behave in some sense like specified solutions of the linear equation (2)Lnz=0as t→∞.The global nature of these results is unusual as compared to most theorems of this kind, which guarantee the existence of solutions of (1)only for sufficiently large t. The main theorem requires no assumptions regarding oscillation or nonoscillation of solutions of (2).A second theorem is specifically applicable to the situation where (2)is disconjugate on [t0, ∞),and a corollary of the latter applies to the case where Lz=zn.

Inversion of Toeplitz band matrices

An algorithm for inverting Toeplitz matrices is simplified for Toeplitz band matrices. In some cases, the simplification yields formulas for the elements in the first row and column of the inverse, from which the remaining elements can be easily calcu- lated. Two examples are given. In any case, the simplification yields a recursive method for computing the first row and column of the inverse of an nth order Toe- plitz band matrix with O(n) operations, where 0(n ) are required with the more gen- eral algorithm.

Characterization and Properties of ( R , S )-Symmetric, ( R , S )-Skew Symmetric, and ( R , S )-Conjugate Matrices

Let

Interlacement of the even and odd spectra of real symmetric Toeplitz matrices

R\in \mathbb{C}^{m\times m}

Numerical solution of the eigenvalue problem for symmetric rationally generated Toeplitz matrices

and

Spectral evolution of a one-parameter extension of a real symmetric toeplitz matrix

S\in \mathbb{C}^{n\times n}

Explicit inversion formulas for Toeplitz band matrices

be nontrivial involutions; i.e.,

Asymptotic behavior of solutions of Poincaré recurrence systems

R=R^{-1}\ne\pm I_m