On the inverse problem of constructing symmetric pentadiagonal toeplitz matrices from three largest eigenvalues

The inverse problem of constructing a symmetric Toeplitz matrix with
prescribed eigenvalues has been a challenge both theoretically and
computationally in the literature. It is now known in theory that symmetric
Toeplitz matrices can have arbitrary real spectra. This paper addresses a
similar problem--can the three largest eigenvalues of symmetric pentadiagonal
Toeplitz matrices be arbitrary? Given three real numbers ? ? ?, this paper
finds that the ratio ? = ?-?
?-? , including infinity if ? = ?, determines whether
there is a symmetric pentadiagonal Toeplitz matrix with ?, ? and ? as its three
largest eigenvalues. It is shown that such a matrix of size n × n does not exist
if n is even and ? is too large or if n is odd and ? is too close to 1. When such
a matrix does exist, a numerical method is proposed for the construction.