Abstract
We exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a (2k+1)-diagonal Toeplitz matrix with elements in the complex field, either for sequential or for parallel computations. A fast algorithm, to achieve one step of Newton's method, is shown to be suitable to compute the eigenvalues of such a matrix. Bounds to the eigenvalues and necessary and sufficient conditions for positive definiteness, which are easy to check, are given either for matrices with scalar elements or for matrices with blocks. In the case in which the blocks are themselves band Toeplitz matrices such conditions assume a very simple form.
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