  Para-Hermitian polynomial matrices obtained by matrix spectral factorization
lead to functions useful in control theory systems, basis functions in
numerical methods or multiscaling functions used in signal processing. We
introduce a fast algorithm for matrix spectral factorization based on Bauer$'$s
method. We convert Bauer$'$ method into a nonlinear matrix equation (NME). The
NME is solved by two different numerical algorithms (Fixed Point Iteration and
Newton$'$s Method) which produce approximate scalar or matrix factors, as well
as a symbolic algorithm which produces exact factors in closed form for some
low-order scalar or matrix polynomial matrices, respectively. Convergence rates
of the two numerical algorithms are investigated for a number of singular and
nonsingular scalar and matrix polynomials taken from different areas. In
particular, one of the singular examples leads to new orthogonal multiscaling
and multiwavelet filters. Since the NME can also be solved as a Generalized
Discrete Time Algebraic Riccati Equation (GDARE), numerical results using
built-in routines in Maple 17.0 and 6 Matlab versions are presented.
