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.

翻译：通过矩阵谱分解得到的准埃尔米特多项式矩阵可产生控制理论系统中的实用函数、数值方法中的基函数或信号处理中的多尺度函数。我们提出了一种基于Bauer方法的快速矩阵谱分解算法。将Bauer方法转化为非线性矩阵方程(NME)，分别采用两种数值算法（不动点迭代法和牛顿法）求解该方程以得到近似的标量或矩阵因子，同时提出一种符号算法可用于求解某些低阶标量或矩阵多项式矩阵的闭式精确因子。针对来自不同领域的多个奇异和非奇异标量及矩阵多项式，研究了两种数值算法的收敛速度。特别地，其中一个奇异例子导出了新的正交多尺度及多小波滤波器。由于该非线性矩阵方程亦可作为广义离散时间代数Riccati方程(GDARE)求解，本文还给出了基于Maple 17.0和6个Matlab版本内置程序的数值结果。