This is the sequel to the papers by V. B. Khazanov and V. N. Kublanovskaya 'Spectral problems for matrix pencils. Methods and algorithms. I and IT.A review of methods and algorithms for the solution of spectral problems for singular matrix pencils with linear and nonlinear (polynomial) dependence on the spectral parameter is presented. The methods are constructed using new principles which differ from those presented in Parts I and II of the paper. Here, we apply, as a rule, equivalent transformations with very simple unimodular matrices constructed using elementary plane rotation and reflection matrices. This property will be exploited when selecting a permutation matrix in the ZW decomposition algorithm which ensures the validity of inequalities (1.3).

The paper describes numerical solution of spectral problems for general matrices both regular and singular with rational dependence of the spectral parameter.We suggest algorithms which enable us to compute singular points (zeroes and poles) and construct a minimal basis of the null space consisting of polynomial solutions of rational matrices by solving spectral problems for polynomial matrices. These algorithms are based on factorization of a rational matrix into the product of two matrices one of which is a polynomial matrix while the other matrix is a rational one inverse to the regular polynomial matrix. We also suggest three algorithms for computing this factorization. In the first two algorithms the polynomial and rational matrices can possess reducible singular points. In these algorithms the rational matrices are diagonal (scalar, in particular). The third algorithm constructs the so-called minimal factorization which is irreducible. The ultimate spectrum of the polynomial matrix, i.e. the spectrum of the first factor of the factorization coincides with finite zero-type singular points of the original rational matrix. The ultimate spectrum of the regular polynomial matrix whose inverse is the second factor of the minimal factorization coincides with finite poles of the original rational matrix.We also present an algorithm for constructing a minimal basis made up of polynomial solutions of polynomial and rational matrices.All the algorithms suggested are based on transformations with simplest unimodular matrices constructed by using elementary orthogonal matrices of plane rotations and reflections.This paper is a logical sequel to the three published papers under the common title 'Spectral Problems for Matrix Pencils. Methods and Algorithms: I, II, IIP.The paper treats the solution of spectral problems for matrices whose entries are rational functions of the spectral parameter (of rational matrices) and logically continues the three previous papers under the common title 'Spectral Problems for Matrix Pencils. Methods and Algorithms. I, II and ΙΙΓ. Parts I and II are written by V. B. Khazanov and V. N. Kublanovskaya, and Part III is written by V. A. Belyi, V. B. Khazanov, and V. N. Kublanovskaya [3,8,9].In this paper the solution of spectral problems for rational matrices is reduced to the solution of similar spectral problems for polynomial matrices by using certain factorizations of the rational matrix. Here, we do not consider methods and algorithms for solving spectral problems for polynomial matrices referring the reader to the above-mentioned papers [3,8,9]. The paper suggests factorization algorithms which do not require linearization of the original spectral problem. As a rule, we utilize equivalent transformations in terms of simplest unimodular matrices constructed by using elementary plane rotations or reflections. This paper is organized as follows. Section 1 contains the theoretical background underlying the construction of methods and algorithms suggested. Section 2 considers methods and alg...