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 paper is the logical sequel to Parts I and II of the paper by V. B. Khazanov and V. N. Kublanovskaya 'Spectral problems for matrix pencils. Methods and algorithms' [10, 11]. The paper treats methods and algorithms for solving spectral problems for generaltype polynomial pencils applicable (in contrast with methods and algorithms of Part I) both to linear pencils and to polynomial pencils of degree s > 1. Here, we provide a detailed description of methods and principles underlying their construction, first, because the intensive study of spectral problems for general-type polynomial pencils has started only in recent years and, second, along with well-known results we supply here also some new results presented in 1987 at the USSR Conference 'Topical Problems of Calculus and Applied Mathematics' in Novosibirsk and at the International Symposium on Numerical Analysis (ISNA) in Prague. The methods suggested are based on the idea of using different pencil decompositions which enable us either to separate the regular and the singular parts of the pencil (the ZW and &W decompositions) or to deflate some of the pencil spectral characteristics (the DQ decomposition and different deflation methods). In Chapters 1 and 2, algorithms performing the ZW and MY decompositions of the pencil based on using very simple unimodular matrices are described. Such transformation matrices make it possible to pass from the original pencil to a full column rank pencil whose finite scalar spectral characteristics and left vector spectral characteristics coincide with those of the original pencil, while their right vector spectral characteristics (eigenvectors and Jordan chains) are in one-to-one correspondence. The ZW decomposition somewhat extends the normalized decomposition to polynomial matrix pencils. The knowledge of the resulting unimodular matrix performing the ZW decomposition makes it possible to compute vectors of Jordan chains corresponding to the zero pencil eigenvalue and to construct a basis of the null 20 V. A. Belyi et al. space consisting of polynomial solutions of the pencil whose order can be reduced up to the minimal one by deflating its zero at the infinity if the pencil forming this basis possesses it. In contrast with the Z W decomposition, the AVK decomposition does not raise the degree of the pencil under transformation due to the special choice of unimodular matrices. It allows the extraction of a full column rank pencil which has no zero at the infinity point, the construction of the minimal basis of the null space of the original pencil, i.e. of a fundamental solution row, and also the computation of eigenvalues of the regular part of the pencil and of the corresponding eigenvectors and Jordan chains. The DQ decomposition outlined in Chapter 3 is performed by using only orthogonal transformation matrices (plane rotation or reflection matrices). This makes it possible to deflate from the pencil its zero degree solutions. The application of the DQ decomposition to a recursively constructed sequence of pencils permits the construction of polynomial solutions of the original pencil. Two algorithms for the transformation of polynomial solutions into the minimal basis of the pencil null space are considered. Algorithms to deflate from the pencil its computed polynomial solutions are considered in Chapter 4. The deflation of singular spectral characteristics of the pencil removes ambiguity when computing regular spectral characteristics and ensures a better numerical stability of their computation. We suggest to construct algorithms for computing a fundamental solution row of the original pencil on the basis of an algorithm for computing the least degree polynomial solutions and of a deflation algorithm. An algorithm for the reduction of the pencil to a full column rank pencil using a regular Α-matrix is also suggested which permits the deflation of the right null space of the pencil which is not to be explicitly determined. The spectrum of the pencil of full column rank thus constructed contains the spectrum of the original pencil. Chapter 5 treats algorithms for computing eigenvectors and Jordan chains of full column rank pencils. Their construction is based on principles which differ from those of the algorithm contained in Section 1.4. Methods based on the original pencil linearization are not considered in this paper. The description of such methods can be found, for instance, in [1, 3, 4]. Throughout the paper we make use of the theoretical background and notation given in Parts I and II of our paper [10, 11]. Some theoretical background is recalled just when describing the methods as we believe it to be convenient. 1. THE ZW DECOMPOSITION AND ITS APPLICATION TO SOLVING SPECTRAL PROBLEM 1.1. A ZW decomposition algorithm [2] Theorem 1.1. Let Ό(λ) = C0 + λ€ΐ + —h λ*€8 be a polynomial mxn matrix pencil of rank ρ *ξ min (m, n) and of degree s > 1. Then there exist a permutation matrix Θ and a unimodular matrix W(l) such that the decomposition of the form ®0(λ)ΐν(λ) = Ζ(λ) = [Z0(A) Ο] (1.1) is valid. Here, Ο is a m χ (η — ρ) zero matrix, Ζ0(λ) is a polynomial m χ ρ matrix pencil of full column rank of the form Spectral problems for matrix pencils 21