A new approach to designing direct solution methods for algebraic problems with polynomial and rational occurrences of parameters is suggested. This approach is based on factorizations of polynomial matrices depending on one or several parameters.
We consider different factorizations of polynomial matrices, including the so-called rank factorizations, and describe their applications to developing direct methods for solving the following problems.
Х Spectral problems for polynomial matrices (computing bases of null-spaces with various spectral properties, separating the regular and singular parts of the spectrum, decomposing matrices into factors possessing prescribed spectral properties, computing invariant and complete polynomials and exhausting them from the matrix spectrum, solving certain inverse eigenproblems).
Х Problems for scalar and matrix polynomials (finding the GCD and LCM, and computing partial relative factorizations of polynomials in several variables).
Х Solution of linear algebraic equations with polynomial matrices (computing adjoint matrices, inverses, and pseudo inverses).
Х Solution of nonlinear algebraic equations (reduction to spectral problems for polynomial matrices).
Х Problems for rational matrices (computing bases of null-spaces, computing irreducible factorizations, and separating zeros and poles).
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4. Kublanovskaya V.N., Khazanov V.B. Relative factorization of polynomials in several variables // Z. vychisl. matem. matem. phys. 1996. V. 36, N 3. P. 6-11.
5. Kublanovskaya V.N., Khazanov V.B. Numerical methods for solving parametric problems of algebra. Part 1. One-parameter problems. St. Petersburg, УNaukaФ. 2004.
Note. Abstracts are published in author's edition
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