An approach to solving multiparameter algebraic problems

V. N. Kublanovskaya

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XI, Tome 229 (1995), pp. 191-246

Résumé

An approach to solving the following multiparameter algebraic problems is suggested: (1) spectral problems for singular matrices polynomially dependent on $q\geqslant2$ spectral parameters, namely: the separation of the regular and singular parts of the spectrum, the computation of the discrete spectrum, and the construction of a basis that is free of a finite regular spectrum of the null-space of polynomial solutions of a multiparameter polynomial matrix; (2) the execution of certain operations over scalar and matrix multiparameter polynomials, including the computation of the $GCD$ of a sequence of polynomials, the division of polynomials by their common divisor, and the computation of relative factorizations of polynomials; (3) the solution of systems of linear algebraic equations with multiparameter polynomial matrices and the construction of inverse and pseudoinverse matrices. This approach is based on the so-called $\Delta W-q$ factorizations of polynomial $q$-parameter matrices and extends the method for solving problems for one- and two-parameter polynomial matrices considered in [1?3] to an arbitrary $q\geqslant2$. Bibliography: 12 titles.