Polynomial Matrix—Evaluation, Interpolation, Inversion

Abstract

In many problems in systems theory, (Barnett [1971], Rosenbrock and Storey [1970]) we encounter matrices (called “Polynomial matrices”) whose elements are polynomials over the field of rationals or over the ring of integers, in an indeterminate x or several indeterminates x, y, z,…. The inversion and manipulation of such matrices become involved due to the following reasons:

• (i) The conventional floating point arithmetic is not suitable for this purpose, since we cannot neglect very small terms occurring as the coefficients (elements from a field or a ring) of a polynomial.

• (ii) The integer arithmetic demands excessively long precision operands and the computation becomes slow.

• (iii) The coefficients of the polynomial grow during the inversion (or other transformation operations) since Gaussian type reduction does not allow one to use the division operation when the coefficients of the polynomial matrix elements are integers.