2018/05/11 by Chan, Eunice Y. S., Corless, Robert M., Sevyeri, Leili Rafiee
#15A22 #65D05 #65F15 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1805.04488
We define generalized standard triples X, Y, and L(z) = zC1 - C0, where L(z) is a linearization of a regular matrix polynomial P(z) ∈ ℂn × n[z], in order to use the representation X(z C1~-~C0)-1Y~=~P-1(z) which holds except when z is an eigenvalue of P. This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form H(z) = z A(z)B(z) + C ∈ ℂn × n[z] from generalized standard triples of A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using the coefficients of the expression 1 = ∑k=0^ℓ ek ϕk(z) in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations.