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10.1. SOLVING THE SYSTEM OF QUADRATIC EQUATIONS 163
10.1 Solving the system of quadratic equations
For the co-order k = 2 case, a single parameter ρ 1 shows up in the system of equations
(10.2). We want to use the Stetter-Möller matrix method here to solve this system
of equations and we consider the linear operator which performs multiplication by a
polynomial r(x 1 ,...,x N ) within the quotient space C(ρ 1 )[x 1 ,x 2 ,...,x N ]/I(ρ 1 ). Here
the coefficients are from the field of complex rational functions in ρ 1 and I(ρ 1 ) denotes
the associated ideal generated by the polynomials in the system of equations (10.2),
regarded to be rationally parameterized by ρ 1 .
Multiplication by any polynomial r(x 1 ,...,x N ) within this quotient space is represented
by the linear operator A r(x1,...,x N ). With respect to an identical monomial basis
as in (8.47), this linear operator is represented by the 2 N ×2 N matrix A r(x1,...,x N )(ρ 1 ) T
which depends rationally on ρ 1 . The 2 N eigenvalues of this matrix are the values of
the polynomial r(x 1 ,...,x N ) at the solutions of the system of equations (10.2).
A particularly interesting choice for r(x 1 ,...,x N ) is now provided by the polynomial
ã N−1 in (10.3) to incorporate the requirement that any feasible solution of
(10.2) satisfies ã N−1 =0:
r(x 1 ,...,x N )=ã N−1 . (10.6)
This gives rise to the linear operator AãN−1 which is represented by the 2 N × 2 N
matrix AãN−1 (ρ 1 ) T . The eigenvalues of this matrix coincide with the values of the
polynomial ã N−1 at the solutions of the system of equations (10.2). Because we
require ã N−1 = 0, the eigenvalue λ in AãN−1 (ρ 1 ) T v = λv is required to be zero too.
Thus, we examine the remaining problem AãN−1 (ρ 1 ) T v = 0, which can be regarded as
a rational eigenvalue problem in the unknown ρ 1 . Thus, we are looking for non-trivial
solutions of:
AãN−1 (ρ 1 ) T v = 0. (10.7)
The (real) values of ρ 1 that make the matrix AãN−1 (ρ 1 ) T singular, are of importance
now. For such ρ 1 any associated eigenvector v in its kernel allows for the computation
of a solution (x 1 ,x 2 ,...,x N ) by reading them off from the eigenvector which exhibits
the Stetter structure.
The matrix AãN−1 (ρ 1 ) T is rational in ρ 1 but can be made polynomial in ρ 1 by
premultiplication of each row by the least common denominator (in terms of ρ 1 ) of the
rational entries in ρ 1 in that row. This yields a polynomial eigenvalue problem. This
premultiplication does not change the eigenvalues ρ 1 of the matrix. However, spurious
eigenvalues/solutions can be introduced for values of ρ 1 where ρ(δ i )=1+ρ 1 δ i =0
for some i. To avoid the occurrence of spurious solutions as much as possible, it is
necessary not to multiply the rows of AãN−1 (ρ 1 ) T by any polynomial factor which
does not remove a pole for ρ 1 . The following result specifies a convenient choice of
multiplication factors to achieve this goal.