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164 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
Theorem 10.1. The least common denominator of the rational entries in ρ 1 in the
row of the matrix AãN−1 (ρ 1 ) T corresponding to the coefficients of the normal form
of a product x α1
1 x α2
2 ··· x α N
N ã N−1 (x 1 ,x 2 ,...,x N ) with α 1 ,α 2 ,...,α N ∈{0, 1}, is
generically given by:
ρ(δ 1 ) α1 ρ(δ 2 ) α2 ··· ρ(δ N ) α N
. (10.8)
Proof. Note that ã N−1 (x 1 ,x 2 ,...,x N )=γ 1,1 x 1 + γ 1,2 x 2 + ...+ γ 1,N x N is a homogeneous
polynomial of total degree 1. When multiplied by a monomial x α1
1 x α2
2
···x α N
N
an expression of N terms results, all of total degree |α| + 1. For each
term it holds that at most one of the variables x i is of degree 2. Each reduction
step, in which (for some index i) a quantity x 2 i is replaced by the expression
e(δ i)
ρ(δ (m i) i,1x 1 + m i,2 x 2 + ···+ m i,N x N ), produces new terms of lower degree. These
terms are either reduced (this happens for the term involving m i,i x i and for all the
terms m i,j x j with j such that α j = 0) or they contain another squared variable (this
happens for the terms m i,j x j with j such that α j =1;thenx 2 j occurs). In the latter
case, the variable x i is not present in such a term. Hence, any new term produced
from it by a later reduction step with reintroduces the variable x i , is then reduced.
We conclude that a second replacement of the quantity x 2 i does not occur: the reduction
algorithm produces reduced terms by replacing quantities x 2 i for each index i at
most once per term.
The variables x i for which the index i satisfies α i = 0 initially do not show up
in the monomial. Therefore, by the same argument, the term generated from it do
not require replacement of a quantity x 2 i anywhere in the reduction process. In this
way it follows that the reduction process for the product of ã N−1 and the monomial
x α1
1 xα2 2 ···xα N
N
involves replacements of the quantities x 2 i for precisely those
indices α i = 1. This makes clear that the least common denominator of the entries
in the corresponding row of AãN−1 (ρ 1 ) T is always contained in the product
ρ(δ 1 ) α1 ρ(δ 2 ) α2 ···ρ(δ N ) α N
. It is easily verified by considering an example that generically
this least common denominator is in fact equal to this expression. □
Using the result of this theorem the rational matrix AãN−1 (ρ 1 ) T is made polynomial
in ρ 1 as follows:
ÃãN−1 (ρ 1 ) T =
⎛
ρ(δ 1 ) α1,1 ···ρ(δ N ) α 1,N
⎜
⎝
⎞
⎟
⎠ AãN−1 (ρ 1 ) T ,
(10.9)
. ..
ρ(δ 1 ) α 2 N ,1 ···ρ(δN ) α 2 N ,N
where the 2 N × 2 N matrix Ãã N−1
(ρ 1 ) T is polynomial in ρ 1 . Note that this implies
that the coefficients in the rows of ÃãN−1 (ρ 1 ) T correspond to the coefficients
of the normal form of the expressions ρ(δ 1 ) α1 ρ(δ 2 ) α2 ···ρ(δ N ) α N
x α1
1 x α2
2 ···x α N
N
ã N−1 (x 1 ,x 2 ,...,x N ).