The enabling condition imposes no practical limitation because any algorithm with
M 2 N could be effectively supplanted by exhaustive search over S.
Since I P|I K includes every row of P which is differentiated to nonzero order (e.g.
Vn e S' 3 ]1 r|I > 0), it follows from Eq. 9.18, Eq. 9.20 and Eq. 9.21 that any row n for
which I| ri;[ > M introduces an all zero row into I PI K, making both I PI| I = 0 and
C(m,r) = 0. Therefore, I| irII| M represents a suitable upper bound on I rijI for n # m. This
bound, along with the previously established condition II rilj = 0, implies I r|
(N'- 1) x M and permits the following revision of the Eq. 9.17 definition of C(m, r)
(-l)N'-k-l [P rI ,
C(m,r)= l 11r ll = 0, II r-ll M for n m m
0 otherwise
Further, the conditions in this result can be expressed in terms of S", yielding
(1)N'-k-r ,r S"-0for
C(m,r)-= r r5 = 0,r, E S"- {0 forn#m Eq. 9.23
0 otherwise
At (a= 1, using Eq. 9.6 in Eq. 9.21 yields -
P ( | v)| a = -M i [r(i, v)!(1/2 ] Eq. 9.24
(w rv_ i r s
S(1/2 ) n [r(i, v)!].
w, r. ies
Thus, every element in row v e Kj of I P0I includes the constant factor
(1/2 ) [r(i,v)!].
i S
Substituting Eq. 9.24 into Eq. 9.18 and collecting these common row factors outside the
determinant yields