is rank 1 at aX= 1, which makes the low order derivatives of p with respect to the condi-
tional probabilities P(i I n) have comparatively low rank, and this suggests expanding
--I P- II = I P- II I P- -1 in a multivariate Taylor's series about the point
corresponding to = 1. The result reflected in Eq. 9.8 is the constant term of the series.
9.3 The General Case 0 < a < 1
The state transition matrix of the two and three-operator algorithms is completely
determined by the fixed algorithm parameter M and the N' x N array of conditional prob-
abilities [P(i I n)] for i e S and n e S'. Each element of row n in P consists of a multino-
mial coefficient and a distinct order M product composed of integral powers of the
P(i I n)'s corresponding to row n (Eq. 4.15 or 4.25). Thus, the order k principal minor of
I PI generated by inclusion of rows K = {nin, 2, ,nk} c S' can be written as an order
k x M polynomial (composed of order k x M monomial terms) in the k x N array of vari-
ables [P(i I n)] for i e S,n e K. The corresponding order k principal minor of I Pz has
identical value provided m e K and is zero if m e K. These facts along with the succinct
representation of the P(i | n)'s as rational functions of the objective function and algo-
rithm parameters (Eq. 4.2, 4.11, 4.24) and the degeneration of P to rank 1 at a = 1 (Eq.
9.7) suggest an attempt to expand -.(1l) = | P-Il as a multivariate Taylor's series in the
P(i I n)'s about the point a = 1 where, according to Eq. 9.6, Vi e S, Vn e S':P(i I n) = 1/2.
(Actually, expanding the alternative form )(1)- 4((1)= P-II -I P- II in a new array
of N' x N variables which uniquely determines [P(i I n)] proves more productive). The
constant term in the series is provided by Eq. 9.8 and the highest order terms in the series
are the order (N' 1) x M monomials contributed by the single nonzero order N' 1 prin-
cipal minor of I P~ .
Let r= [r(i,n)] be an N' x N nonnegative integer array having rows of the form
r5 = (r(0, n),r( 1,n), .,r((2 1 ),n)). The nonnegative integer r(i,n) represents the expo-
nent of the factor (P(i I n) 1/2L) appearing in a monomial term of the Taylor's series.