16 
I + (N - B 1 * U B) L 
I 
b 1 
B t 
U B N" 1 
U B N" 1 
L" 1 = N - B 1 U B 
L" 1 = NB" 1 U" 1 (B*)" 1 (N - B 1 U B) 
= NB” 1 U" 1 (B 1 )" 1 N - NB" 1 B 
= NB" 1 (U" 1 - BN" 1 B t )(B t )" 1 N 
then 
L = - N" 1 B t (U" 1 - BN" 1 ^ 1 )" 1 BN" 1 
• • • • 
4.26 
Substituting 4.26 into 4.25 we get: 
M" 1 = N" 1 + N" 1 6 t (U" 1 - BN" 1 ^)" 1 BN" 1 .... 4.27 
. . . • -1 
Equation 4.27 is the relationship sought. The inverses U and 
(U" 1 - BN" 1 ^ 1 )" 1 are of sizes that range from 2x2 to a maximum of 
16 x 16 in a 3x3 sub-block. These inverses are still smaller than the 
size of the entire system (47 x 47). In addition, the use of equation 
4.27 reduces considerably the amount of computations that would be 
required to evaluate M - ^ from the retained observation equations. Also, 
equation 4.27 can be applied in an iterative process, eliminating only 
one point in each cycle, then modifying the new inverse. The modified 
vector of constant terms Q can be easily computed from equation 4.22 as 
Q=P-B t UF ....4.28 
which also possesses the characteristic of stepwise application. 
It is interesting to note that equations 4.27 and 4.28 may be 
of considerable value for the general problem of least squares, particularly 
for large systems of linear equations. Assuming that the general form of 
the observation equations (corresponding to 4.01) is: 
KV + GE + F = 0 
.... 4.29 
the generalized equations corresponding to 4.27 and 4.28 become 
M' 1 = N" 1 + N“ 1 G t (KW” 1 K t - GN“ 1 G t ) -1 GN" 1 .... 4.30