10 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
say, where A*=pAp1. If we have, for period 1, an intermed- 
late output vector, u say, an intermediate input vector, v say, 
and a vector of total outputs, q say, then 
(IV. 51) 
Ag=u 
and 
(IV. 52) 
GA i=v 
We can make an initial estimate, u, say, of » by premultiply- 
ing q by A*. Thus 
(IV. 53) 
A*q = 
=u, 
In general uy#u, but we can force an equality by an appro- 
priate multiplication of the rows of A*. Thus 
(IV. 54) (un, -A*)g =u 
[f we regard the term in brackets as an estimate of A, it can 
be seen that it satisfies the row conditions but not the column 
conditions. These can be satisfied by a substitution for A from 
(IV. 54) into (IV. 52), followed by an appropriate multiplica- 
tion of the columns of A*. Thus 
(IV. 55) 
A $7 AN 1 
gA* wu," li=v, 
and 
‘IV. 56) 
A An 1 , AN I. 
g(vvy~'A* uu, i =v 
[f we regard the term in brackets as a revised estimate of A, 
we can see that it now satisfies the column conditions, but not, 
in general, the row conditions. We can, however, repeat the 
cycle of operations until we obtain convergence with both the 
"1] Stone - pag. 68