WORKING METHODS FOR COMPUTERS         469
have used as weights, or (with less accuracy) by dividing any single
product-sum by the corresponding single saturation coefficient. If
we carry out the latter division before multiplying, the weight when
divided by the corresponding saturation coefficient becomes equal
to unity, since the saturation coefficient is the weight.1
Let us begin by following this procedure, since it renders the
essential comparison more obvious to the eye.
(a)  1st Approximation (Table VI ^ a)
, i. We begin, as in simple summation, by inserting the estimated
values for the self-correlations ('7216, '6351, etc.),2 and adding up
the several columns as they stand, i.e. giving, for the first approxi-
mation, an equal weight—namely, unity—to each row.
2.	Instead of converting the totals straight away into saturation
coefficients, each total is simply divided by the largest in the set:
(2-91 -f- 2-91 == l-oo ;   2-12 -=- 2-91 = 73 ;   etc.) :   though we can
no longer make all the weights equal to unity, we can at least make
the largest equal to unity, thus simplifying both comparisons and
calculations.    These ratios are now to be used as provisional weights
for the next approximation.    We cannot presume that they are
correct to more than the first decimal place.    Hence to save labour
we merely take 1*0, 7, -6, etc.    We then proceed as in Table VI, b.
(ft) zncl Approximation (Table 71, ft)
3.	(a) Multiply each row of correlations by the corresponding
new weight;   (7216 X l-o = 722, etc,;   -58 X 7 = -406;  etc.).
(b} Add each column of products : (722 + "406 + •   •   • + '3°8 =
2*176;   etc).   The largest of the totals (2-176) will form a first
1 The theory underlying these numerical methods is succinctly set out in Frazer,
Duncan, and Collar, Elementary Matrices, 1938, pp. 133-144, Historically, most of
the iterative methoas of solving the ' characteristic equation' of matrices appear
to be developments of Bernouilli's method of solving polynomial equations (equiva-
lent to repeated premultiplication of the related matrix by an arbitrary row-vector) :
in practice they have been employed more particularly to deal with problems arising
from the principle of least squares. To psychologists the best-known use of the
method is Retelling's iterative procedure for determining * principal components':
as already noted, however, my own application differs somewhat from fiotelling's
and also from the earlier iterative method of Thurstone (whose method of * principal
axes' apparently "served as a point of departure for (Hotelling's) paper" [7p]
p, 48 ; [04], p. 120). Other procedures for obtaining latent roots and vectors will
be found in mathematical books and papers on the subject (for refs. cf. [81]). Usually,
however, they are devised to deal with small matrices to a high degree of accuracy;
and. being in addition of more general application, prove too elaborate for the needs
of the computing psychologist. The advanced mathematician will find a technical
summary of methods in a paper by Aitken [in], in which he has introduced many
ingenious devices of his own.
* The figures adopted are the sums of the squares of the saturations ascertained by
simple summation with successive approximation (Table VII, a, ii): since they yield
a matrix of minimal rank, and since the method of weighted summation will yield a
matrix of the minimal rank, the same variances hold good for both methods (p. 449)