by 



E(u; J 



H(o-') 

 1 + 4 «*», V 



-1/2 /[x io- ',^)+K(e-', -,/,)] d,^ 



(10) 



where a- ' is the given value of tr : from the discrete tr -set for which ^ ., at 

 w; is nearest to zero. The desired value of A;; may be readily calculated. 'Values 



of 



less than 



cannot give a contribution to E at the particular uij/ and 



hence the corresponding coefficients of A-motrix must be zero. The single integral 

 equation is finally reduced to a set of n-linear algebraic equations in n-unknowns. 



Muij)- E A;jH(o-j) =0 



i = 1, n 



01) 



The E's ore given and the terms of the coefficient matrix A;; have been determined 

 in the most appropriate manner. 



It is now left to solve these equations for the desired unknowns, H ( o" j). 

 Affecting this solution is not the easy matter that it would appear, for merely reducing 

 a Fredholm equation of the first kind to a set of linear algebraic equations is not a 

 magic vehicle to simple solution of the original equation. The biggest problem that 

 arises is that the set of algebraic equations may be "ill-conditioned", which is another 

 way of saying that various of the equations may be dependent or nearly so. A simple 

 test for ill-conditioning is given by Redish (1961). Geometrically stated, the test is 

 as follows: Each equation of set (11) may be thought to represent a hyper-plane. If 

 two or more of these hyper-planes are nearly parallel, the set is said to be illconditioned. 

 To calculate the actual angle between the hyperplanes, the equations are first normal- 

 ized by dividing each equation by the square root of the sum of the square of its coeffic- 

 ients. The new coefficients, A'jj say, are now the direction cosines of the normals of 

 the hyper-planes. The angle (H)|m between the }*^ and m*" hyper-planes can be calcu- 

 lated since 



©Im = cos 



-1 



X 



j = l 



A- ,j A' 



The nearer the angle @ |m is to zero, the more poorly conditioned (dependent) are 

 the 1 and m equations. 



To suitably condition the equations (11) (i.e., to maximize @ i^^) it is necessary 

 to choose the values a : in a judicious manner. This is necessitated because of the 

 complicated relation between er and « and because some ^ -values will not be able 



29 



