3f 3f 

 o o 



since this system of equations Is linear In the correction terms 

 Aa and Ab, these terms can be derived by the method of least-squares. 



At this point, one variation from the description given by 

 Scarborough (1930) Is Introduced. Because the linearization of the sys- 

 tem of equation Is only valid for small Aa and Ab, It Is possible to 

 derive correction terms which yield solutions that extend beyond the 

 local neighborhood of linearization. Under these circumstances. It Is 

 possible that the method will not converge to a valid solution. During 

 any particular Iteration, this non-convergence would appear as an 

 Increase In the total sum of squares of the residuals over the previous 

 Iterations. 



To ensure a decrease In the total residuals (and therefore a con- 

 verging solution) during each Iteration, the correction terms are scaled 

 by a term a to yield, 



a " a + otAa 

 o 



b - b + oAb 

 o 



The scaling term a Is first set equal to one, and the calculated 

 total residuals compared to those calculated In the previous Iteration. 

 If the residuals do not decrease, the scale factor a Is halved until a 

 decrease In the total residuals Is observed. These new a and b become 

 the new "Initial estimates" a^ and b^, which are then used In the next 

 Iteration. The process continues until the values of both Aa and Ab 

 reach some suitable minimum, and a final a and b are derived. It Is 



129 



