Appendix: Newton’s method
Consider finding the root of the single equation in one variable, 
where 
takes the form shown in Figure 9. Newton’s method involves taking an arbitrary starting point, 
and drawing the tangent, with slope 
. By approximating the function by the tangent, the new value is given by the point of intersection of this tangent with the 
axis, at 
. Selecting as the next starting point leads quickly to the required root.
- Figure 9 – Newton’s method
-
From the triangle in Figure 9:
(34) 
Hence, starting from 
, the sequence of iterations is:
(35)
Convergence is reached when 
and 
depends on the accuracy required. Newton’s method is easily adapted to deal with a set of equations, 
where is a vector. The method involves repeatedly solving the following matrix equation, where 
now denotes the vector in the 
th iteration and is a vector containing the 
values.
(36)
