The basic theory behind Newton's method was already developed in the section " Complex roots" where we developed the theory to solve two non-linear equations of the form
The same can be extended to the systems with more than two equations. But before that, let us develop the same using a slightly different technique. If we assume the above two equations as change of coordinate system from (x, y) to (u, v) then any small increments du and dv are expressed in terms of dx and dy as
Now consider the system f 1(x, y) = 0 and f2(x, y) = 0 with (s, t) be the analytic solution and (x0, y0) be the initial guess. Now, if
then dv = v - v0 = f2(s, t) - f2(x0, y0 ) = -f2(x0, y0) Þ by the above relation at (x0 , y0) or in-general
Once we know then the iterative process can be written as,
for i = 0, 1, 2, . . .
Now for a system of n equations the above formula can be written in vector form as
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