Newton's Method
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


u = f1(x, y)
v = f2(x, y)

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
du =   ¶f1  ¶x +   ¶f1  ¶y
 ¶x  ¶y

 
dv =   ¶f2  ¶x +   ¶f2  ¶y
 ¶x  ¶y

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


du = u - u0        dv = v - v0
dx = s - x0         dy = t - y0

then
du   =   u - u0   =   f1(s, t) - f1(x0, y0 )   =   -f1(x0, y0)
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,


xi+1 = xi +dxi
yi+1 = yi + dyi

for i = 0, 1, 2, . . . 
Now for a system of n equations the above formula can be written in vector form as


Xi+1 = Xi + dXi

where 
dXi = (J-1 F) at x = xi

Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations