The Newton method can be used to find the solutions of

f(x)=0 

Suppose x₀ is a sufficiently close approximation to a solution.  Then a better approximation x₁is found by:

                   f(x₀)
x₁= x₀ -  ¾¾¾
                  f ’(x₀)

The specified interval is explored for changes in the sign of the function.  If one is detected in a given subinterval, then a root must lie inside that subinterval. The center of it is then chosen as a first approximation and Newton's method applied to obtain the sequence of approximations:  x₀, x₁, x₂, ... , x_n.  The x_n is assumed correct if it differs from the previous value in the sequence by less than the specified error.

SYSTEMS OF SIMULTANEOUS NONLINEAR EQUATIONS

Suppose that, instead of a single nonlinear equation, we have a systems system of simultaneous nonlinear equations. We can solve it using the generalized Newton method, which is just a generalization of what we did to find the roots of a one-variable function. To illustrate, suppose we want to solve the system:

F(x,y,z) = 0  ;  G(x,y,z) = 0  ;  H(x,y,z) = 0         (I)

Assume (x₀,y₀,z₀) is close to a solution.  Now expand all three equations into their Taylor series.  Neglecting terms of order higher than the first, we get:

F_x*(x-x₀) + F_y*(y-y₀) + F_z*(z-z₀) = - F(x₀,y₀,z₀)
G_x*(x-x₀) + G_y*(y-y₀) + G_z*(z-z₀) = - G(x₀,y₀,z₀)          (II)
H_x*(x-x₀) + H_y*(y-y₀) + H_z*(z-z₀) = - H(x₀,y₀,z₀)

            ¶F        ¶F         ¶F
where  F_x = ———, F_y = ———— , F_z = ——— , all evaluated at (x₀,y₀,z₀).
            ¶x        ¶y         ¶z

Similarly for G_x, G_y, G_z, H_x, H_y, H_z.

Solving system (II) for (x-x₀), (y-y₀) and (z-z₀), we obtain corrections on the initial approximation, that is, if its solutions are D_x, D_y and D_z, where

x-x₀=D_x;  y-y₀=D_y ;  z-z₀=D_z

then a closer approximation to the solution is

x₁=x₀+D_x ;  y₁=y₀+D_y ;  z₁=z₀+D_z

We now use x₁, y₁ and z₁ as initial approximations and again set a system of linear equations then solve it to obtain x₂, y₂, and z₂.  This process is continued until all the variables in the current approximation in the sequence differ from the values of the previous approximation by less than the specified error and all equations differ from zero by less than said error.