Solution of Equations
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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 -
 
                  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, ... , xn.  The xn 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:

Fx*(x-x0) + Fy*(y-y0) + Fz*(z-z0) = - F(x0,y0,z0)
Gx*(x-x0) + Gy*(y-y0) + Gz*(z-z0) = - G(x0,y0,z0)          (II)
Hx*(x-x0) + Hy*(y-y0) + Hz*(z-z0) = - H(x0,y0,z0)

            F        F         F
where  Fx = , Fy = , Fz = , all evaluated at (x0,y0,z0).
           
x        y         z

Similarly for Gx, Gy, Gz, Hx, Hy, Hz.

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

x-x0=Dx y-y0=Dy z-z0=Dz

then a closer approximation to the solution is

x1=x0+Dx y1=y0+Dy z1=z0+Dz

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.

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