1. Solving Nonlinear Equation
Chapter 6
Solving Nonlinear Equation
System

2. Content Nonlinear equation system Successive substitution
Newton-Raphson
Example

3. Introduction (1) Nonlinear equation in n variables f1(x1,x2,…,xn) = 0
fn(x1,x2,…,xn) = 0
For example:
x21+x1x = 10
x2+3x1x = 57
Solve by graphical
method ?

4. Introduction (2)
x21+x1x2 = 10
x2+3x1x = 57

5. Successive sub…(1)
Each one of the nonlinear equations can be solved for one of the unknowns.
Example
x21+x1x = 10
x2+3x1x = 57
Initial guess with
x1=1.5, x2 = 3.5
(1)
(2)
Substitute with x1=1.5, x2 = 3.5 into (1)
substitute with new x1 and x2 = 3.5 into (2)

6. Successive sub…(2) Example (cont’d) Substitute so on…. not converge
Now try with these two eqs
This will converge to x1=2 and x2 =3, which are
answers!!!!
Problem is u should formulate eqns in appropriate
forms.

7. Newton-Raphson (1) For nonlinear equation in n variables where
If the derivatives of f exists, then we can estimate x from
is the initial guess vector x

8. Newton-Raphson (2) is the vector x of ith estimation
is the (updated) vector x of
i+1th estimation
is the function of x of
ith estimation
Named Jacobian matrix