1. Numerical Solution of Nonlinear Equations
Lecture 4
Chapter 2
(continued)
Numerical Solution of Nonlinear Equations

2. Numerical methods for the solution of Nonlinear equations
Single Nonlinear Equations
Fixed-Point Iteration.
bisection method
Newton–Raphson method
Secant method
System of Nonlinear Equations
Newton Method

3. 3- Newton–Raphson (tangents) method
The Newton–Raphson formula can be derived from the Taylor series expansion of f (x) about x:

4. Geometric interpretation
The equation of the tangent to the curve y = f(x) at P(x(0), y(0)) which is x
f(x) x* x0 y0 x3 x1 y1 x2

6. 4- Secant method
The secant method can be regarded as a modification of the Newton method in the sense that the derivative is replaced by a difference approximation based on the successive estimates
By this approximation, the iterative formula of becomes

8. We observe that the Secant method will take iterations more than required when using Newton-Raphson method.

9. Newton Method for a System of Nonlinear Equations
Consider the system of two equations
by taking the Taylor series expansion up to first-order about some estimate point (x0,y0)
This can be arranged into a matrix–vector form as

10. we solve for (x, y) to get the updated vector estimate let,
Jacobian matrix

11. For 2x2 system we prefer to apply Cramer’s rule.

12. Steps of Newton–Raphson method for simultaneous, nonlinear equations:

13. The Jacobian matrix is defined as
From the plot we can get a rough estimate of the coordinates of an intersection point: (x0,y0 ) =(0.5, 1.5), which we use as the starting values.

15. the coordinates of the four intersection points are
±( , ) and ± ( , )

16. Apply Bisection Method
Matlab Functions
Apply Bisection Method
fzero
Find root of continuous function of one variable
Syntax
x = fzero(fun,x0)
>>
ans =

17. Apply Newton-Raphson Method
fsolve
Solve system of nonlinear equations
Equation
Solves a problem specified by
F(x)=0
for x where x is a vector and F(x) is a function that returns a vector value.
Syntax
x = fsolve(fun,x0)
Apply Newton-Raphson Method
>>
Equation solved.
fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.
<stopping criteria details>
x =

18. % For system of nonlinear equations
>>
Equation solved.
fsolve completed because the vector of function values is near zero
as measured by the default value of the function tolerance, and
the problem appears regular as measured by the gradient.
<stopping criteria details>
x =

19. End of Chapter 2