Numerical methods presentation 11 iteration method

iteration method

1. Iteration method
What is iteration: Iteration is the act of repeating a process, to generate a (possibly
unbounded) sequence of outcomes, with the aim of approaching a desired goal, target or
result. Each repetition of the process is also called an "iteration", and the results of one
iteration are used as the starting point for the next iteration.
Iterative method;
%% Solve a numerical problem ( e.g., finding the root of a system of equation) by finding
successieve approximations to solution from an initial guess.
%% Stoping criteria; the relative error
Ea= [Xi-1-Xi/Xi+1] 100%
Is smaller than a pre-specified value
Solving using iteration method
%% Iteration is the process of repeatedly using a previous result to obtain a new result
Method
%% Rearrange the equation to make the highest power the subject .
%% Use the power root to leave X on the own on the LHS.
%% Make X on the LHS Xn+1.
%% Makes X on the RHS Xn
%% Now that the function is in the form
Xn+1= f ( Xn )
%% We can use the value for x1 to calculate x2.
Than we can use the value x3 and so on………..

2. Solving the iterative method
Example: find the root of the equation
F(x)=x^3-2x+3
Given that there is a solution close to x= -2
Step 01: rearrange the equation
-2x+3=0
X^3=2x-3
X= (2x-3)
Xn+1= 2xn-3)
Step 02: we can now input x1=-2(taken from equation )
X1=-2 .
X2= = -1.9129
X3= 3√(2(-1.9129)-3)= -1.8969
X4= 2(-1.8969)-3)= -1.8940
X5= 2(-1.8940)-3)= -1.8934
X6= 2(-1.8934)-3)= -1.8933
X7= (2(-1.8933)-3)= -1.8933

This gives us the solution
X= -1.8933 (ANS)

3. Or another method
Problem 01: find the root of e^-x –x using the method of iteration, starting guess x1=0.
Solution:
Here , f(x)=e^-x –x = 0
Then X= -x.
The error is ꜫa=
Now,
N0 of
iteration
xi Xi+1 ꜫa
0 0 1 100%
1 1 0.3678 171.89%
2 0.3678 0.6922 46.87%
3 0.6922 0.5 36.44%
4 0.5 0.607 17.63%
5 0.607 0.544 11.58%
6 0.544 0.580 6.21%
7 0.580 0.560 3.57%
8 0.560 0.571 1.92%
9 0.571 0.565 1.1%
10 0.565 0.568 0.53%

So the root of the equation is 0.568 and the .