ECS 511
NUMERICAL ANALYSIS & FINITE ELEMENT METHOD
TOPIC 1a
ROOT OF
EQUATIONS
- Bracketing Method
LEARNING OUTCOMES
1. Students should be able to identify the difference between bracketing
and open methods for root location (CO1-PO2).
2. Students should have mastered the techniques and capable of
choosing the best method (or methods) for any problem (CO1-PO2).
# CO2-PO5 in lab session
Simple second order equation (quadratic formula);
Eqt. 1.1
To solve;
Eqt. 1.2
Complex higher order equation;
Eqt. 1.3
NONLINEAR EQUATIONS SOLUTIONS
3𝑥 2 + 2𝑥 + 1 = 0 (quadratic eqtn.)
4𝑥 3 + 3𝑥 2 + 2𝑥 + 1 = 0
ROOT OF EQUATIONS
Equations in area of engineering in the form of f(x) = 0 need to be solved.
A solution to the equation which is also known as a root of the equation is a value
of x that satisfies the equation.
Graphically, the solution is the point where the function f(x) intersect the x axis.
An equation might have no solution or one or several solutions of root value.
BISECTION METHOD
The Bracketing method requires two initial
guess points (XL and XU) that bracket the root
(XR), the root is located within an interval of
lower (XL) and an upper bound (XU).
If f(XL).f(XU) <0 then it is guaranteed that exist at
least a root in the interval XL and XU.
Bisection method procedure;
Root, XR
f(XL)
f(XU)
BISECTION METHOD
Lower
bound, XL
Upper subinterval
Lower subinterval
Other termination criteria;
1) If the answer of XR is close to the analytical answer by having a small percentage of true error, 𝜀𝑡 or;
2) If the answer of XR is close to the previous answer in previous iteration by having a small percentage of
approximate error, 𝜀𝑎 .
Upper
bound, XU
BISECTION METHOD: EXAMPLE 1
Employ the Bisection method to find the solution of this equation by using initial guesses of X l = 0.5
and Xu = 1.0;
𝑓 𝑥 = −7𝑥 5 − 2.5𝑥 3 + 10𝑥 − 1
(Analytical answer = 0.98401)
BISECTION METHOD: EXAMPLE 1
EXERCISE 1:
Use the bisection method to find the root of the function:
𝑓 𝑥 = 𝑥 3 − 4𝑥 − 9
Terminate the calculation when the approximate error is less
than 2%.
-
Are the initial guesses
given?
#If not, you have to select
the initial guesses.
EXERCISE 1:
Use the bisection method to find the root of the function:
𝑓 𝑥 = 𝑥 3 − 4𝑥 − 9
Terminate the calculation when the approximate error is less than 2%.
-
Are the initial guesses given?
#If not, you have to select the initial guesses.
(x)
f(x)
EXERCISE 2:
Please solve at least 1 question from the book/ previous final
exam/ ufuture.
SOLUTION EXERCISE 2:
THANK YOU