KKKM3043: PENGIRAAN
BERANGKA (NUMERICAL
COMPUTATIONS)
ROOTS OF EQUATIONS
Dr. Nashrah Hani Jamadon
Motivation
• Remember that in order to solve
!
𝑓 𝑥 = 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0,
You need to use the quadratic formula?
−𝑏 ± 𝑏 ! − 4𝑎𝑐
𝑥=
2𝑎
𝑥 = roots
How about
!
a𝑥
+
"
𝑏𝑥
+
#
𝑐𝑥
+
$
𝑑𝑥
+ 𝑒𝑥 + 𝑓 = 0
sin 𝑥 + 𝑥 = 0
Þ𝑥 =?
Þ𝑥 =?
NON-COMPUTER METHODS TO
DETERMINE ROOTS
•When it crosses the x-axis, such value
represents the x (the root) value when ƒ(x)
=0
•Although graphical method is useful in
obtaining rough estimation values of
roots, it is still limited due to lack of
precision
NON-COMPUTER METHODS TO
DETERMINE ROOTS
• Solution? Use trial and error method
• This technique consists of guessing a value of
x and evaluate whether ƒ(x) is zero. If it is not,
repeat the step again
• If not (as is almost always the case), another
guess is made
• The process is repeated until a guess is
obtained that results in an f(x) that is close to
zero
ROOTS OF EQUATIONS AND
ENGINEERING PRACTICE
Dependent variable = ƒ (independent variable, parameters, forcing functions)
Fundamental
practise
Energy balance
Newton’s laws of
motion
Heat balance
Mass balance
Force balance
Dependent variable
Independent variable
Parameters
Changes in the kinetic and
potential-energy states of system
Time & position
Thermal properties, mass of
material & system geometry
Time & position
Mass of material, system
geometry, drag coefficient,
etc
Time & position
Thermal properties of
material and geometry of
system
Time & position
Chemical behavior of
material, mass transfer
coefficient, geometry of
system
Time & position
Strength of material,
structural properties,
geometry of system
Acceleration, velocity or location
Temperature
Concentration or quantity of mass
Magnitude & direction of forces
Roots of equations by
BRACKETING METHODS
•Two initial guesses for the root are
required
•The guesses must be in ‘bracket’ or be on
either side of the root
METHODS IN
BRACKETING
METHODS
1) GRAPHICAL
2) BISECTION
3) FALSE-POSITION
st
1
method:
GRAPHICAL METHOD
• A simple method to estimate the root of
equation ƒ(x) = 0, is to make a plot of the
function and observe where it crosses the
x-axis
Example:
By using the graphical method, determine
the drag coefficient, c needed for a
parachutist with the mass of 68.1 kg to
have a velocity of 40 m/s after a free-falling
at time of 10 s. Take note the acceleration
due to gravity is 9.81 m/s2
Solution: Use the formula:
f(c) =
!"
#
1−𝑒
$
!
"
%
− v (Eq.1)
Make a table with different values of drag
coefficient, c that give you different values of
ƒ(c).
c
ƒ(c)
For example;
4
34.190
8
17.712
12
6.114
16
-2.230
20
-8.368
What we know:
the c-axis between values of 12
• Crosses
and 16
using a graph paper, the root may be
• By
estimated to be 14.75
the true root value is 14.8011
• However,
(when f(c) = 0)
order to estimate the validity of this
• In
estimated root, insert into Eq.1:
𝑔𝑚
%
f(c) =
1−𝑒
𝑐
&
(
'
−v
𝑔𝑚
f(c) =
1 − 𝑒!
𝑐
9.81 68.1
f 14.75 =
14.75
9.81 68.1
v=
14.75
1−
1−
"
# $
−v
%&.()
! *+.% %,
𝑒
%&.()
! *+.% %,
𝑒
− 40 = 0.10
= 40.1 m/s
Estimation value
The answer (if the estimated root, c=14.75),
the final difference is 0.100 (v = 40.1 – 40.0)
which is close to zero (0)
• Graphical method is somehow a limited practise due to
such method is not that precise
• However, this method is still reliable to predict rough
values of roots
• There are various forms of graphs. Therefore, the location
of roots may varies.
GRAPHICAL METHOD (roots)
Example 5.2
𝑓 𝑥 = sin 10𝑥 + cos 3𝑥
Compute with excel