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Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Numerical Methods
Nonlinear System
by
Norhayati Rosli & Nurfatihah Mohd Hanafi Faculty of Industrial Sciences & Technology
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Description
AIMS
EXPECTED OUTCOMES
REFERENCES
1. Norhayati Rosli, Nadirah Mohd Nasir, Mohd Zuki Salleh, Rozieana Khairuddin,
Nurfatihah Mohamad Hanafi, Noraziah Adzhar. Numerical Methods, Second Edition,
UMP, 2017 (Internal use)
2. Chapra, C. S. & Canale, R. P. Numerical Methods for Engineers, Sixth Edition,
McGraw–Hill, 2010.
1. Students should be able to find the Jacobian matrix of the nonlinear system.
2. Students should be able to solve nonlinear system by using Newton Raphson
method.
This chapter is aimed to solve nonlinear system by using Newton Raphson method.
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Content
Introduction
Newton Rapshon Method
1
2
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Nonlinear
System 1
2
INTRODUCTION
Nonlinear system of
equations refer to the
equations which cannot
be written as a linear
combination of the
unknown variables or
functions .
Nonlinear systems
with multiple
variables can be
solved by using
Newton Raphson’s
method.
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Newton Raphson Method Formula
NEWTON RAPHSON METHOD
Suppose we have two nonlinear equations with two variables, 𝑓1(𝑥, 𝑦)
and 𝑓2 𝑥, 𝑦 , the Newton Raphson formula is
2, 1,
1, 2,
11, 2, 1, 2,
i i
i i
i ii i i i
f ff f
y yx x
f f f f
x y y x
1, 2,
2, 1,
11, 2, 1, 2,
i i
i i
i ii i i i
f ff f
x xy yf f f f
x y y x
where the denominator can be simplified by write it as the determinant
of the Jacobian, |𝐉|
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Newton Raphson Method Formula (Cont.)
NEWTON RAPHSON METHOD (Cont.)
The Jacobian matrix, 𝐉 is given by
1 1
2 2
f f
x y
f f
x y
J
The determinant of Jacobian matrix, 𝐉 is
1 1
2 2
f f
x y
f f
x y
J
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Newton Raphson Method Procedures
NEWTON RAPHSON METHOD (Cont.)
Find the Jacobian matrix, 𝐉 Step 2
Rearrange the nonlinear systems so that the equation
becomes
Step 1
1 1
2 2
f f
x y
f f
x y
J
1
2
( , ) 0
( , ) 0
f x y
f x y
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Newton Raphson Method Procedures (Cont.)
NEWTON RAPHSON METHOD (Cont.)
Find the determinant of Jacobian matrix, 𝐉
1 1
2 2
f f
x y
f f
x y
J
Step 3
Step 4 The calculation process starts by iterating the next values of
𝑥 and 𝑦.
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Example 1
Use the multiple equations Newton–Raphson’s method to determine the roots
of the following nonlinear equations
Initiate two iterations with initial guesses of 𝑥 = 1.5 and 𝑦 = 2.5.
Solution
Rearrange the nonlinear systems
Find the Jacobian matrix, 𝐉
1
2
2
2 15
3 10
f x xy
f xy x y
2
1 2 2
6 3
y x
y xy x x
J
NEWTON RAPHSON METHOD (Cont.)
2
2 15
3 10
z xy
xy x y
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
NEWTON RAPHSON METHOD (Cont.)
Solution (Cont.)
Find the determinant of the Jacobian
2
2
1 2 2(1 2 )( 3 ) (2 )( 6 )
6 3
y xy x x x y xy
y xy x x
J
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
NEWTON RAPHSON METHOD (Cont.)
Solution (Cont.)
Do iteration to calculate the values of 𝑥 and 𝑦
First iteration: 𝑖 = 0, 𝑥0 = 1.5, 𝑦0 = 2.5
1
2
2
2
1.5 2(1.5)(2.5) 15 6
(1.5)(2.5) 3(1.5) (2.5) 10 10.625
1 2(2.5) 2(1.5) 6 3
2.5 6(1.5)(2.5) 1.5 3(1.5) 25 8.25
| | (6)(8.25) 3(25) 25.5
f
f
J
J
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
NEWTON RAPHSON METHOD (Cont.)
Solution (Cont.)
First iteration: 𝑖 = 0, 𝑥0 = 1.5, 𝑦0 = 2.5 (Cont.)
1
1
( 6)(8.25) 10.625(3)1.5 1.6912
25.5
(10.625)(6) ( 6)(25)2.5 10.8824
25.5
x
y
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
NEWTON RAPHSON METHOD (Cont.)
Solution (Cont.)
Second iteration: 𝑖 = 1, 𝑥1 = −1.6912, 𝑦1 = 10.8824
1
2
53.4998
64.9718
22.7648 3.3824
99.5435 92.6542
| | 2445.9503
f
f
J
J
2
2
2.5128
9.3098
x
y
Numerical Methods
by Norhayati Rosli
http://ocw.ump.edu.my/course/view.php?id=449
Author Information
Norhayati Binti Rosli,
Senior Lecturer,
Applied & Industrial Mathematics
Research Group,
Faculty of Industrial Sciences &
Technology (FIST),
Universiti Malaysia Pahang,
26300 Gambang, Pahang.
SCOPUS ID: 36603244300
UMPIR ID: 3449
Scholars: https://scholar.google.com/c
itations?user=SLoPW9oAAAAJ&hl=en
e-mail: [email protected]
Nurfatihah Binti Mohd Hanafi
e-mail:
