Numerical Methods
Introduction
INTRODUCTION
Mathematical model in science and engineering involve equations
that need to be solved.
Equation of one variable can be formulated as
𝑓𝑥
=0
(1)
Equation (1) can be in the form of linear and nonlinear.
Solving equation (1) means that finding the values of x that satisfying
equation (1).
INTRODUCTION (Cont.)
1
Transcendental
equations
A non-algebraic
equation of
trigonometric,
exponential and
logarithm function
3
Equation (1) may
belong to one of the
following types of
equations
2
Algebraic
equations
Polynomial
equations
INTRODUCTION (Cont.)
Example 1: Algebraic Equation
4𝑥 − 3𝑥2𝑦 − 15 = 0
Example 2: Polynomial Equation
𝑥2 + 2𝑥 − 4 = 0
Example 3: Transcendental Equation
sin 2𝑥 − 3𝑥 = 0
INTRODUCTION (Cont.)
Finding Roots for Quadratic Equations
𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐
1
Factorization
Analytical
Methods
Quadratic Formula
3
2
Completing the Square
INTRODUCTION (Cont.)
All above mentioned methods to solve quadratic equations are analytical
methods
The solution obtained by using analytical methods is called exact solution
Due to the complexity of the equations in modelling the real-life system, the
exact solutions are often difficult to be found.
Thus require the used of numerical methods.
The solution that obtained by using numerical methods is called numerical
solution.
Why use Numerical Methods?
• To solve problems that cannot be solved exactly
1
2
x
e

−
2
u
−
2
du
INTRODUCTION (Cont.)
Three types of Numerical Methods shall be considered to find the
roots of the equations:
1
3
Open Methods
1
Newton Raphson Method
2
Secant Method
Finding
Roots
using
Numerical
Methods
2
Incremental Search
Bracketing Methods
1
Bisection Method
2
False Position Method
Prior to the numerical methods, a graphical method of finding roots of the
equations are presented.
Steps in Solving an
Engineering Problem
How do we solve an engineering
problem?
Problem Description
Mathematical Model
Solution of Mathematical Model
Using the Solution
Mathematical Procedures
• Nonlinear Equations
• Curve Fitting
– Interpolation
– Regression
• Other Advanced Mathematical Procedures:
– Partial Differential Equations
– Optimization
Nonlinear Equations
How much of the floating ball is under water?
Diameter=0.11m
Specific Gravity=0.6
−4
x − 0.165 x + 3.993  10 = 0
3
2
Nonlinear Equations
How much of the floating ball is under the water?
f ( x ) = x 3 − 0.165 x 2 + 3.993  10−4 = 0
Interpolation
What is the velocity of the rocket at t=7 seconds?
Time (s)
Vel (m/s)
5
106
8
177
12
600
Regression
Thermal expansion coefficient data for cast steel
Regression (cont)