This document outlines the contents of a numerical methods course. It introduces the instructor and provides an overview of the course structure. The course contents are divided into three parts that cover topics like mathematical modeling, roots of equations, integration, differentiation, and ordinary differential equations. Evaluation criteria including exams, projects, and grading are also presented. Prerequisites for the course and references are listed. An example problem on modeling the velocity of a bungee jumper is provided and solved both analytically and numerically to demonstrate the concepts.

1. Introduction Course Contents Numerical Methods
NUMMET H
ECE Computational Numerical Methods
Engr. Melvin Kong Cabatuan
De La Salle University
Manila, Philippines
May 2014
Engr. Melvin Kong Cabatuan N UMMET H

2. Introduction Course Contents Numerical Methods
Self Introduction
Engr. Melvin K. Cabatuan, ECE
Masters of Engineering, NAIST (Japan)
Thesis: Cognitive Radio (Wireless Communication)
ECE Reviewer/Mentor (Since 2005)
2nd Place, Nov. 2004 ECE Board Exam
Test Engineering Cadet, ON Semiconductors
DOST Academic Excellence Awardee 2004
Mathematician of the Year 2003
DOST Scholar (1999-2004)
Panasonic Scholar, Japan (2007-2010)
Engr. Melvin Kong Cabatuan N UMMET H

3. Introduction Course Contents Numerical Methods
1 Introduction
2 Course Contents
Evaluation Criteria
Pre-requisite
References
3 Numerical Methods
Mathematical Modeling
Problem Solving
Example
Engr. Melvin Kong Cabatuan N UMMET H

4. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part I
1 Mathematical Modeling & Engineering
Problem Solving
2 Approximation and Round-off Errors
3 Truncation Errors and the Taylor Series
4 Roots of Equations
Engr. Melvin Kong Cabatuan N UMMET H

5. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part I
1 Mathematical Modeling & Engineering
Problem Solving
2 Approximation and Round-off Errors
3 Truncation Errors and the Taylor Series
4 Roots of Equations
Engr. Melvin Kong Cabatuan N UMMET H

6. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part I
1 Mathematical Modeling & Engineering
Problem Solving
2 Approximation and Round-off Errors
3 Truncation Errors and the Taylor Series
4 Roots of Equations
Engr. Melvin Kong Cabatuan N UMMET H

7. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part I
1 Mathematical Modeling & Engineering
Problem Solving
2 Approximation and Round-off Errors
3 Truncation Errors and the Taylor Series
4 Roots of Equations
Engr. Melvin Kong Cabatuan N UMMET H

8. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part II
1 Linear Algebraic Equations
2 Curve Fitting
Engr. Melvin Kong Cabatuan N UMMET H

9. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part II
1 Linear Algebraic Equations
2 Curve Fitting
Engr. Melvin Kong Cabatuan N UMMET H

10. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part III
1 Numerical Integration & Differentiation
with Engineering Applications
2 Ordinary Differential Equations &
Engineering Applications
Engr. Melvin Kong Cabatuan N UMMET H

11. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part III
1 Numerical Integration & Differentiation
with Engineering Applications
2 Ordinary Differential Equations &
Engineering Applications
Engr. Melvin Kong Cabatuan N UMMET H

12. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part III
1 Numerical Integration & Differentiation
with Engineering Applications
2 Ordinary Differential Equations &
Engineering Applications
Engr. Melvin Kong Cabatuan N UMMET H

13. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Course Contents - Part III
1 Numerical Integration & Differentiation
with Engineering Applications
2 Ordinary Differential Equations &
Engineering Applications
Engr. Melvin Kong Cabatuan N UMMET H

14. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Evaluation Criteria
Quiz Average: 45%
Final Exam: 40%
Project: 10%
Teacher‘s Evaluation: 5%
Total: 100%
PASSING GRADE: 65%
Engr. Melvin Kong Cabatuan N UMMET H

15. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Pre-requisite
1 LBYEC12 (Hard)
2 CONTSIG (Soft)
3 Mathematical Background
4 C++ or MATLAB/SCILAB
Engr. Melvin Kong Cabatuan N UMMET H

16. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Pre-requisite
1 LBYEC12 (Hard)
2 CONTSIG (Soft)
3 Mathematical Background
4 C++ or MATLAB/SCILAB
Engr. Melvin Kong Cabatuan N UMMET H

17. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Pre-requisite
1 LBYEC12 (Hard)
2 CONTSIG (Soft)
3 Mathematical Background
4 C++ or MATLAB/SCILAB
Engr. Melvin Kong Cabatuan N UMMET H

18. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
Pre-requisite
1 LBYEC12 (Hard)
2 CONTSIG (Soft)
3 Mathematical Background
4 C++ or MATLAB/SCILAB
Engr. Melvin Kong Cabatuan N UMMET H

19. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
References
1 Canale, R., & Chapra, S. (2009). Numerical
Methods for Engineers (6 ed.), New York,
McGraw-Hill
2 Fausett, L.V. (2008). Applied Numerical
Analysis using Matlab. USA: Pearson
Prentice Hall.
3 Online Resources
Engr. Melvin Kong Cabatuan N UMMET H

20. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
References
1 Canale, R., & Chapra, S. (2009). Numerical
Methods for Engineers (6 ed.), New York,
McGraw-Hill
2 Fausett, L.V. (2008). Applied Numerical
Analysis using Matlab. USA: Pearson
Prentice Hall.
3 Online Resources
Engr. Melvin Kong Cabatuan N UMMET H

21. Introduction Course Contents Numerical Methods Evaluation Criteria Pre-requisite References
References
1 Canale, R., & Chapra, S. (2009). Numerical
Methods for Engineers (6 ed.), New York,
McGraw-Hill
2 Fausett, L.V. (2008). Applied Numerical
Analysis using Matlab. USA: Pearson
Prentice Hall.
3 Online Resources
Engr. Melvin Kong Cabatuan N UMMET H

22. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Why study numerical methods?
Engr. Melvin Kong Cabatuan N UMMET H

23. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Why study numerical methods?
Numerical Methods expand the types of problems you
can address, i.e. handling large systems of equations,
nonlinearities, and complicated geometries.
Engr. Melvin Kong Cabatuan N UMMET H

24. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Why study numerical methods?
Numerical Methods expand the types of problems you
can address, i.e. handling large systems of equations,
nonlinearities, and complicated geometries.
Numerical Methods allow you to use "canned" software
with insight.
Engr. Melvin Kong Cabatuan N UMMET H

25. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Why study numerical methods?
Numerical Methods expand the types of problems you
can address, i.e. handling large systems of equations,
nonlinearities, and complicated geometries.
Numerical Methods allow you to use "canned" software
with insight.
Numerical Methods enable you to design your own
programs to solve problems without having to buy or
commission expensive software.
Engr. Melvin Kong Cabatuan N UMMET H

26. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Why study numerical methods?
Numerical Methods expand the types of problems you
can address, i.e. handling large systems of equations,
nonlinearities, and complicated geometries.
Numerical Methods allow you to use "canned" software
with insight.
Numerical Methods enable you to design your own
programs to solve problems without having to buy or
commission expensive software.
Numerical Methods are an efficient vehicle for learning
to use computers and also reinforce your understanding in
mathematics.
Engr. Melvin Kong Cabatuan N UMMET H

27. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Numerical Methods/ Analysis
Purpose
To find approximate solutions to
complicated mathematical problems using
arithmetic operations.
Engr. Melvin Kong Cabatuan N UMMET H

28. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Numerical Methods/ Analysis
Insight
c Numerical methods solve hard problems
by doing lots of easy steps. d
Engr. Melvin Kong Cabatuan N UMMET H

29. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Numerical Methods/ Analysis
Insight
c Computers are great tools, but w/o
fundamental understanding of
engineering problems, they will be
useless! d
Engr. Melvin Kong Cabatuan N UMMET H

30. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Engr. Melvin Kong Cabatuan N UMMET H

31. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Engr. Melvin Kong Cabatuan N UMMET H

32. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Solution
Elaborate and complicated
method.
Engr. Melvin Kong Cabatuan N UMMET H

33. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Solution
Elaborate and complicated
method.
Interpretation
In-depth analysis limited
time-consuming solution.
Engr. Melvin Kong Cabatuan N UMMET H

34. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Solution
Elaborate and complicated
method.
Interpretation
In-depth analysis limited
time-consuming solution.
Formulation
In-depth exposition of the
problem.
Engr. Melvin Kong Cabatuan N UMMET H

35. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Solution
Elaborate and complicated
method.
Interpretation
In-depth analysis limited
time-consuming solution.
Formulation
In-depth exposition of the
problem.
Solution
Easy-to-use computer
method.
Engr. Melvin Kong Cabatuan N UMMET H

36. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Mathematical Modeling
Traditional vs. Modern Approach
Formulation
Fundamental laws
explained briefly.
Solution
Elaborate and complicated
method.
Interpretation
In-depth analysis limited
time-consuming solution.
Formulation
In-depth exposition of the
problem.
Solution
Easy-to-use computer
method.
Interpretation
More time for in-depth
analysis due to ease of
calculation.
Engr. Melvin Kong Cabatuan N UMMET H

37. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Engineering Problem Solving
Engr. Melvin Kong Cabatuan N UMMET H

38. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Engineering Problem Solving
Engr. Melvin Kong Cabatuan N UMMET H

39. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Engineering Problem Solving
Engr. Melvin Kong Cabatuan N UMMET H

40. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Engineering Problem Solving
Engr. Melvin Kong Cabatuan N UMMET H

41. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Example: Bungee-jumping
Predict the velocity of a jumper as a
function of time during the free-fall part of
the jump.
Engr. Melvin Kong Cabatuan N UMMET H

42. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Example: Bungee-jumping
Predict the velocity of a jumper as a
function of time during the free-fall part of
the jump.
F = FD + FU
F = Net force acting on the body
FD = Force due to gravity = mg
FU = Force due to air resistance = −cv
(c = drag coefficient)
Engr. Melvin Kong Cabatuan N UMMET H

43. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Example: Bungee-jumping
Predict the velocity of a jumper as a
function of time during the free-fall part of
the jump.
dv
dt = g − c
m v
c This is a first order ordinary linear differential equation. d
Engr. Melvin Kong Cabatuan N UMMET H

44. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical Solution
If the jumper is initially at rest (v = 0 at
t = 0), dv/dt can be solved to give the
result:
v(t) =
gm
c
1 − e−(c/m)t
Engr. Melvin Kong Cabatuan N UMMET H

45. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical Solution
v(t) =
gm
c
1 − e−(c/m)t
g = 9.8 m/s2
, c = 12.5 kg/s, m = 68.1 kg
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
Engr. Melvin Kong Cabatuan N UMMET H

46. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Numerical Solution
dv
dt
= g −
c
m
v
dv
dt
∼=
∆v
∆t
=
v(ti+1) − v(ti)
ti+1 − ti
........
dv
dt
= lim
∆t→0
∆v
∆t
v(ti+1) − v(ti)
ti+1 − ti
= g −
c
m
v(ti)
v(ti+1) = v(ti) + [g −
c
m
v(ti)](ti+1 − ti)
Engr. Melvin Kong Cabatuan N UMMET H

47. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Numerical Solution
v(ti+1) = v(ti) + [g −
c
m
v(ti)](ti+1 − ti)
@ ∆t = 2 sec
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39
Engr. Melvin Kong Cabatuan N UMMET H

48. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical vs. Numerical Solution
Engr. Melvin Kong Cabatuan N UMMET H

49. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical vs. Numerical Solution
Analytical
Solution
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
Engr. Melvin Kong Cabatuan N UMMET H

50. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical vs. Numerical Solution
Analytical
Solution
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
Numerical
@ ∆t = 2 sec
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39
Engr. Melvin Kong Cabatuan N UMMET H

51. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical vs. Numerical Solution
Analytical
Solution
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
Numerical
@ ∆t = 2 sec
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39
Numerical
@ ∆t = 0.01 sec
t (sec.) V (m/s)
0 0
2 16.41
4 27.83
8 41.13
10 44.90
12 47.51
∞ 53.39
Engr. Melvin Kong Cabatuan N UMMET H

52. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analytical vs. Numerical Solution
Analytical
Solution
t (sec.) V (m/s)
0 0
2 16.40
4 27.77
8 41.10
10 44.87
12 47.49
∞ 53.39
Numerical
@ ∆t = 2 sec
t (sec.) V (m/s)
0 0
2 19.60
4 32.00
8 44.82
10 47.97
12 49.96
∞ 53.39
Numerical
@ ∆t = 0.01 sec
t (sec.) V (m/s)
0 0
2 16.41
4 27.83
8 41.13
10 44.90
12 47.51
∞ 53.39
c Minimize the error by using smaller step size, ∆t. d
Engr. Melvin Kong Cabatuan N UMMET H

53. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Analogy
Engr. Melvin Kong Cabatuan N UMMET H
n = 3

54. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Key Insights
Engr. Melvin Kong Cabatuan N UMMET H

55. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Key Insights
Numerical Methods find approximate solutions to
complicated problems using arithmetic operations.
Engr. Melvin Kong Cabatuan N UMMET H

56. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Key Insights
Numerical Methods find approximate solutions to
complicated problems using arithmetic operations.
c Solving hard problems with lots of easy steps. d
Engr. Melvin Kong Cabatuan N UMMET H

57. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Key Insights
Numerical Methods find approximate solutions to
complicated problems using arithmetic operations.
c Solving hard problems with lots of easy steps. d
Computers are great tools, but w/o fundamental
understanding of engineering problems, they will be useless!
Engr. Melvin Kong Cabatuan N UMMET H

58. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
Key Insights
Numerical Methods find approximate solutions to
complicated problems using arithmetic operations.
c Solving hard problems with lots of easy steps. d
Computers are great tools, but w/o fundamental
understanding of engineering problems, they will be useless!
You can minimize the error in numerical solutions by using
smaller step size, ∆t.
Engr. Melvin Kong Cabatuan N UMMET H

59. Introduction Course Contents Numerical Methods Mathematical Modeling Problem Solving Example
END
c Thank you for your attention d
Engr. Melvin Kong Cabatuan N UMMET H