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software.
✓ Understanding these elementary steps of a larger numerical simulation
is useful:
• to understand the way computers work and how they can be used to
solve problems efficiently.
• to be able to modify/create your own routines.
• to understand the errors generated by a code you didn’t write and
its limits.
Goal of the class
✓ The goal of this class is
- to provide you with an introduction to a set of fundamental recipes,
- to enable you to choose the right techniques when facing a
particular mathematical equation to solve.
✓ The goal of the class is not to have you write a long code for a
particular engineering problem.
✓ Instead, the routines you will write will be direct applications of the
fundamental concepts studied in class.
Why MATLAB?
✓ MATLAB is a convenient and user-friendly software compared to
other scientific programming languages (C++, Fortran,...)
✓ MATLAB has good graphic capabilities.
✓ MATLAB has a lot of pre-optimized built-in functions for numerical
methods.
- In this class, we will write short routines to apply the numerical
methods we’ll study.
- Some routines that we’ll write already exist in MATLAB.
- The goal of the class is to understand how it works, so in general, we
won’t use these built-in MATLAB functions unless specified
otherwise (e.g. validate our result).
The numerical methods we will study
✓ We will introduce the following fundamental techniques:
- Systems of linear equations
- Solution of non-linear equations
- Data regression
- Function interpolation
- Numerical integration
- Numerical differentiation
- Solution of Ordinary Differential Equations
✓ These techniques are found in engineering applications by themselves
or through more elaborate numerical methods that are built on these
fundamental ones.
Systems of linear equations
What is the electric current inside resistor R2?
In general, we will have to solve systems of equations of the form:
!""#""$
a11x1 + a12x2 + . . . + a1nxn = b1
a21x1 + a22x2 + . . . + a2nxn = b2
... ...
am1x1 + am2x2 + . . . + amnxn = bm
!" A · x = b with A =%&&&'
a11 a12 . . . a1n
a21 a22 . . . a2n
...
...
am1 am2 . . . amn
()))*
, x =%&&&'
x1
x2
...
xn
()))*
and b =%&&&'
b1
b2
...
bn
()))*
i1
i2
i3
!"#2R1i1 −
R2i2 =
!
1
−
!
2
R2i2 − 2R1i3 = !2 − !3
i1 + i2 + i3 = 0
Non-linear equations
Given the geometry and the density of the ball
relative to the water, what is the immersed
height?
In general, we will learn how to solve numerically equations of the form f(x)=0 when f is not
linear.
−−51 −0.5 0 0.5 1
0
5
10
15
20
f(x) = x6 − 7x5 + 9x2 − 1
Find x such that f(x) = 0?
x2(3R − x) =
4!ballR3
!water
1050 200 250 300
50
100
150
Price (in thousands $)
New home sales
Data regression
How to represent the general trend
of the data?
What is the best linear fit?
In general, experimental measurements include some noise (due to the
precision of the instruments, ...).
Data regression allows to extract from the data the general behavior by offering
an approximation of the data set.
−2 −1 −0.5 0 0.5 1
−1
0
1
2
x
y
Interpolation
In contrary to the regression problems, we are looking here for a function that
fits exactly the given points.
Interpolation is particularly useful to understand and compute integrals.
How can you fit the data exactly by
a polynomial?
How to determine the value of the
function at x=0.15 for example?
00 2 4 6 8 10
5
10
15
20
days
Power usage
Integration
Knowing the instantaneous power consumption of a household, how can you
find the total power consumption?
E = ! t
0 P(t!)dt!
Differentiation
From its GPS system, an aircraft is able to determine its instantaneous
position. From this data, how can you compute numerically its velocity
(relative to the ground)?
v =
dx
dt !
x(t + !t) − x(t)
!t
Ordinary Differential Equations
dx
dt
= f (x, t), x(t = 0) = x0
In general, we will learn how to solve numerically equations of the form:
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