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INTRODUCTION TO MATLAB

1. I NSTALLATION The National University of Singapore has a Total Academic Headcount licence for

.

Students may use it for academic, research, and learning. The license allows students to install on personally-owned computers. 1. If you are NOT using NUS network, you are required to use

:

◦ Please sign in with your NUSNET ID in the format: nusstu\nusnetid and password. If you are using NUS network, please proceed to Step 2. 2. Click the link

to download

. It

is required to sign in with your NUSNET ID in the format: nusstu\nusnetid and password. Then follow the instructions. 3. Create a

account using your NUS email address.

(i) ⋄ Email address: Your NUS email account (e.g., [email protected]) ⋄ Location: Singapore

⋄ How will you use MathWorks software? Student use ⋄ Are you at least 13 years or older? Yes (ii) You will receive an email from

with title “Verify Email Ad-

dress”. Click the link in the email to verify your account. (iii) Finish creating your profile. Then you should be able to see the following information: ◦ Your account has been created and license 40707750 has been associated with your account.

4. Click the Download bottom to download and run the installer. (i) When prompted, log in with your

Account (your NUS email account).

(ii) Select your licence (40707750, Student, Academic — Total Headcount). (iii) Choose installation folder. (iv) Select products to install. 1

INTRODUCTION TO MATLAB

2

2. B ASIC O PERATIONS First of all, we learn some basic operations in

.

environment behaves like a super-complex calculator. You can enter the commands at the >> command prompt. The answer appears by pressing We can use the following arithmetic operators: addition , division

, exponentiation

.

, subtraction

, multiplication

.

For example, to add two numbers 123 and 321, we simply type >>

and then press

:

Similarly, we can apply other operators to use

as an ordinary calculator:

>>

>>

>>

>>

You may add a semicolon

at the end of the statement; then

will hide the output.

For example, >> >>

Note that the symbol

is now defined as 3. We may remove it from the memory by using

or remove all variables from the memory by window, we can use

. If we want to clear the command

.

3. B ASIC F UNCTIONS The function >>

computes the principle square root of the number x. For example,

INTRODUCTION TO MATLAB

By default,

3

displays four decimal digits to its answers. But we can change the for-

mat for numeric display. (The percent symbol

is used for indicating a comment line.)

>> >>

>> >>

>> >>

We can also use the following functions in (i) Trigonometric functions:

:

,

(ii) Inverse trigonometric functions: (iii) Exponential and logarithmic functions:

,

,

,

,

, ,

, ,

. .

(base e ),

(base 10).

For trigonometric and inverse trigonometric functions, the angles are measured in radian (π radian = 180◦ ), and the constant π = 3.1415926 · ·· is defined by

(

is case-sensitive).

>>

>>

4. S OLVING B ASIC A LGEBRAIC E QUATIONS The command (i) Use (ii) Use Note that

is used for solving algebraic equations. We shall

to declare the variables. to solve the specific equations. is used to assign a value to a variable; and

shall be used for equality.

4.1. Single Variable Equations. For example, x 2 + x − 1 = 0. >> >>

INTRODUCTION TO MATLAB

We may use

or

to evaluate each element of the symbolic input to d

digits (the default is 32 digits). >>

Sometimes the exact solutions cannot be specifically displayed. For example, x 4 − 7x 3 + 3x 2 − 5x + 9 = 0. >>

Then evaluate using floating points to obtain the decimal expressions of the four roots: >>

Here the symbol

refers to the imaginary unit i where i = e πi /2 such that i 2 = −1.

Alternatively, one may use

to get the decimal expression directly:

>>

can also be used to solve symbolic equations. For example, ax 2 + bx + c = 0. >> >>

4

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4.2. Multi-Variable Equations.

5

can also solve multi-variable equations, for which

the equations shall be put together in square brackets

, so do the variables. Moreover, since

the output consists multiple variables, we shall assign the solution as a vector. For example, 3x + y = 10,

x + y = 20.

>> >>

The solution of the a general linear system in variables x and y ( ax + by = e cx + d y = f

can be obtained as follows: >> >>

5. S UMS If we need to find the sum of a sequence, use the command

. The format is

>>

For example, for the arithmetic sequence {a n } given by a n = a + (n − 1)d , the sum of its first

n terms is (2a + (n − 1)d )n/2. >> >>

>>

For the geometric sequence {a n } given by a n = ar n−1 , the sum of its first n terms is ( a(r n − 1)/(r − 1) if r 6= 1, Sn = an if r = 1.

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>> >>

∼ Moreover, we can find the sum to infinity, which is defined by

in

. Recall that

for the geometric sequence {a n } given by a n = ar n−1 , the sum to infinity is ( a/(1 − r ) if |r | < 1, S∞ = does not exist if |r | ≥ 1. >> >> >>

6. V ECTOR A vector a i + b j + c k or (a, b , c) can be defined as

or simply

.

The addition, subtraction and multiplication with numbers can be evaluated using and

respectively.

For example, let u = (1,2,3) and v = (4,5,6). >> >> >>

>>

>>

The dot product u • v of two vectors u and v is defined by >>

We can use the formula |v | = >>

p

v • v to find the norm of v :

.

,

INTRODUCTION TO MATLAB

Alternatively,

provides a command

7

for the norm of a vector.

>>

The cross product u × v is defined by

.

>>

7. F UNCTION 7.1. Standard Function. To define a function, we shall (i) give the name of the function, (ii) use to declare the name of the variable, (iii) provide the expression of the function. For example, define f (x) = x 2 : >>

Then f (2) can be evaluated by >>

Another example: g (x) = sin( x 3 )/x 2 >>

Multi-variable functions can be defined similarly by declaring more variables. For example, p h(x, y) = x 2 + y 2 : >>

To evaluate h(5,12): >>

7.2. Piecewise Function. The absolute value function f (x) = |x| is a piecewise function: ( x if x ≥ 0, f (x) = −x if x < 0, which can be defined in

:

>> >>

The general format is to use the >>

as

INTRODUCTION TO MATLAB

In the example above, the first condition is to 0”. The value

8

and this means “x is greater than or equal

is the expression of the function when the condition

defines f (x) = x when x ≥ 0. The second condition

is satisfied. It

and the second value

defines

f (x) = −x when x < 0.

In order to evaluate the value of the function at given point, for example, f (−2), we use >>

8. C URVE P LOTTING 8.1. Standard Function. Once a function is defined, a specified interval in the form

can be used to plot its graph over

. 2

For example, we plot f (x) = x on the interval (−100,100): >> >>

The command can be used even if the function is undefined somewhere on the specific interval. For example, g (x) = 1/x on (−1,1) is defined at x = 0. >> >>

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8.2. Multiple Curves. In order to plot more curves in the same coordinate system, we use the commands

and

. For example,

f (x ) = sin x (We can use

and g ( x ) = cos t ,

−2π < x < 2π.

to start a new line without breaking the command.)

>> >> >> >> >> >>

We can draw multiple graphs on the same plot. white

, black

, blue

, red

, cyan

, green

provides some basic colour options: , magenta

, yellow

to distinguish

INTRODUCTION TO MATLAB

different graphs. For example, f (x ) = x 3 − x + 1, g (x) = x 4 − 3x 2 + x, h(x) = x 5 + 0.3x 4 − 2.8x 3 − 0.3x 2 + 1.8x . >> >> >> >> >> >> >> >>

8.3. Parametric Equations. A curve may be defined using a pair of parametric equations: x = x(t ) In order to use

and

y = y( t ).

, we shall first define the functions for the x- and y-coordinates.

Recall that a unit circle can be parameterized by x = cos t , >> >> >>

y = sin t ,

0 ≤ t ≤ 2π.

10

INTRODUCTION TO MATLAB

can draw a butterfly: >> >> >>

8.4. Piecewise Function. There are two ways to plot a piecewise function. Once a piecewise function is already defined,

f (x) = >> >> >>

(

ex

can directly plot its graph:

if x < 0,

cos x if x ≥ 0.

11

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12

Alternatively, we can plot different branches on different intervals using the same colour. >> >> >> >> >> >>

8.5. Implicit Function. The command

can be used to plot graphs defined by im-

plicit functions, i.e., f (x, y) = 0. The format is: >>

For example, to plot x 3 + y 3 = 3x y, we shall first define the function f (x, y) = x 3 + y 3 − 3x y : >> >>

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9. L IMITS The limit of a function describes the behavior of a function near a point or at infinity. The limit lim f (x) can be easily evaluated with the command x→a

>>

For example, lim p x→0

>>

x 1 + 3x − 1

:

:

>>

The one-sided limits can be evaluated by specifying the direction

or

. For ex-

ample, let the floor function ⌊x⌋ denote the greatest integer smaller or equal to x. Then lim ⌊x⌋ = 1,

x→2−

lim ⌊x ⌋x = 2,

x→2+

and

lim ⌊x⌋ does not exist. x→2

>> >>

>>

>>

We can find the limit at infinity >> >>

or negative infinity

³ a ´x . For example, lim 1 + : x→∞ x

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Sometimes the limit depends on the value of the parameters. For example,     0 if a < 0, lim 2ax =

x→∞

We may use

1 if a = 0,    ∞ if a > 0.

to declare the value of the parameter.

>> >> >>

>> >>

>> >>

10. D ERIVATIVE The derivative of a function f at point a is defined as the limit f ′ (x) = lim

h→0

f (x + h) − f (x) . h

For example, if f (x) = x 2 , then f ′ (x) = 2x : >> >> >>

On the other hand,

includes the command

find the derivative of f (x) = cos(x 2 ) with respect to x : >> >> >>

for differentiation. For example,

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p x − 3x x with respect to x : p In order to differentiate g (x) = x >> >> >>

The answer seems complicated. Fortunately

provides the command

which

may help in this situation: >>

Suppose we are looking for f (4) (x), the 4th order derivative of f (x) with respect to 4, of course we may use >>

However, it may be a bit cumbersome. The following shorter commands have the same effect: >>

>>

For example, we can find the local extreme values of f (x) = x 3 − 3x 2 + x − 2. >> >> >> >>

(i) Solve f ′ (x) = 0 to obtain the stationary points. >>

(ii) Check the sign of f ′′ (x) at the stationary points. (Note that you can use in the scripts.) >>

and

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>>

p p Hence, f has a local maximum at 1 − 6/3 and a local minimum at 1 + 6/3. We can plot f (x), f ′ (x) and f ′′ (x) in the same coordinate system. >> >> >> >> >>

11. I NTEG RAL 11.1. Indefinite Integral. For a function f (x), its indefinite integral is a function F (x) such that F ′ (x) = f (x), denoted by F (x) =

Z

f (x) d x.

All indefinite integrals differ by a constant only. So we also use Z F (x ) = f ( x ) d x + C to represent the entire family of indefinite integrals of f (x), where C is an arbitrary constant. In

, we can use the command to find an indefinite integral. Z 1 1 x d x = tan−1 x + For example, +C. 2 2 2 (1 + x ) 2(x + 1) 2 >> >> >>

INTRODUCTION TO MATLAB

17

Note that the constant C is dropped in the answer. Another example: Z 5 15 3 1 sin6 x d x = sin 6x + C . x− sin 2x + sin 4x − 192 16 64 64 >> >> >>

11.2. Definite Integral. The definite integral

Zb

f (x) d x represents the net area bounded be-

a

tween the graph of y = f (x) and the x-axis from x = a to x = b. The command

can also be

used to find definite integral by specifying the lower limit a and upper limit b in the form >>

For example,

Z1 0

>>

π 1 1 dx = + . 2 2 (1 + x ) 8 4

>> >>

Another example:

Zπ 0

sin6 x d x =

5 π. 16

>> >> >>

We can verify the fundamental theorem of calculus: Z d x f (t ) d x = f (x). dx a >> >>

It is also possible to integrate over an infinite interval by specifying the lower limit and/or the upper limit as

or

. For example, Z∞ p 2 e −x d x = π. −∞

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>>

12. D IFFERENTIAL E QUATION 12.1. General Solution. In order to solve an ordinary differential equation of the form

dy = dx

f (x, y), we shall first declare that y is a function in x, and then use . dy = 1 + x + y. Note that the equal sign is represented by . For example, dx >> >>

Here

refers to an undermined constant. So the general solution of the given equation is x

y = C e − x − 2, where C is an arbitrary constant. Ordinary differential equations with higher order can be solved similarly. dy d2y +2 + 2y = e x . For example, 2 dx dx >> >>

The general solution is y = C 2 e −x sin x + C 3 e −x cos x, where C 2 ,C 3 are arbitrary constants. 12.2. Particular Solution. Suppose an initial condition is given. We can use >>

For example,

dy = 1 + x + y such that y = 1 at x = 0. dx

>> >>

For higher order differential equations, we need more initial conditions. We use >>

The condition y ′ (a) = b shall be defined as >>

For example, >>

if we set

dy d2y +8 − 9y = 0, where y(1) = 2 and y ′ (1) = 0. 2 dx dx

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>> >>

>>

12.3. System of Differential Equations. The command

can also be used for a system

of differential equations. Note that the output consists multiple functions. So the output must be in vector form: >>

For example, the following linear system consists of two functions x(t ) and y(t ) with two equations and two initial conditions: (

x ′ (t ) = 3x + 4y

y ′ (t ) = −4x + 3y

where x(0) = 2 and

y(0) = 3.

>> >>

13. P LOT TING

IN THE

S PACE

13.1. Curve in the Space. A curve in the x y z-space can be parametrized as

r ( t ) = x ( t )i + y(t )j + z(t )k,

a ≤ t ≤ b.

Plotting a space curve in the x y z-space is similar to plotting a parametrized curve in the x y plane. The only difference is to replace

by

. For example,

r (t ) = (e −t/10 sin 5t ) i + (e −t/10 cos 5t ) j + t k. >> >> >> >>

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13.2. Level Curves. A surface is defined as a two-variable function z = f (x, y). Its level curve at

z = c is the intersection of the surface with the horizontal plane at z = c; that is, f (x, y) = c. The command

plots the level curve of a two-variable function.

>>

For example, f (x, y) = sin x + cos y : >> >>

It is also possible plot the level curves of two surfaces using

and

13.3. Surface. A surface z = f (x, y) in two variables can be plotted in x y z-space using For example, f (x, y) = sin x + cos y . >> >>

.

.

INTRODUCTION TO MATLAB

13.4. Parametrized Surface. A surface can be parametrized by two parameters u and v :

r (u, v ) = x(u , v ) i + y(u, v ) j + z(u, v ) k. We can still use

to plot parametrized surface in the form

>>

For example,

r (u, v ) = u cos v i + u sin v j + v k. >> >> >> >>

21

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13.5. Implicit Function. A surface can also be defined by an implicit function, i.e., f (x, y, z) = 0.

It is almost the same as plotting a curve defined by implicit function in the x y-plane. Simply use

instead of

.

For example, plot the sphere x 2 + y 2 + z 2 = 4z and the paraboloid z = x 2 + y 2 . >> >> >> >> >> >>

14. M ULTI -VARIABLE D ERIVATIVES 14.1. Partial Derivatives. For a multi-variable function, we can find its partial derivatives by specifying the variables to be differentiated with respect to properly. ∂f ∂f ∂f For example, for f (x, y, z) = x ln(x y 2 z 3 ), its partial derivatives are given by , and ∂z ∂x ∂y >> >> >>

>>

>>

INTRODUCTION TO MATLAB

The command

23

provides the gradient vector ∇ f of a function f so that we can

obtain all the derivatives at once (the output is a column vector). >>

We may verify the mixed derivative theorem:

∂2 f ∂y ∂x

=

∂2 f : ∂x ∂y

>>

>>

Recall that the directional derivative of f (x, y, z) along a vector v at (a,b,c) is given by ∇ f (a,b,c) • v /|v |. For example, at (1,2,3), the dire...