ce206 matlab fundamentals

7/31/2019 CE206 MATLAB Fundamentals

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CE 206: Engineering

ompu a on ess ona

1.50 Credits, 3hrs/week

Dr. Tanvir Ahmed

Assistant Professor


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Course Outline

Introduction to hi-level computational programming tools- MATLAB, Mathematica etc.

Application to numerical analysis- as c ma r x compu a ons- Solving system of linear equations- Solving non-linear equations

- Solving differential Equations- Interpolation and curve-fitting- Numerical differentiation- Numerical integration

- Solving mechanics problems- Numerical solution of e uation of motion etc.

CE 206: Engg. Computation Sessional Dr. Tanvir Ahmed


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References and Resources

The MATLAB help file documentation- Provided with the MATLAB software- www.mathworks.com

Course materials in m website

- Lecture slides for CE206- An introduction to MATLAB David Griffith

- .

You will also need:- Your CE205 Course references/Lecture slides



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MATLAB basics

MATLAB can be thought of as a super-powerful graphingcalculator with many more buttons (i.e. built-in functions)

You can build up your own set of functions suited for a particular

It is an interpreted programming language

-commands are executed line by line

It is ca able of ra hicall re resentincomputational results simultaneously.

- Not possible in C++, Fortran



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MATLAB as a calculator

Basic arithmetic operators (+, , *, /) used in conjunction withbrackets ( )

-5/(4.8+5.32)^2

ans =

2)32.58.4(

5

+

Built-in-0.0488

(3+4i)*(3-4i) )43)(43( ii +

functions

ans =252/cos

s n as ncos() acos()

ans =

6.1230e-017

log() log10()

exp() sqrt() exp(acos(0.3))

ans =

)3.0(cos

e abs() round()


.


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Numbers and Formats

All computations in MATLAB are done in double precision (15


significant figures)


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Variables

Variable names and their types do not have to be declared inMATLAB.

a statement s term nate w t a sem co on ; , t e resu tsare suppressed

,digits, and underscores.

The name of variable is not accepted if it is reserved word.

Example=- = * = ^ ,

y =

-65104



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Variables

Commands involving variables

whos: lists the names and sizes of defined variables

clear: clears all variables clear name: clears the variable name

clc: clears the command window

Avoid using

ans: default variable name for the result.

pi: = 3.1415926 eps: = . e- , sma es va ue y w c wo num ers

can differ

inf: , infinity


NAN or nan: not-a-number


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Vectors

- Entries within a row are separated by spaces or commas.-Rows are separated by semicolons.- Vector properties using size( ) and length( )

a =a =

1 2 3 4 5

s ze aans =

1 5

b =2

>> length(a)ans =

6

8

5



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Other methods of vector creation

The colon operator: generally a : b : c produces a vector ofentries starting with the value a, incrementing by the value bun ge s o c

>> 3:7ans =

3 4 5 6 7

>> 0.32:0.1:0.6

ans =0.3200 0.4200 0.5200

>> -1.4:-0.3:-2ans =-1.4000 -1.7000 -2.0000

linspace (a,b,n)


, .


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Graphics: plotting functions

>> x = linspace (0,1,11);>> y = sin(3*pi*x);xy

3sin=

,10 xor

Increasing the number of elements in x



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Multiplots, titles, labels, linestyles and colors

>> plot(x,y,'w-',x,cos(3*pi*x),'g--')>> legend('Sin curve','Cos curve')>> title 'Multi- lot '>> xlabel('x axis'), ylabel('y axis')>> grid



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Subplot example

>> su p ot , p ot x,y>> xlabel('x'),ylabel('sin 3 pi x')

>> subplot(222), plot(x,cos(3*pi*x))x a e x ,y a e cos p x>> subplot(223), plot(x,sin(6*pi*x))>> xlabel('x'),ylabel('sin 6 pi x')

, ,

>> xlabel('x'),ylabel('cos 6 pi x')

subplot(m, n, p)

subplot splits the figure window into an mxn array of smallth . -is at the top left, then the numbering continues across therow



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Subplot example



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Formatted text on plots

p o e rs erms n e sequence xn g ven y xn =

[1 + 1/n]n and then graph the function (x) = x3 sin2(3x) on the interval-1 x 1

>> set(0,'Defaultaxesfontsize',16);^

Default font

changed. .>> subplot (211)>> plot(n,x,'.',[0 max(n)],exp(1)*[1 1],...-- , mar ers ze ,>> title('x_n = (1+1/n)^n','fontsize',12)>> xlabel('n'), ylabel('x_n')

changed

Subscri t>> legend('x_n','y = e^1 = 2.71828...',4)>> %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

>> subplot (212)

superscript

LineWidth>> x = -2:.02:2; y = x.^3.*sin(3*pi*x).^2;>> plot(x,y,'linewidth',2)>> le end ' = x^3sin^2 3 i x' 4

changed


>> xlabel('x)

characters


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Formatted text on plots

p o e rs erms n e sequence xn g ven y xn =

[1 + 1/n]n and then graph the function (x) = x3 sin2(3x) on the interval-1 x 1



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Adding additional plots

hold on and hold off

hold on tells MATLAB to keep the current data plotted and addthe results of an further lot commands to the ra h. Hold offreleases the hold on the figure

= *

y = sin(x);

plot(x,y,'b')

grid on;

hold on;

plot(x,exp(-x),'r:*');


h l i d il il


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Other plotting commands: semilogx, semilogy

1010

.semilogx(10.^x,x)

10

.semilogy(x, 10.^x)

108

8

9

106

5

6

7

102

104

2

3

4

0 1 2 3 4 5 6 7 8 9 110

0

100

102

104

106

108

1010

0

1


O h l i d l l


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Other plotting commands: loglog

x = logspace(-1,2);

50

og og x,exp x ,'-s'grid on

1040

1030

10

1020

-1 0 1 210

0

10


M t i


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Matrices

A 2-D array, or matrix, of data is entered row by row, with spaces(or commas) separating entries within the row and semicolonsseparating the rows:

>> A = [1 2 3; 4 5 6; 7 8 9]

=1 2 34 5 6

7 8 9

A(j,k) gives jth row, kth column, A([1,3], :) all of rows 1 and 3A(:, 1) first column


N i l A C t ti


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Numerical Array Concatenation

a=[1 2;3 4]a =

1 2

3 4

Use square

brackets [ ]

cat_a=[a, 2*a; 3*a, 4*a; 5*a, 6*a]

cat_a =

3 4 6 83 6 4 8 4*a5 10 6 12

15 20 18 24

Use [ ] to combine existing arrays as matrix elements


M M t i


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More on Matrices

zeros(n) Returns a n n matrix of zeros

zeros(m,n) Returns a m n matrix of zeros

rand m n

eye(m,n) Returns a m n Identity matrix

ones(n) Returns a n

n matrix of onesones(m,n) Returns a m n matrix of ones

size(A)

For a m n matrix A, returns the row

vector [m,n] containing the number of rowsan co umns n ma r x

length(A)Returns the larger of the number of rows orcolumns in A


Diagonal Matrix


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Diagonal Matrix

First define a vector containing the values of the diagonalentries (in order) then use the diag function to form the matrix

>> = - , D = ag

d =-3 4 2D =

-3 0 0

0 4 00 0 2

This command is useful when we need to construct very large

matrices.

If A is a matrix, diag(A) extracts the diagonal elements of thematrix.


Sparse Matrix


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Sparse Matrix

These are generally large matrices that have a very smallproportion of non-zero entries

>> i = [1, 3, 5]; j = [2,3,4];

>> v = [10 11 12];>> S = sparse (i,j,v)S =(1,2) 10 - -

(3,3) 11(5,4) 12

=

only 3 non-zero values:S(1,2) = 10,

T =

0 10 0 0

S(3,3) = 11 andS(5,4) = 12

0 0 11 00 0 0 0 The full command displays the sparse


matr x

Matrix operations


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Matrix operations

Transpose B = A

Subtraction

C = A+B C = A-B

ScalarMultiplication

= , w ere s a sca ar

Matrix= *

Multiplication

Matrix Inverse B = inv(A), A must be a square matrix

Matrix powers B = A * A, A must be a square matrix

Determinant det(A), A must be a square matrix

Operators * ^ and / have two modes of operation-


-element-wise

Standard matrix product operation


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Standard matrix product operation


Elementwise matrix operations


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Elementwise matrix operations

To do element-wise operations, use the dot .(.*, ./,^ . .


for loop


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for loop

Used when we want to repeat a segment of the code several times

Exam le: Test the assertion that theratio of the two successive values in theFibonacci series approach the golden

ratio o (5-1)/2. i.e. fn/fn-1 = (5-1)/2 where n = 3,4,5..

>> F(1) = 0; F(2) = 1;>> for i = 3:20F(i) = F(i-1) + F(i-2);end

>> plot(1:19, F(1:19)./F(2:20),'o' )>> o on, x a e 'n'>> plot(1:19, F(1:19)./F(2:20),'-' )>> legend('Ratio of terms f_{n-1}/f_n')


p o , sqr - , , --

while loop


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while loop

when we want to repeat a segment of the code several times until

some logical condition is satisfied.

The while loop is used when we do not know for certain how many

iterations may be neededExample: What is the greatest value of n that can be used in the sum

and get a value less than 100?

>> S = 1; n = 1;>> while S+ (n+1)^2 < 100

n = n+1; S = S + n^2;en>> [n, S]ans =


Logical and relational operators


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Logical and relational operators

== Equal

~= Not equal

> Greater than

-2.0000 3.1416 5.0000

-5.0000 -3.0000 -1.0000

< Less than

x xans =0 1 0

>= reater or equa

> x > 3 | x == -3ans =

& && And

Or

0 1 10 1 0

Matlab represents true and false by means of the integers 0 and 1


Flow control using if/else/elseif


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Flow control usingif/else/elseif

Example: Given *

count=0;

for n=1:length(x) , , ,of the entries are positive?

count=count+1;

end


en

The find command for vectors


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The findcommand for vectors

returns a list of the positions (indices) of the elements of avector satisfying a given condition.

Example: Produce a plot for y = e-x2 sin(3x) and mark all the points that havea value of y greater than 0.2

-

0.8

1

.>> y = sin(3*pi*x).*exp(-x.^2);>> k = find(y > 0.2)

0.4

0.6

=

Columns 1 through 129 10 11 12 13 22 23

-0.2

0

0.224 25 26 27 36Columns 13 through 15

37 38 39

-0.8

-0.6

-0.4

>> lot x ':' hold on

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1


>> plot(x(k),y(k),'o)

Avoiding loops: efficient coding


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o d g oops: e c e t cod g

Example 1: Given x= sin(linspace(0,10*pi,100)), how many of the entries arepositive?

count=0;for n=1:length(x)

count=count+1;end

Count=length(find(x>0));

en

Example 2 find the sum: 12 + 22 + 32 + . +1002

sum_sq=0;sum_sq = sum_sq + n^2;

end

Sum_sq=sum((1:100).^2)


Avoiding loops: efficient coding


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g p g

Built-in functions (e.g. find, sum) will make it faster to writeand execute

count=0;

for n=1:len th xif x(n)>0count=count+1;

end

Vectorized code is more efficient for MATLAB

Use indexing and matrix operations to avoid loops


Advanced graphics: plotting surfaces


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g p p g

- The commandmeshgridis used to construct the (x, y) gridpoints

at certain intervals- ,

-Use a surface plot feature (e.g.mesh,surf) to plot the surface

= 2 2-

[X,Y] = meshgrid(2:.2:4, 1:.2:3);

Saddle

Z = (X-3).^2-(Y-2).^2;mesh(X,Y,Z)title('Saddle'), 0.5

1

xlabel('x'),ylabel('y')

-0.5

0

42.5

3-1


2

2.5

3

.

1

1.5

xy

Advanced graphics: surf and shading


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g p gSaddle

, . , .

Z = (X-3).^2-(Y-2).^2;surf(X,Y,Z)

0.5

1

t t e 'Sa eshading faceted

-0.5

0

shading flat

colorma ( ra ) 2.53

3.5

4

1.5

2

2.5

3-1

xy

Saddle

1

Saddle

0

0.5

0

0.5

42.5

3-1

-0.5

42.5

3-1

-0.5


2

2.5

3

.

11.5

2

xy2

2.5

3

.

11.5

2

xy

Advanced graphics: contour plot


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g p pa es e same argumen s as sur ormes

Example: Plot the contour of the surface defined by z = (sinx) (cosy) for x and - 3

x=-pi:0.1:pi;y=-pi:0.1:pi;

1

2

, =mes gr x,y ;Z =sin(X).*cos(Y);contour(X,Y,Z,'LineWidth',2) -1

0

-3 -2 -1 0 1 2 3

-3

-2

1

0

0.5

4-1

-0.5

mesh(X,Y,Z); hold on;contour(X,Y,Z)

CE 206: Engg. Computation Sessional Dr. Tanvir Ahmed-4

-2

0

2

-4

-20

Specialized plotting functions


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p p g

polar-to make polar plotsolar 0 0 01 2* i cos 0 0 01 2* i *2

bar-to make bar graphs

, ,

quiver-to add velocity vectors to a plot

=, ,

quiver(X,Y,rand(10),rand(10));

stairs- lot iecewise constant functions

stairs(1:10,rand(1,10));

-

fill([0 1 0.5],[0 0 1],'r');


Scripting


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p g

Script files are ASCII (text) files containing MATLAB commands.

Commonly known as m-files because they have a .m extensionCE 206: Engg. Computation Sessional Dr. Tanvir Ahmed

Function m-files


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Example: Write a function m-file which calculates the area of a triangle having sides

a, b and c using the formula: Area = Where s = (a+b+c)/2))()(( csbsass

List of out ut s

The function name. Also the name of the m-file

where the function definition will be stored.

function [A] = area(a,b,c)List of inputs

Compute t e area o a tr ang e w ose

% sides have length a, b and c.% Inputs:

Purpose of thefunction and how itcan be used. Mainl

% a,b,c: Lengths of sides% Output:

% A: area of trian le

to aid the futureusers

s = (a+b+c)/2;A = sqrt(s*(s-a)*(s-b)*(s-c)); The code


Function overloading


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MATLAB functions are generally overloaded

Can take a variable number of inputs

You can overload your own functions by having variable input

and out ut ar uments

The following function plots a sine wave with frequency f1 on the range [0, 2],

function plotSin(f1,f2)= * *figure

if nargin == 1

built in

function ,elseif nargin == 2

disp('Two inputs were given');contains thenumber of


ennputs

ce206 matlab fundamentals

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