Sit04 Mit Matlab

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EE 475 575 Digital Signal Processing, Spring 2010 (Dr. Mousavinezhad)

These MATLAB tutorials are from MITs Open Courseware Site:

http://ocw.mit.edu


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MIT OpenCourseWare

http://ocw.mit.edu

6.094 Introduction to MATLAB

January (IAP) 2009

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
http://ocw.mit.edu/http://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/


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6.094Introduction to Programming in MATLAB

Sourav DeyDanilo epanovi

Ankit PatelPatrick Ho

IAP 2009

Lecture 1: Variables, Operations, andPlotting


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Course Layout

Lectures (7pm-9pm)1: Variables, Operations and Plotting2: Visualization & Programming3: Solving Equations, Fitting4: Advanced Methods


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Course Layout

Problem Sets / Office HoursOne per day, should take about 3 hours to do

Submit doc or pdf (include pertinent code)

Requirements for passing

Attend all lecturesComplete all problem sets (FAIL, Check or +)

PrerequisitesBasic familiarity with programmingBasic linear algebra, differential equations, and

probability


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Outline

(1) Getting Started

(2) Making Variables

(3) Manipulating Variables

(4) Basic Plotting


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Getting Started

To get MATLAB Student Version for yourself https://msca.mit.edu/cgi-bin/matlab

Use VPN client to enable off-campus accessNote: MIT certificates are required

Open up MATLAB for Windows

Through the START Menu

On Athena

add matlabmatlab &


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Command Window

Current directory

Workspace

Command History

Courtesy of The MathWorks, Inc. Used with permission.


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Customization

File PreferencesAllows you personalize your MATLAB experience



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MATLAB Basics

MATLAB can be thought of as a super-powerfulgraphing calculator

Remember the TI-83 from calculus?

With many more buttons (built-in functions)

In addition it is a programming language MATLAB is an interpreted language, like

Scheme Commands executed line by line


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Conversing with MATLAB

who

MATLAB replies with the variables in your workspace

what

MATLAB replies with the current directory andMATLAB files in the directory

why

help

The most important function for learning MATLAB onyour own

More on help later


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Outline

(1) Getting Started



(4) Basic Plotting


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Variable Types

MATLAB is a weakly typed languageNo need to initialize variables!

MATLAB supports various types, the most often used are 3.84

64-bit double (default) a

16-bit char

Most variables youll deal with will be arrays or matrices ofdoubles or chars

Other types are also supported: complex, symbolic, 16-bitand 8 bit integers, etc.


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Naming variables

To create a variable, simply assign a value to a name: var1=3.14

myString=hello world

Variable names first character must be a LETTER

after that, any combination of letters, numbers and _CASE SENSITIVE! (var1 is different fromVar1)

Built-in variablesi and j can be used to indicate complex numberspi has the value 3.1415926

ans stores the last unassigned value (like on a calculator)

Inf and -Inf are positive and negative infinity

NaN represents Not a Number


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Hello World

Here are several flavors of Hello World to introduce MATLAB

MATLAB will display strings automatically Hello 6.094

To remove ans =, use disp() disp('Hello 6.094')

sprintf() allows you to mix strings with variables

class=6.094; disp(sprintf('Hello %g', class))

The format is C-syntax


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Scalars

A variable can be given a value explicitly a = 10

shows up in workspace!

Or as a function of explicit values and existing variables c = 1.3*45-2*a

To suppress output, end the line with a semicolon

cooldude = 13/3;


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Arrays

Like other programming languages, arrays are animportant part of MATLAB

Two types of arrays

(1) matrix of numbers (either double or complex)

(2) cell array of objects (more advanced data structure)

MATLAB makes vectors easy!Thats its power!


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Row Vectors

Row vector: comma or space separated values betweenbrackets row = [1 2 5.4 -6.6];

row = [1, 2, 5.4, -6.6];

Command window:

Workspace:



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Column Vectors

Column vector: semicolon separated values betweenbrackets column = [4;2;7;4];

Command window:

Workspace:



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Matrices

Make matrices like vectors

Element by element a= [1 2;3 4];

By concatenating vectors or matrices (dimension matters) a = [1 2];

b = [3 4];

c = [5;6];

d = [a;b];

e = [d c];

f = [[e e];[a b a]];

1 2

3 4a

=


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save/clear/load

Use save to save variables to a file save myfile a b

saves variables a and b to the file myfile.mat myfile.mat file in the current directory Default working directory is

\MATLAB\work

Create own folder and change working directory to it MyDocuments\6.094\day1

Use clear to remove variables from environment clear a b

look at workspace, the variables a and b are gone

Use load to load variable bindings into the environment load myfile look at workspace, the variables a and b are back

Can do the same for entire environment

save myenv; clear all; load myenv;


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Exercise: Variables

Do the following 5 things:Create the variable r as a row vector with values 1 4

7 10 13

Create the variable c as a column vector with values13 10 7 4 1Save these two variables to file varExclear the workspace load the two variables you just created

r=[1 4 7 10 13];

c=[13; 10; 7; 4; 1];

save varEx r c

clear r c

load varEx


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Outline

(1) Getting Started



(4) Basic Plotting


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Basic Scalar Operations

Arithmetic operations (+,-,*,/) 7/45

(1+i)*(2+i)

1 / 0

0 / 0

Exponentiation (^) 4^2

(3+4*j)^2

Complicated expressions, use parentheses ((2+3)*3)^0.1

Multiplication is NOT implicit given parentheses 3(1+0.7) gives an error

To clear cluttered command window Clc


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Built-in Functions

MATLAB has an enormous library of built-in functions

Call using parentheses passing parameter to function sqrt(2)

log(2), log10(0.23)

cos(1.2), atan(-.8)

exp(2+4*i)

round(1.4), floor(3.3), ceil(4.23)

angle(i); abs(1+i);


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Help/Docs

To get info on how to use a function: help sin

Help contains related functions To get a nicer version of help with examples and easy-to-

read descriptions: doc sin

To search for a function by specifying keywords: doc + Search tab

lookfor hyperbolic

One-word description of what

you're looking for


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Exercise: Scalars

Verify that e^(i*x) = cos(x) + i*sin(x) for a few values of x.

x = pi/3;

a = exp(i*x)

b = cos(x)+ i*sin(x)

a-b


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size & length

You can tell the difference between a row and a columnvector by:

Looking in the workspace

Displaying the variable in the command windowUsing the size function

To get a vector's length, use the length function


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transpose

The transpose operators turns a column vector into a rowvector and vice versa a = [1 2 3 4]

transpose(a)

Can use dot-apostrophe as short-cut a.'

The apostrophe gives the Hermitian-transpose, i.e.transposes and conjugates all complex numbers

a = [1+j 2+3*j] a'

a.'

For vectors of real numbers .' and ' give same result

Addi i d S b i


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Addition and Subtraction

Addition and subtraction are element-wise; sizes mustmatch (unless one is a scalar):

The following would give an error c = row + column

Use the transpose to make sizes compatible

c = row + column c = row + column

Can sum up or multiply elements of vector s=sum(row);

p=prod(row);

[ ]

[ ]

[ ]

12 3 32 11

2 11 30 32

14 14 2 21

+

=

12 3 9

1 1 2

10 13 23

0 33 33

=

El t Wi F ti


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Element-Wise Functions

All the functions that work on scalars also work on vectors t = [1 2 3];

f = exp(t);

is the same as f = [exp(1) exp(2) exp(3)];

If in doubt, check a functions help file to see if it handlesvectors elementwise

Operators (* / ^) have two modes of operation element-wise standard

O t l t i


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Operators: element-wise

To do element-wise operations, use the dot. BOTHdimensions must match (unless one is scalar)! a=[1 2 3];b=[4;2;1];

a.*b, a./b, a.^b all errors

a.*b, a./b, a.^(b) all valid

[ ]4

1 2 3 2

1

1 4 4

2 2 4

3 1 3

3 1 3 1 3 1

.* ERROR

.*

.*

=

=

=

1 1 1 1 2 3 1 2 3

2 2 2 1 2 3 2 4 6

3 3 3 1 2 3 3 6 9

3 3 3 3 3 3

.*

.*

=

=

2 2

2 2

1 2 1 22

3 4 3 4.^

Can be any dimension

=

O t t d d


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Operators: standard

Multiplication can be done in a standard way or element-wise Standard multiplication (*) is either a dot-product or an outer-

product

Remember from linear algebra: inner dimensions must MATCH!! Standard exponentiation (^) implicitly uses *

Can only be done on square matrices or scalars

Left and right division (/ \) is same as multiplying by inverse

Our recommendation: just multiply by inverse (more on thislater)

[ ]

4

1 2 3 2 11

1

1 3 3 1 1 1

*

*

=

=

1 1 1 1 2 3 3 6 9

2 2 2 1 2 3 6 12 18

3 3 3 1 2 3 9 18 27

3 3 3 3 3 3

*

*

=

=

1 2 1 2 1 22

3 4 3 4 3 4^ *

Must be square to do powers

=

E e cise Vecto Ope ations


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Exercise: Vector Operations

Find the inner product between [1 2 3] and [3 5 4] a=[1 2 3]*[3 5 4]

Multiply the same two vectors element-wiseb=[1 2 3].*[3 5 4]

Calculate the natural log of each element of the resultingvector c=log(b)

Automatic Initialization


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Automatic Initialization

Initialize a vector of ones, zeros, or random numbers o=ones(1,10)

row vector with 10 elements, all 1 z=zeros(23,1)

column vector with 23 elements, all 0 r=rand(1,45)

row vector with 45 elements (uniform [0,1]) n=nan(1,69)

row vector of NaNs (useful for representing uninitializedvariables)

The general function call is:var=zer os(M,N) ;

Number of rows Number of columns


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Exercise: Vector Functions


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Exercise: Vector Functions

Make a vector that has 10,000 samples off(x) = e^{-x}*cos(x), for x between 0 and 10.

x = linspace(0,10,10000);

f = exp(-x).*cos(x);

Vector Indexing


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Vector Indexing

MATLAB indexing starts with 1, not 0We will not respond to any emails where this is the

problem.

a(n) returns the nth element

The index argument can be a vector. In this case, eachelement is looked up individually, and returned as a vectorof the same size as the index vector. x=[12 13 5 8];

a=x(2:3); a=[13 5];

b=x(1:end-1); b=[12 13 5];

[ ]13 5 9 10

a(1) a(2) a(3) a(4)

Matrix Indexing


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Matrix Indexing

Matrices can be indexed in two ways using subscripts (row and column) using linear indices (as if matrix is a vector)

Matrix indexing: subscripts or linear indices

Picking submatricesA = rand(5) % shorthand for 5x5 matrix

A(1:3,1:2) % specify contiguous submatrix

A([1 5 3], [1 4]) % specify rows and columns

14 33

9 8

b(1)

b(2)

b(3)

b(4)

14 33

9 8

b(1,1)

b(2,1)

b(1,2)

b(2,2)

Advanced Indexing 1


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Advanced Indexing 1

The index argument can be a matrix. In this case, eachelement is looked up individually, and returned as a matrixof the same size as the index matrix.

a=[-1 10 3 -2];

b=a([1 2 4;3 4 2]);

To select rows or columns of a matrix, use the :

d=c(1,:); d=[12 5];

e=c(:,2); e=[5;13];

c(2,:)=[3 6]; %replaces second row of c

1 10 2

3 2 10b

=

12 5

2 13

c =

Advanced Indexing 2


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Advanced Indexing 2

MATLAB contains functions to help you find desired valueswithin a vector or matrix vec = [1 5 3 9 7]

To get the minimum value and its index: [minVal,minInd] = min(vec);

To get the maximum value and its index:

[maxVal,maxInd] = max(vec); To find any the indices of specific values or ranges

ind = find(vec == 9);

ind = find(vec > 2 & vec < 6);

find expressions can be very complex, more on this later

To convert between subscripts and indices, use ind2sub,and sub2ind. Look up help to see how to use them.

Exercise: Vector Indexing


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Exercise: Vector Indexing

Evaluate a sine wave at 1,000 points between 0 and 2*pi. Whats the value at

Index 55 Indices 100 through 110

Find the index of the minimum value, the maximum value, and

values between -0.001 and 0.001

x = linspace(0,2*pi,1000);

y=sin(x); y(55)

y(100:110)

[minVal,minInd]=min(y)

[maxVal,maxInd]=max(y)

inds=find(y>-0.001 & y


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BONUS Exercise: Matrices

Make a 3x100 matrix of zeros, and a vector x that has 100 valuesbetween 0 and 10mat=zeros(3,100);

x=linspace(0,10,100);

Replace the first row of the matrix with cos(x)mat(1,:)=cos(x);

Replace the second row of the matrix with log((x+2)^2)mat(2,:)=log((x+2).^2);

Replace the third row of the matrix with a random vector of thecorrect sizemat(3,:)=rand(1,100);

Use the sum function to compute row and column sums of mat(see help) rs = sum(mat,2);

cs = sum(mat); % default dimension is 1

Outline


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Outline

(1) Getting Started



(4) Basic Plotting

Plotting Vectors


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Plotting Vectors

Example x=linspace(0,4*pi,10);

y=sin(x);

Plot values against their indexplot(y);

Usually we want to plot y versus xplot(x,y);

MATLAB makes visualizing datafun and easy!

What does plot do?


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What does plot do?

plot generates dots at each (x,y) pair and then connects the dotswith a line

To make plot of a function look smoother, evaluate at more points x=linspace(0,4*pi,1000);

plot(x,sin(x));

x and y vectors must be same size or else youll get an errorplot([1 2], [1 2 3])

error!!

10 x values:

0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1000 x values:

Plot Options


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Plot Options

Can change the line color, marker style, and line style byadding a string argumentplot(x,y,k.-);

Can plot without connecting the dots by omitting line style

argumentplot(x,y,.)

Look at help plot for a full list of colors, markers, andlinestyles

color marker line-style

Other Useful plot Commands


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p

Much more on this in Lecture 2, for now some simplecommands

To plot two lines on the same graph hold on; To plot on a new figure

figure;

plot(x,y);

Play with the figure GUI to learn more add axis labels add a title add a grid zoom in/zoom out

Exercise: Plotting


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g

Plot f(x) = e^x*cos(x) on the interval x = [0 10]. Use a redsolid line with a suitable number of points to get a goodresolution.

x=0:.01:10;

plot(x,exp(x).*cos(x),r);

End of Lecture 1


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(1) Getting Started



(4) Basic Plotting

Hope that wasnt too much!!

MIT OpenCourseWare

http://ocw.mit.edu
http://ocw.mit.edu/http://ocw.mit.edu/


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6.094 Introduction to MATLABJanuary (IAP) 2009

http://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/terms


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Lecture 2: Visualization and Programming



IAP 2009

Outline


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(1) Plotting Continued

(2) Scripts

(3) Functions

(4) Flow Control

Cartesian Plots


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We have already seen the plot functionx=-pi:pi/100:pi;

y=cos(4*x).*sin(10*x).*exp(-abs(x));

plot(x,y,'k-');

The same syntax applies for semilog and loglog plots

semilogx(x,y,'k');semilogy(y,'r.-');

loglog(x,y);

For example:x=0:100;

semilogy(x,exp(x),'k.-');0 10 20 30 40 50 60 70 80 90 100

100

1010

1020

1030

1040

1050

Playing with the Plot


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to select linesand delete orchange

propertiesto zoom in/out

to slide the plotaround

to see all plottools at once


Line and Marker Options


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Everything on a line can be customizedplot(x,y,'--rs','LineWidth',2,...

'MarkerEdgeColor','k',...

'MarkerFaceColor','g',...

'MarkerSize',10)

See doc line for a full list ofproperties that can be specified

-4 -3 -2 -1 0 1 2 3 4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Labels


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Last time we saw how to add titles and labels using the GUI. Canalso do it command-line:title('Stress-Strain');

xlabel('Force (N)');

For multiple lines, add a legend entry for each linelegend('Steel','Aluminum','Tungsten');

Can specify font and size for the textylabel('Distance (m)','FontSize',14,...

'FontName','Helvetica');

use ... to break long commands across multiple lines

To put parameter values into labels, need to use num2str andconcatenate:str = [Strength of ' num2str(d) 'cm diameter rod'];

title(str)

Axis


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A grid makes it easier to read valuesgrid on

xlim sets only the x axis limitsxlim([-pi pi]);

ylim sets only the y axis limitsylim([-1 1]);

To specify both at once, use axis:axis([-pi pi -1 1]);

sets the x axis limits between -pi and pi and the y axis limitsbetween -1 and 1

Can specify tickmarksset(gca,'XTick', linspace(-pi,pi,3))

see doc axes for a list of properties you can set this way more on advanced figure customization in lecture 4

Axis Modes


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Built-in axis modes

axis square

makes the current axis look like a boxaxis tight

fits axes to dataaxis equalmakes x and y scales the same

axis xy

puts the origin in the bottom left corner (default)

axis ij

puts the origin in the top left corner (for viewing matrices)

Multiple Plots in one Figure


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Use the figure command to open a new figurefigure

or activate an open figurefigure(1)

To have multiple axes in one figuresubplot(2,3,1) or subplot(231)

makes a figure with 2 rows and three columns of axes, andactivates the first axis for plotting

each axis can have labels, a legend, and a titlesubplot(2,3,4:6)

activating a range of axes fuses them into one

To close existing figuresclose([1 3])

closes figures 1 and 3

close all closes all figures (useful in scripts/functions)

Copy/Paste Figures


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Figures can be pasted into other apps (word, ppt, etc) Edit copy options figure copy templateChange font sizes, line properties; presets for word and ppt

Edit copy figure to copy figure Paste into document of interest


Saving Figures


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Figures can be saved in many formats. The common onesare:

.fig preserves allinformation

.bmp uncompressedimage

.eps high-qualityscaleable format

.pdfcompressed

imageCourtesy of The MathWorks, Inc.

Used with permission.

Figures: Exercise


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Open a figure and plot a sine wave over two periods withdata points at 0, pi/8, 2pi/8 . Use black squares asmarkers and a dashed red line of thickness 2 as the line

figureplot(0:pi/4:4*pi,sin(0:pi/4:4*pi),'rs--',...

'LineWidth',2,'MarkerFaceColor','k');

Save the figure as a pdf

View with pdf viewer.

Visualizing matrices


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Any matrix can be visualized as an imagemat=reshape(1:10000,100,100);

imagesc(mat);

colorbar

imagesc automatically scales the values to span the entire

colormap

Can set limits for the color axis (analogous to xlim, ylim)caxis([3000 7000])

Colormaps


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You can change the colormap:imagesc(mat) default map isjet

colormap(gray)

colormap(cool)

colormap(hot(256))

See help hot for a list

Can define custom colormapmap=zeros(256,3);

map(:,2)=(0:255)/255;

colormap(map);

Images: Exercise


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Construct a Discrete Fourier Transform Matrix of size 128using dftmtx

Display the phase of this matrix as an image using a hot

colormap with 256 colors

dMat=dftmtx(128);

phase=angle(dMat);

imagesc(phase);

colormap(hot(256));

3D Line Plots


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-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-10

-5

0

5

10

We can plot in 3 dimensions just as easily as in 2time=0:0.001:4*pi;

x=sin(time);

y=cos(time);

z=time;

plot3(x,y,z,'k','LineWidth',2);

zlabel('Time');

Use tools on figure to rotate it Can set limits on all 3 axes

xlim, ylim, zlim

Surface Plots


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It is more common to visualize surfaces in 3D

Example:

surfputs vertices at specified points in space x,y,z, and

connects all the vertices to make a surface

The vertices can be denoted by matrices X,Y,Z

How can we make these matrices loop (DUMB) built-in function: meshgrid

( ) ( ) ( )

[ ] [ ]

f x, y sin x cos y

x , ; y ,

=

surf


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Make the x and y vectorsx=-pi:0.1:pi;

y=-pi:0.1:pi;

Use meshgrid to make matrices (this is the same as loop)[X,Y]=meshgrid(x,y);

To get function values,evaluate the matricesZ =sin(X).*cos(Y);

Plot the surfacesurf(X,Y,Z)

surf(x,y,Z);

surf Options


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See help surffor more options There are three types of surface shading

shading faceted

shading flat

shading interp

You can change colormaps

colormap(gray)

contour


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You can make surfaces two-dimensional by using contourcontour(X,Y,Z,'LineWidth',2)

takes same arguments as surf

color indicates height can modify linestyle properties can set colormap

hold on

mesh(X,Y,Z)

Exercise: 3-D Plots


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Plot exp(-.1(x^2+y^2))*sin(xy) for x,y in [2*pi,2*pi]with interpolated shading and a hot colormap:

x=-2*pi:0.1:2*pi;y=-2*pi:0.1:2*pi;

[X,Y]=meshgrid(x,y);

Z =exp(-.1*(X.^2+Y.^2)).*sin(X.*Y);

surf(X,Y,Z);

shading interp

colormap hot

Specialized Plotting Functions


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MATLAB has a lot of specialized plotting functions polar-to make polar plots

polar(0:0.01:2*pi,cos((0:0.01:2*pi)*2))

bar-to make bar graphsbar(1:10,rand(1,10)); quiver-to add velocity vectors to a plot

[X,Y]=meshgrid(1:10,1:10);

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

stairs-plot piecewise constant functionsstairs(1:10,rand(1,10));

fill-draws and fills a polygon with specified verticesfill([0 1 0.5],[0 0 1],'r'); see help on these functions for syntax doc specgraph for a complete list

Outline


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(2) Scripts

(3) Functions(4) Flow Control

Scripts: Overview


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Scripts arewritten in the MATLAB editor saved as MATLAB files (.m extension)

evaluated line by line

To create an MATLAB file from command-lineedit myScript.m

or click


Scripts: the Editor


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* Means that it's not savedLine numbers

Debugging tools

Comments

MATLABfile path

Help file

Possible breakpointsCourtesy of The MathWorks, Inc. Used with permission.

Scripts: Good Practice


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Take advantage of "smart indent" option

Keep code clean

Use built-in functionsVectorize, vectorize, vectorizeWhen making large matrices, allocate space first

Use nan or zeros to make a matrix of the desired size

Keep constants at the top of the MATLAB file

COMMENT!Anything following a % is seen as a comment The first contiguous comment becomes the script's help fileComment thoroughly to avoid wasting time later

Hello World


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Here are several flavors of Hello World to introduce MATLAB

MATLAB will display strings automatically

Hello 6.094

To remove ans =, use disp()disp('Hello 6.094')

sprintf() allows you to mix strings with variablesclass=6.094;

disp(sprintf('Hello %g', class))

The format is C-syntax


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User-defined Functions


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Functions look exactly like scripts, but for ONE difference Functions must have a function declaration

Help file

Function declarationInputsOutputs


User-defined Functions


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Some comments about the function declaration

No need for return: MATLAB returns the variables whosenames match those in the function declaration

Variable scope: Any variables created within the functionbut not returned disappear after the function stops running

Can have variable input arguments (see help varargin)

function [x, y, z] = funName(in1, in2)

Must have the reservedword: function

Function name shouldmatch MATLAB filenameIf more than one output,

must be in brackets

Inputs must be specified

Functions: Exercise


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Take the script we wrote to calculate the student's overallscore and make it into a function

The inputs should be the scores row vector the weight row vector, with the same length as scores

The output should beA scalar: the overall score

Assume the user knows the input constraints (no need to

check if the inputs are in the correct format\size)

Name the function overallScore.m

Functions: Exercise


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Functions


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We're familiar withzeros

size

length

sum

Look at the help file for size by typing

help size

The help file describes several ways to invoke the functionD = SIZE(X) [M,N] = SIZE(X) [M1,M2,M3,...,MN] = SIZE(X)M = SIZE(X,DIM)

Functions


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MATLAB functions are generally overloadedCan take a variable number of inputsCan return a variable number of outputs

What would the following commands return:a=zeros(2,4,8);

D=size(a)

[m,n]=size(a)

[x,y,z]=size(a)

m2=size(a,2)

Take advantage of overloaded methods to make your codecleaner!

Outline


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(2) Scripts


Relational Operators

tl


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MATLAB uses mostlystandard relational operators equal == not equal ~= greater than > less than < greater or equal >= less or equal


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Basic flow-control, common to all languages MATLAB syntax is somewhat unique

IFifcond

commands

end

ELSEifcond

commands1

elsecommands2

end

ELSEIFifcond1

commands1

elseifcond2commands2

else

commands3end

No need for parentheses: command blocks are between

reserved words

Conditional statement:

evaluates to true or false

for


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for loops: use for a definite number of iterations MATLAB syntax:

for n=1:100commands

end

The loop variable Is defined as a vector Is a scalar within the command block

Does not have to have consecutive values The command block

Anything between the for line and the end

Loop variable

Command block

while

Th hil i lik l f l


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The while is like a more general for loop:Don't need to know number of iterations

The command block will execute while the conditionalexpression is true

Beware of infinite loops!

WHILE

while cond

commandsend

Exercise: Control-Flow

W it f ti t l l t th f t i l f i t N i


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Write a function to calculate the factorial of an integer N using aloop (you can use a for or while loop). If the input is less than 0,return NaN. Test it using some values.

function a = factorial(N)

if N


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find is a very important functionReturns indices of nonzero valuesCan simplify code and help avoid loops

Basic syntax: index=find(cond)x=rand(1,100);

inds = find(x>0.4 & x0.4 returns a vector with 1 where true and 0 where false x


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Given x= sin(linspace(0,10*pi,100)), how many of theentries are positive?

Using a loop and if/elsecount=0;

for n=1:length(x)

ifx(n)>0count=count+1;

end

end

Being more clevercount=length(find(x>0));

length(x) Loop time Find time

100 0.01 0

10,000 0.1 0

100,000 0.22 0

1,000,000 1.5 0.04

Avoid loops like the plague!

Built-in functions will make it faster to write and execute

Efficient Code

Avoid loops whenever possible


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Avoid loops whenever possible This is referred to as vectorization

Vectorized code is more efficient for MATLAB

Use indexing and matrix operations to avoid loops For example:a=rand(1,100);

b=zeros(1,100);

for n=1:100

if n==1

b(n)=a(n);

else

b(n)=a(n-1)+a(n);

end

endSlow and complicated

a=rand(1,100);

b=[0 a(1:end-1)]+a;

Efficient and clean


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End of Lecture 2



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(2) Scripts


Vectorization makescoding fun!

MIT OpenCourseWare

http://ocw.mit.edu


January (IAP) 2009
http://ocw.mit.edu/http://ocw.mit.edu/


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6 094Introduction to Programming in MATLAB


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Lecture 3 : Solving Equations and Curve Fitting

Sourav Dey

Danilo epanoviAnkit PatelPatrick Ho

IAP 2009

Outline

(1) Linear Algebra


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(1) Linear Algebra

(2) Polynomials

(3) Optimization(4) Differentiation/Integration

(5) Differential Equations

Systems of Linear Equations

Given a system of linear equations MATLAB makes linearalgebra fun!


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Given a system of linear equations x+2y-3z=5 -3x-y+z=-8 x-y+z=0

Construct matrices so the system is described by Ax=bA=[1 2 -3;-3 -1 1;1 -1 1];b=[5;-8;0];

And solve with a single line of code!x=A\b; x is a 3x1 vector containing the values of x, y, and z

The \ will work with square or rectangular systems. Gives least squares solution for rectangular systems. Solutiondepends on whether the system is over or underdetermined.

MATLAB makes linearalgebra fun!

More Linear Algebra

Given a matrix


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Given a matrixmat=[1 2 -3;-3 -1 1;1 -1 1];

Calculate the rank of a matrixr=rank(mat);

the number of linearly independent rows or columns

Calculate the determinantd=det(mat);mat must be square if determinant is nonzero, matrix is invertible

Get the matrix inverseE=inv(mat);

if an equation is of the form A*x=b with A a square matrix,x=A\b is the same as x=inv(A)*b

Matrix Decompositions

MATLAB has built-in matrix decomposition methods


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MATLAB has built in matrix decomposition methods

The most common ones are

[V,D]=eig(X) Eigenvalue decomposition

[U,S,V]=svd(X)

Singular value decomposition[Q,R]=qr(X)

QR decomposition

Exercise: Linear Algebra

Solve the following systems of equations:


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So e t e o o g syste s o equat o s

System 1: x+4y=34 -3x+y=2

System 2: 2x-2y=4 -x+y=3 3x+4y = 2

Exercise: Linear Algebra

Solve the following systems of equations:


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g y q

System 1: x+4y=34 -3x+y=2

System 2: 2x-2y=4 -x+y=3 3x+4y = 2

A=[1 4;-3 1];

b=[34;2];rank(A)

x=inv(A)*b;

A=[2 -2;-1 1;3 4];

b=[4;3;2];

rank(A)

rectangular matrixx1=A\b;

gives least squares solution

A*x1

Outline

(1) Linear Algebra


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( ) g

(2) Polynomials



Polynomials

Many functions can be well described by a high-orderl i l


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y y gpolynomial

MATLAB represents a polynomials by a vector of coefficients if vector P describes a polynomial ax3+bx2+cx+d

P=[1 0 -2] represents the polynomial x2-2

P=[2 0 0 0] represents the polynomial 2x3

P(1) P(2) P(3) P(4)

Polynomial Operations

P is a vector of length N+1 describing an N-th order polynomialTo get the oot of pol nomi l


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To get the roots of a polynomialr=roots(P)

r is a vector of length N

Can also get the polynomial from the rootsP=poly(r)

r is a vector length N

To evaluate a polynomial at a pointy0=polyval(P,x0)

x0 is a single value; y0 is a single value

To evaluate a polynomial at many pointsy=polyval(P,x)

x is a vector; y is a vector of the same size

Polynomial Fitting

MATLAB makes it very easy to fit polynomials to data


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Given data vectors X=[-1 0 2] and Y=[0 -1 3]

p2=polyfit(X,Y,2); finds the best second order polynomial that fits the points

(-1,0),(0,-1), and (2,3) see help polyfit for more information

plot(X,Y,o, MarkerSize, 10);hold on;

x = linspace(-2,2,1000);

plot(x,polyval(p2,x), r--);

Exercise: Polynomial Fitting

Evaluate x^2 over x=-4:0.1:4 and save it as y.


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Add random noise to these samples. Use randn. Plot thenoisy signal with . markers

fit a 2nd degree polynomial to the noisy data

plot the fitted polynomial on the same plot, using the samex values and a red line

Exercise: Polynomial Fitting

Evaluate x^2 over x=-4:0.1:4 and save it as y. x 4:0 1:4;


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x=-4:0.1:4;

y=x.^2;

Add random noise to these samples. Use randn. Plot thenoisy signal with . markersy=y+randn(size(y));

plot(x,y,.);

fit a 2nd degree polynomial to the noisy data[p]=polyfit(x,y,2);

plot the fitted polynomial on the same plot, using the samex values and a red linehold on;

plot(x,polyval(p,x),r)


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Nonlinear Root Finding

Many real-world problems require us to solve f(x)=0 Can use fzero to calculate roots for any arbitrary function


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Can use fzero to calculate roots for any arbitrary function

fzero needs a function passed to it. We will see this more and more as we delve into solvingequations.

Make a separate function filex=fzero('myfun',1)

x=fzero(@myfun,1)

1 specifies apoint close tothe root


Minimizing a Function

fminbnd: minimizing a function over a bounded interval x=fminbnd('myfun' -1 2);


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x=fminbnd( myfun , 1,2);

myfun takes a scalar input and returns a scalar outputmyfun(x) will be the minimum of myfun for -1x 2

fminsearch: unconstrained intervalx=fminsearch('myfun',.5)

finds the local minimum of myfun starting at x=0.5

Anonymous Functions

You do not have to make a separate function file


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Instead, you can make an anonymous function

x=fzero(@(x)(cos(exp(x))+x^2-1), 1 );

x=fminbnd(@(x) (cos(exp(x))+x^2-1),-1,2);

input function to evaluate

Optimization Toolbox

If you are familiar with optimization methods, use theoptimization toolbox


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p

Useful for larger, more structured optimization problems

Sample functions (see help for more info)linprog

linear programming using interior point methodsquadprog

quadratic programming solver

fmincon constrained nonlinear optimization

Exercise: Min-Finding

Find the minimum of the function f(x) =cos(4*x).*sin(10*x).*exp(-abs(x)) over the range pi to


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( ) ( ) p( ( )) g ppi. Use fminbnd. Is your answer really the minimumover this range?

Exercise: Min-Finding

Find the minimum of the function f(x) =cos(4*x).*sin(10*x).*exp(-abs(x)) over the range pi to


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( ) ( ) p( ( )) g ppi. Use fminbnd. Is your answer really the minimumover this range?

function y = myFun(x)

y=cos(4*x).*sin(10*x).*exp(-abs(x));

fminbnd(myFun, -pi, pi);

Outline

(1) Linear Algebra

(2) P l i l


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(2) Polynomials



Numerical Differentiation

MATLAB can 'differentiate' numericallyx=0:0.01:2*pi;

0 4

0.6

0.8

1


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y=sin(x);

dydx=diff(y)./diff(x);

diff computes the first difference

Can also operate on matricesmat=[1 3 5;4 8 6];

dm=diff(mat,1,2)

first difference of mat along the 2nd dimension, dm=[2 2;4 -2] see help for more details

2D gradient[dx,dy]=gradient(mat);

0 100 200 300 400 500 600 700-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Numerical Integration

MATLAB contains common integration methods


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Adaptive Simpson's quadrature (input is a function)

q=quad('derivFun',0,10); q is the integral of the function derivFun from 0 to 10

q2=quad(@sin,0,pi)

q2 is the integral of sin from 0 to pi

Trapezoidal rule (input is a vector)x=0:0.01:pi;

z=trapz(x,sin(x));

z is the integral of sin(x) from 0 to piz2=trapz(x,sqrt(exp(x))./x)

z2 is the integral of from 0 to pixe x


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ODE Solvers: Method

Given a differential equation, the solution can be found byintegration:


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Evaluate the derivative at a point and approximate by straight line Errors accumulate! Variable timestep can decrease the number of iterations

ODE Solvers: MATLAB

MATLAB contains implementations of common ODE solvers


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Using the correct ODE solver can save you lots of time andgive more accurate resultsode23

Low-order solver. Use when integrating over small intervalsor when accuracy is less important than speed

ode45High order (Runge-Kutta) solver. High accuracy and

reasonable speed. Most commonly used.ode15s

Stiff ODE solver (Gear's algorithm), use when the diff eq'shave time constants that vary by orders of magnitude


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ODE Function

The ODE function must return the value of the derivative ata given time and function value


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Example: chemical reaction Two equations

ODE file:

y has [A;B] dydt has

[dA/dt;dB/dt]

A B

10

50

10 50

10 50

dAA B

dt

dB A Bdt

= +

=

Courtesy of The MathWorks, Inc.

Used with permission.


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ODE Function: viewing results

The code on the previous slide produces this figure


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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (s)

Amount

ofchemical(g)

Chem reaction

A

B

Higher Order Equations

Must make into a system of first-order equations to useODE solvers

Nonlinear is OK!


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Pendulum example:

( )

( )

( )

0g sinL

gsin

L

let

gsin

L

x

dx

dt

+ =

=

=

=

=

=

&&

&&

&

&

v

v &

&


Plotting the Output

We can solve for the position and velocity of the pendulum:[t,x]=ode45('pendulum',[0 10],[0.9*pi 0]);

d l i l t ti l ( t t )


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assume pendulum is almost vertical (at top)plot(t,x(:,1));

hold on;

plot(t,x(:,2),'r');

legend('Position','Velocity');

0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8

Position

Velocity

Position in terms ofangle (rad)

Velocity (m/s)

Plotting the Output

Or we can plot in the phase plane:plot(x(:,1),x(:,2));

l b l('P iti ')


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xlabel('Position');

yLabel('Velocity');

The phase plane is just a plot of one variable versus theother:

-3 -2 -1 0 1 2 3-8

-6

-4

-2

0

2

4

6

8

Position

Velocity

Velocity is greatestwhen theta=0

Velocity=0 whentheta is the greatest

ODE Solvers: Custom Options

MATLAB's ODE solvers use a variable timestep Sometimes a fixed timestep is desirable

[t y] ode45('chem' [0:0 001:0 5] [0 1]);


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[t,y]=ode45('chem',[0:0.001:0.5],[0 1]);

Specify the timestep by giving a vector of times The function will be evaluated at the specified points Fixed timestep is usually slower (if timestep is small) and

possibly inaccurate (if timestep is too large)

You can customize the error tolerances using odesetoptions=odeset('RelTol',1e-6,'AbsTol',1e-10);

[t,y]=ode45('chem',[0 0.5],[0 1],options);

This guarantees that the error at each step is less thanRelTol times the value at that step, and less than AbsTol

Decreasing error tolerance can considerably slow the solver

See doc odeset for a list of options you can customize

Exercise: ODE

Use ODE45 to solve this differential equation on the ranget=[0 10], with initial condition y(0) = 10: dy/dt=-t*y/10.Plot the result.


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Plot the result.

Exercise: ODE

Use ODE45 to solve this differential equation on the ranget=[0 10], with initial condition y(0) = 10: dy/dt=-t*y/10.Plot the result.


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function dydt=odefun(t,y)

dydt=-t*y/10;

[t,y]=ode45(odefun,[0 10],10);

plot(t,y);

End of Lecture 3

(1) Linear Algebra(2) Polynomials


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( ) y



We're almost done!

Issues with ODEs

Stability and accuracy if step size is too large, solutions might blow up if step size is too small, requires a long time to solve


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if step size is too small, requires a long time to solve use odeset to control errors

decrease error tolerances to get more accurateresults

increase error tolerances to speed up computation

(beware of instability!) Main thing to remember about ODEs Pick the most appropriate solver for your problem If ode45 is taking too long, try ode15s

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6.094

Introduction to Programming in MATLAB


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Lecture 4: Advanced Methods

Sourav Dey

Danilo

epanovi


IAP 2009

Outline

(1) Probability and Statistics(2) Data Structures


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(3) Images and Animation(4) Debugging

(5) Symbolic Math

(6) Other Toolboxes

Statistics

Whenever analyzing data, you have to compute statisticsscores = 100*rand(1,100);


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Built-in functionsmean, median, mode

To group data into a histogram

hist(scores,5:10:95);makes a histogram with bins centered at 5, 15, 2595

N=histc(scores,0:10:100);

returns the number of occurrences between the specifiedbin edges 0 to


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rand

draws from the uniform distribution from 0 to 1randn

draws from the standard normal distribution (Gaussian)

random can give random numbers from many more distributions see doc random for help the docs also list other specific functions

You can also seed the random number generatorsrand(state,0)

Changing Mean and Variance

We can alter the given distributionsy=rand(1,100)*10+5; gives 100 uniformly distributed numbers between 5 and 15


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y=floor(rand(1,100)*10+6);

gives 100 uniformly distributed integers between 10 and15. floor or ceil is better to use here than round

y=randn(1,1000)y2=y*5+8

increases std to 5 and makes the mean 8-25 -20 -15 -10 -5 0 5 10 15 20 250

50

100

150

200

250

300

350

400

-25 -20 -15 -10 -5 0 5 10 15 20 250

10

20

30

40

50

60

70

80

90

Exercise: Probability

We will simulate Brownian motion in 1 dimension. Call the scriptbrown Make a 10,000 element vector of zeros Write a loop to keep track of the particles position at each time


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Start at 0. To get the new position, pick a random number, and if

its 0.5, go right. Store each new position inthe kth position in the vector Plot a 50 bin histogram of the positions.

Exercise: Probability

We will simulate Brownian motion in 1 dimension. Call the scriptbrown Make a 10,000 element vector of zeros Write a loop to keep track of the particles position at each time

h k d b d f


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Start at 0. To get the new position, pick a random number, and if

its 0.5, go right. Store each new position inthe kth position in the vector Plot a 50 bin histogram of the positions.

x=zeros(10000,1);

for n=2:10000 if rand


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(3) Images and Animation(4) Debugging

(5) Symbolic Math

(6) Other Toolboxes

Advanced Data Structures

We have used 2D matricesCan have n-dimensions Every element must be the same type (ex. integers,

doubles characters )


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doubles, characters)

Matrices are space-efficient and convenient for calculation

Sometimes, more complex data structures are moreappropriate

Cell array: it's like an array, but elements don't have to bethe same type

Structs: can bundle variable names and values into onestructure

Like object oriented programming in MATLAB

Cells: organization

A cell is just like a matrix, but each field can containanything (even other matrices):

3x3 Matrix 3x3 Cell Array


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One cell can contain people's names, ages, and the ages oftheir children

To do the same with matrices, you would need 3 variables

and padding

3x3 Matrix

1.2 -3 5.5

-2.4 15 -10

7.8 -1.1 4

3x3 Cell Array

32

27 1

18

J o h n

M a r y

L e o

2

4

[ ]


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Structs

Structs allow you to name and bundle relevant variables Like C-structs, which are objects with fields

To initialize an empty struct:


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p y

s=struct([]); size(s) will be 1x1 initialization is optional but is recommended when using large

structs

To add fieldss.name = 'Jack Bauer';

s.scores = [95 98 67];

s.year = 'G3'; Fields can be anything: matrix, cell, even struct Useful for keeping variables together

For more information, see doc struct


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Exercise: Cells

Write a script called sentGen Make a 3x2 cell, and put peoples names into the firstcolumn, and adjectives into the second column

Pick two random integers (values 1 to 3)


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Display a sentence of the form [name] is [adjective]. Run the script a few times

Exercise: Cells

Write a script called sentGen Make a 3x2 cell, and put peoples names into the firstcolumn, and adjectives into the second column



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Display a sentence of the form [name] is [adjective]. Run the script a few times

c=cell(3,2);c{1,1}=John;c{2,1}=Mary-Sue;c{3,1}=Gomer;

c{1,2}=smart;c{2,2}=blonde;c{3,2}=hot

r1=ceil(rand*3);r2=ceil(rand*3);disp([ c{r1,1}, is , c{r2,2}, . ]);

Outline


(3) Images and Animation


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(4) Debugging

(5) Symbolic Math

(6) Other Toolboxes

Importing/Exporting Images

Images can be imported into matlabim=imread('myPic.jpg');

MATLAB supports almost all image formats


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MATLAB supports almost all image formatsjpeg, tiff, gif, bmp, png, hdf, pcx, xwd, ico, cur, ras, pbm,

pgm, ppm see help imread for a full list and details

To write an image, give an rgb matrix or indices andcolormapimwrite(mat,jet(256),'test.jpg','jpg');

see help imwrite for more options

Animations

MATLAB makes it easy to capture movie frames and playthem back automatically

The most common movie formats are:


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The most common movie formats are: avi animated gif

Avi good when you have natural frames with lots of colors andfew clearly defined edges

Animated gif

Good for making movies of plots or text where only a fewcolors exist (limited to 256) and there are well-definedlines

Making Animations

Plot frame by frame, and pause in betweenclose allfor t=1:30

imagesc(rand(200));


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imagesc(rand(200));

colormap(gray);

pause(.5);

end


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Handles

Every graphics object has a handleh=plot(1:10,rand(1,10)); gets the handle for the plotted line

h2=gca;


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g ;

gets the handle for the current axish3=gcf;

gets the handle for the current figure

To see the current property values, use getget(h);yVals=get(h,'YData');

To change the properties, use setset(h2,'FontName','Arial','XScale','log');

set(h,'LineWidth',1.5,'Marker','*');

Everything you see in a figure is completely customizable

through handles


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display

When debugging functions, use disp to print messagesdisp('starting loop')disp('loop is over')

disp prints the given string to the command window


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p p g g

It's also helpful to show variable values

disp(strcat(['loop iteration ',num2str(n)])); strcat concatenates the given stringsSometimes it's easier to just remove some semicolons

Debugging

To use the debugger, set breakpointsClick on next to line numbers in MATLAB files Each red dot that appears is a breakpointRun the program The program pauses when it reaches a breakpoint


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The program pauses when it reaches a breakpointUse the command window to probe variablesUse the debugging buttons to control debugger

Two breakpoints

Where the program is now

Clear breakpoint

Step to next

Stop execution; exit


Exercise: Debugging

Use the debugger to fix the errors in the following code:


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Performance Measures

It can be useful to know how long your code takes to run To predict how long a loop will take To pinpoint inefficient code


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You can time operations using tic/toc:tic

CommandBlock1

a=toc;CommandBlock2

b=toc;

tic resets the timer Each toc returns the current value in secondsCan have multiple tocs per tic

Performance Measures

For more complicated programs, use the profilerprofile on Turns on the profiler. Follow this with function calls

profile viewer


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Displays gui with stats on how long each subfunction took


Outline




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( ) g

(4) Debugging

(5) Symbolic Math

(6) Other Toolboxes

What are Toolboxes?

Toolboxes contain functions specific to a particular field for example: signal processing, statistics, optimization

It's generally more efficient to use MATLAB's toolboxes


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rather than redefining the functions yourself saves coding/debugging time some functions are compiled, so they run fasterHOWEVER there may be mistakes in MATLABs functions

and there may also be surprises

MATLAB on Athena contains all the toolboxes

Here are a few particularly useful ones for EECS

Symbolic Toolbox

Dont do nasty calculations by hand! Symbolics vs. Numerics

Advantages Disadvantages


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Symbolic Analytical solutionsLets you intuit

things aboutsolution form

Sometimes can't besolvedCan be overlycomplicated

Numeric Always get asolution

Can make solutionsaccurateEasy to code

Hard to extract adeeper understanding

Num. methodssometimes failCan take a while tocompute

Symbolic Variables

Symbolic variables are a type, like double or char

To make symbolic variables, use syma=sym('1/3');


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b=sym('4/5');

fractions remain as fractionsc=sym('c','positive');

can add tags to narrow down scope see help sym for a list of tags

Or use symssyms x y real

shorthand for x=sym('x','real'); y=sym('y','real');

Symbolic Expressions

Multiply, add, divide expressionsd=a*b does 1/3*4/5=4/15;


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expand((a-c)^2);

multiplies out

factor(ans)

factors the expression


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More Symbolic Operations

We can do symbolics with matrices toomat=sym('[a b;c d]');

mat2=mat*[1 3;4 -2];


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compute the productd=det(mat)

compute the determinant

i=inv(mat) find the inverse

You can access symbolic matrix elements as beforei(1,2)

Exercise: Symbolics

The equation of a circle of radius r centered at (a,b) isgiven by: (x-a)^2 + (y-b)^2 = r^2.

Expand this equation into the form Ax^2 + Bx+Cxy + Dy +Ey^2 = F and find the expression for the coefficients in


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terms of a,b, and r.

Exercise: Symbolics

The equation of a circle of radius r centered at (a,b) isgiven by: (x-a)^2 + (y-b)^2 = r^2.

Expand this equation into the form Ax^2 + Bx+Cxy + Dy +Ey^2 = F and find the expression for the coefficients in


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terms of a,b, and r.

syms a b r x y

pretty(expand((x-a).^2 + (y-b).^2))

Outline




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(4) Debugging

(5) Symbolic Math

(6) Other Toolboxes

Signal Processing Toolbox

MATLAB is often used for signal processing (fft) What you can do:

filter design statistical signal processing

L l t f


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Laplace transforms

Related ToolboxesCommunicationsWaveletsRF Image Processing

Control System Toolbox

The control systems toolbox contains functions helpful foranalyzing systems with feedback

Simulation of LTI system function

Di t ti ti ti


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Discrete time or continuous time You will be exposed to it in 6.003 Can easily study step response, etc. modal analysis.

Related toolboxes:System IdentificationRobust Control modern control theoryModel Predictive Control

Statistics Toolbox

For hardcore statistics and data-analysis Principal component analysis Independent component analysis Tests of significance (chi squared, t-tests)


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Related ToolboxesSpline for fittingBioinformaticsNeural Networks

Optimization Toolbox

For more hardcore optimization problems that occur inOR, business, engineering

linear programming interior point methods

quadratic methods


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quadratic methods


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Central File Exchange

The website the MATLAB Central File Exchange!! Lots of people's code is there Tested and rated use it to expand MATLAB's functionality

http://www mathworks com/matlabcentral/


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http://www.mathworks.com/matlabcentral/

MATLAB Final Exam

Brownian Motion stop-animation integrating loops,randomization, visualization

Make a function brown2d(numPts), where numPts is the

number of points that will be doing Brownian motion


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number of points that will be doing Brownian motion Plot the position in (x,y) space of each point (start initially

at 0,0). Set the x and y limits so theyre consistent. After each timestep, move each x and y coordinate by

randn*.1 Pause by 0.001 between frames Turn on the DoubleBuffer property to remove flicker

set(gcf,DoubleBuffer,on);

Ask us for help if needed!

End of Lecture 4

(1) Data Structures(2) Symbolics

(3) Probability

(4) Toolboxes


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(4) Toolboxes

THE END

Monte-Carlo Simulation

A simple way to model complex stochastic systems Use random numbers to control state changes

BA60

472 2

20 20


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This system represents a complex reaction

The numbers by the arrows show the propensity of thesystem to go from one state to another If you start with 1 molecule of A, how does the system

behave with time?

E

D C

60

60

60 47 47

47

2

2

2

20 20

Example: Monte-Carlo

This MATLAB file will track the behavior of the molecule


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Example: Monte-Carlo

We can run the code 1000 times to simulate 1000 moleculess=zeros(200,5);

for n=1:1000

st=MC(200);

for state=0:4 s(: state+1)= s(: state+1)+(st==state);


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s(:,state+1)= s(:,state+1)+(st==state);

end

end

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0 20 40 60 80 100 120 140 160 180 2000

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1000E

MIT OpenCourseWarehttp://ocw.mit.edu


January (IAP) 2009



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6.094

Introduction to Programming in MATLAB

Lecture 5: Simulink


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IAP 2009


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Getting Started

Create a new file

Examine the Simulink Library Browser

Click on a library:Sources


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y

Drag a block into Simulink:Constant

Visualize the block by going intoSinks

Drag aScopeinto Simulink

Connections

Click on the carat/arrow on the right of theconstant box

Drag the line to the scopeYoull get a hint saying you can quickly connect


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Drag the line to the scopeYoull get a hint saying you can quickly connect

blocks by hitting CtrlConnections between lines represent signals

Click the play button

Double click on the scope.This will open up a chart of the variable over the

simulation time

Simulink Math

Everything is visual in Simulink!

Click on the library Continuous

Drag the integrator block between the constant andthe scope


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p

Play and click on scope.

What happens?

Simulink has a built in ODE solverThe equation that represents your model is solved by

SimulinkWeve represented

Behind the curtain

Go toSimulation->Configuration Parameters

at the top menuSee ode45? Change the solver type here


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So whats going on?

The toolboxes Simulink provides you are full ofmodeling tools

By selecting components that correspond to yourmodel, you can design a simulation


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y g p p y, y g


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Building systems

Sources Step input, white noise, custom input, sine

wave, ramp input,

Provides input to your system

Sinks


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Scope: Outputs to plot

simout: Outputs to a MATLAB vector on workspace

MATLAB mat file


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Modifying Scopes

Within the scope:Autoscale fits the axes

to the curve automatically

Axes properties lets you

customize the axes


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Changing the number of axes:

Left click on icon

Change the number

of axes field


Courtesy of The MathWorks,

Inc. Used with permission.


First System

Drag a summingjunction between theconstant andintegrator


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Change the signs to|+-

Click on the opencarat under the minussign and connect it tothe integrator output


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Example Systems

ODE

d3

y/dt3

+ a*d2

y/dt2

+b* dy/dt + c*y = F

Classical Control System


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Courtesy of The

MathWorks, Inc.Used with permission.

Example: Nervous System

Neural circuits in animals often exhibit oscillatory behavior

Use Simulink to model one behavior of this type:Locomotion

Limbs go Left-right, left-right, left-right


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Locomotive behaviors are generated by central patterngenerators, which oscillate on their own naturally

When connected to an appendage, the central patterngenerator will adapt its frequency and move the

appendage. OpenRIOCPGDemo.mdl

Model based on Iwasaki, T., Zheng, M. (2006a). Sensory feedbackmechanism underlying entrainment of central pattern generator to

mechanical resonance. Biological Cybernetics, 94(4), 245-261


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Playing with the model

Look at scopesWhat are the output signals?

Delete signalsEspecially the signal after the feedback gain

Change gains


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Muscular actuator gainsSwitch feedback gain from negative to positive

Look inside subsystemsWhats inside the CPG?Whats inside the neuron firing dynamics?

Toolboxes

Simulink has many advanced toolboxesControl SystemsNeural NetworksSignal Processing

SimMechanicsVirtual Reality


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yReal Time

Hopefully youll get to use some of these powerful tools!