Introduction to Matlab Mit

6.094Introduction to Programming in MATLAB

Danilo Šćepanović

IAP 2010

Lecture 1: Variables, Scripts,

and Operations

Course Layout

• Lectures1: Variables, Scripts and Operations2: Visualization and Programming3: Solving Equations, Fitting4: Images, Animations, Advanced Methods5: Optional: Symbolic Math, Simulink

Course Layout

• Problem Sets / Office HoursOne per day, should take about 3 hours to doSubmit doc or pdf (include code, figures)No set office hours but available by email

• Requirements for passingAttend all lecturesComplete all problem sets (-, √, +)

• PrerequisitesBasic familiarity with programmingBasic linear algebra, differential equations, and probability

Outline

(1) Getting Started(2) Scripts(3) Making Variables(4) Manipulating Variables(5) Basic Plotting

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 WindowsThrough the START Menu

• On Athena» add matlab

» matlab &

Command Window

Current directory

Workspace

Command History

Courtesy of The MathWorks, Inc. Used with permission.

Making Folders

• Use folders to keep your programs organized

• To make a new folder, click the ‘Browse’ button next to ‘Current Directory’

• Click the ‘Make New Folder’ button, and change the name of the folder. Do NOT use spaces in folder names. In the MATLAB folder, make two new folders: IAPMATLAB\day1

• Highlight the folder you just made and click ‘OK’• The current directory is now the folder you just created• To see programs outside the current directory, they should be in

the Path. Use File-> Set Path to add folders to the path

Customization

• File PreferencesAllows you personalize your MATLAB experience


MATLAB Basics

• MATLAB can be thought of as a super-powerful graphing calculator

Remember the TI-83 from calculus? With many more buttons (built-in functions)

• In addition it is a programming languageMATLAB is an interpreted language, like JavaCommands executed line by line

Help/Docs

• help

The most important function for learning MATLAB on your own

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

Help lists related functions at the bottom and links to the doc

• 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

Outline


Scripts: Overview

• Scripts are collection of commands executed in sequencewritten in the MATLAB editorsaved as MATLAB files (.m extension)

• To create an MATLAB file from command-line» edit helloWorld.m

• or click


Scripts: the Editor

* Means that it's not savedLine numbers

Comments

MATLAB file path

Help file

Possible breakpoints

Debugging toolsReal-time error check


Scripts: Some Notes

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

• Note that scripts are somewhat static, since there is no input and no explicit output

• All variables created and modified in a script exist in the workspace even after it has stopped running

Exercise: Scripts

Make a helloWorld script

• When run, the script should display the following text:

• Hint: use disp to display strings. Strings are written between single quotes, like 'This is a string'

Hello World!I am going to learn MATLAB!

Exercise: Scripts

Make a helloWorld script

• When run, the script should display the following text:

• Hint: use disp to display strings. Strings are written between single quotes, like 'This is a string'

• Open the editor and save a script as helloWorld.m. This is an easy script, containing two lines of code:» % helloWorld.m

» % my first hello world program in MATLAB

» disp('Hello World!');

» disp('I am going to learn MATLAB!');

Hello World!I am going to learn MATLAB!

Outline


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 you’ll deal with will be vectors or matrices of doubles or chars

• Other types are also supported: complex, symbolic, 16-bit and 8 bit integers, etc. You will be exposed to all these types through the homework

Naming variables

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

» myString=‘hello world’

• Variable namesfirst character must be a LETTERafter that, any combination of letters, numbers and _CASE SENSITIVE! (var1 is different from Var1)

• Built-in variables. Don’t use these names!i and j can be used to indicate complex numbers

pi 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’

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;

Arrays

• Like other programming languages, arrays are an important 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!That’s its power!

Row Vectors

• Row vector: comma or space separated values between brackets» row = [1 2 5.4 -6.6]

» row = [1, 2, 5.4, -6.6];

• Command window:

• Workspace:


Column Vectors

• Column vector: semicolon separated values between brackets » column = [4;2;7;4]

• Command window:

• Workspace:


size & length

• You can tell the difference between a row and a column vector by:

Looking in the workspaceDisplaying the variable in the command windowUsing the size function

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

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]];

» str = ['Hello, I am ' 'John'];

Strings are character vectors

1 23 4

a ⎡ ⎤= ⎢ ⎥⎣ ⎦

save/clear/load

• Use save to save variables to a file» save myFile a b

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

» \MATLABMake sure you’re in the desired folder when saving files. Right now, we should be in:

» MATLAB\IAPMATLAB\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;

Exercise: Variables

Get and save the current date and time• Create a variable start using the function clock• What is the size of start? Is it a row or column?• What does start contain? See help clock• Convert the vector start to a string. Use the function

datestr and name the new variable startString• Save start and startString into a mat file named

startTime

Exercise: Variables

Get and save the current date and time• Create a variable start using the function clock• What is the size of start? Is it a row or column?• What does start contain? See help clock• Convert the vector start to a string. Use the function

datestr and name the new variable startString• Save start and startString into a mat file named

startTime

» help clock

» start=clock;

» size(start)

» help datestr

» startString=datestr(start);

» save startTime start startString

Exercise: Variables

Read in and display the current date and time• In helloWorld.m, read in the variables you just saved using

load

• Display the following text:

• Hint: use the disp command again, and remember that strings are just vectors of characters so you can join two strings by making a row vector with the two strings as sub-vectors.

I started learning MATLAB on *start date and time*

Exercise: Variables

Read in and display the current date and time• In helloWorld.m, read in the variables you just saved using

load


• Hint: use the disp command again, and remember that strings are just vectors of characters so you can join two strings by making a row vector with the two strings as sub-vectors.

» load startTime

» disp(['I started learning MATLAB on ' ... startString]);

I started learning MATLAB on *start date and time*

Outline


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 command window» clc

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);

Exercise: Scalars

You will learn MATLAB at an exponential rate! Add the following to your helloWorld script:

• Your learning time constant is 1.5 days. Calculate the number of seconds in 1.5 days and name this variable tau

• This class lasts 5 days. Calculate the number of seconds in 5 days and name this variable endOfClass

• This equation describes your knowledge as a function of time t:

• How well will you know MATLAB at endOfClass? Name this variable knowledgeAtEnd. (use exp)

• Using the value of knowledgeAtEnd, display the phrase:

• Hint: to convert a number to a string, use num2str

/1 tk e τ−= −

At the end of 6.094, I will know X% of MATLAB

Exercise: Scalars

» secPerDay=60*60*24;

» tau=1.5*secPerDay;

» endOfClass=5*secPerDay

» knowledgeAtEnd=1-exp(-endOfClass/tau);

» disp(['At the end of 6.094, I will know ' ... num2str(knowledgeAtEnd*100) '% of MATLAB'])

Transpose

• The transpose operators turns a column vector into a row vector and vice versa» a = [1 2 3 4+i]

» transpose(a)

» a'

» a.'

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

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

Addition and Subtraction

• Addition and subtraction are element-wise; sizes must match (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 91 1 210 13 230 33 33

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥− =⎢ ⎥ ⎢ ⎥ ⎢ ⎥− −⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

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 function’s help file to see if it handles vectors elementwise

• Operators (* / ^) have two modes of operationelement-wisestandard

Operators: element-wise

• To do element-wise operations, use the dot: . (.*, ./, .^). BOTH dimensions 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 21

1 4 42 2 43 1 33 1 3 1 3 1

.* ERROR

.*

.*

⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦× × = ×

1 1 1 1 2 3 1 2 32 2 2 1 2 3 2 4 63 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

⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

Operators: standard

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

productRemember from linear algebra: inner dimensions must MATCH!!

• Standard exponentiation (^) can only be done on square matrices or scalars

• Left and right division (/ \) is same as multiplying by inverseOur recommendation: just multiply by inverse (more on this later)

[ ]4

1 2 3 2 111

1 3 3 1 1 1

*

*

⎡ ⎤⎢ ⎥ =⎢ ⎥⎢ ⎥⎣ ⎦

× × = ×

1 1 1 1 2 3 3 6 92 2 2 1 2 3 6 12 183 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

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Exercise: Vector Operations

Calculate how many seconds elapsed since the start of class

• In helloWorld.m, make variables called secPerMin, secPerHour, secPerDay, secPerMonth (assume 30.5 days per month), and secPerYear (12 months in year), which have the number of seconds in each time period.

• Assemble a row vector called secondConversion that has elements in this order: secPerYear, secPerMonth, secPerDay, secPerHour, secPerMinute, 1.

• Make a currentTime vector by using clock• Compute elapsedTime by subtracting currentTime from

start

• Compute t (the elapsed time in seconds) by taking the dot product of secondConversion and elapsedTime (transpose one of them to get the dimensions right)


» secPerMin=60;

» secPerHour=60*secPerMin;

» secPerDay=24*secPerHour;

» secPerMonth=30.5*secPerDay;

» secPerYear=12*secPerMonth;

» secondConversion=[secPerYear secPerMonth ... secPerDay secPerHour secPerMin 1];

» currentTime=clock;

» elapsedTime=currentTime-start;

» t=secondConversion*elapsedTime';


Display the current state of your knowledge• Calculate currentKnowledge using the same relationship as

before, and the t we just calculated:


/1 tk e τ−= −

At this time, I know X% of MATLAB


Display the current state of your knowledge• Calculate currentKnowledge using the same relationship as

before, and the t we just calculated:


» currentKnowledge=1-exp(-t/tau);

» disp(['At this time, I know ' ... num2str(currentKnowledge*100) '% of MATLAB']);

/1 tk e τ−= −

At this time, I know X% of MATLAB

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 uninitialized variables)

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

Number of rows Number of columns

Automatic Initialization

• To initialize a linear vector of values use linspace» a=linspace(0,10,5)

starts at 0, ends at 10 (inclusive), 5 values

• Can also use colon operator (:)» b=0:2:10

starts at 0, increments by 2, and ends at or before 10increment can be decimal or negative

» c=1:5

if increment isn’t specified, default is 1

• To initialize logarithmically spaced values use logspacesimilar to linspace, but see help

Exercise: Vector Functions

Calculate your learning trajectory• In helloWorld.m, make a linear time vector tVec that has

10,000 samples between 0 and endOfClass• Calculate the value of your knowledge (call it

knowledgeVec) at each of these time points using the same equation as before:

/1 tk e τ−= −

Exercise: Vector Functions

Calculate your learning trajectory• In helloWorld.m, make a linear time vector tVec that has

10,000 samples between 0 and endOfClass• Calculate the value of your knowledge (call it

knowledgeVec) at each of these time points using the same equation as before:

» tVec = linspace(0,endOfClass,10000);

» knowledgeVec=1-exp(-tVec/tau);

/1 tk e τ−= −

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, each element is looked up individually, and returned as a vector of 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 10a =

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

Matrix Indexing

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

• Matrix indexing: subscripts or linear indices

• Picking submatrices» A = rand(5) % shorthand for 5x5 matrix

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

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

14 339 8⎡ ⎤⎢ ⎥⎣ ⎦

b(1)

b(2)

b(3)

b(4)

14 339 8⎡ ⎤⎢ ⎥⎣ ⎦

b(1,1)

b(2,1)

b(1,2)

b(2,2)

Advanced Indexing 1

• 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

12 52 13

c ⎡ ⎤= ⎢ ⎥−⎣ ⎦

Advanced Indexing 2

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

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

max works the same way

• 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: Indexing

When will you know 50% of MATLAB?• First, find the index where knowledgeVec is closest to 0.5.

Mathematically, what you want is the index where the value of is at a minimum (use abs and min).

• Next, use that index to look up the corresponding time in tVec and name this time halfTime.

• Finally, display the string: Convert halfTime to days by using secPerDay

0.5knowledgeVec −

I will know half of MATLAB after X days

Exercise: Indexing

When will you know 50% of MATLAB?• First, find the index where knowledgeVec is closest to 0.5.

Mathematically, what you want is the index where the value of is at a minimum (use abs and min).

• Next, use that index to look up the corresponding time in tVec and name this time halfTime.

• Finally, display the string: Convert halfTime to days by using secPerDay

» [val,ind]=min(abs(knowledgeVec-0.5));

» halfTime=tVec(ind);

» disp(['I will know half of MATLAB after ' ... num2str(halfTime/secPerDay) ' days']);

0.5knowledgeVec −

I will know half of MATLAB after X days

Outline

(1) Getting Started(2) Scripts (3) Making Variables(4) Manipulating Variables(5) Basic Plotting

Did everyone sign in?

Plotting

• Example» x=linspace(0,4*pi,10);

» y=sin(x);

• Plot values against their index» plot(y);

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

MATLAB makes visualizing data fun and easy!

What does plot do?

• plot generates dots at each (x,y) pair and then connects the dots with 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 you’ll get an error» plot([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:

Exercise: Plotting

Plot the learning trajectory• In helloWorld.m, open a new figure (use figure)• Plot the knowledge trajectory using tVec and

knowledgeVec. When plotting, convert tVec to days by using secPerDay

• Zoom in on the plot to verify that halfTime was calculated correctly

Exercise: Plotting

Plot the learning trajectory• In helloWorld.m, open a new figure (use figure)• Plot the knowledge trajectory using tVec and

knowledgeVec. When plotting, convert tVec to days by using secPerDay

• Zoom in on the plot to verify that halfTime was calculated correctly

» figure

» plot(tVec/secPerDay, knowledgeVec);

End of Lecture 1


Hope that wasn’t too much!!

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6.094 Introduction to MATLAB® January (IAP) 2010

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6.094Introduction to programming in MATLAB


IAP 2010

Lecture 2: Visualization and Programming

Homework 1 Recap

• How long did it take to do required problems?• Did anyone do optional problems?• Was level of guidance appropriate?• Unanswered Questions?

• Some things that came up:• Use of semicolon – never required if one command per line.

You can also put multiple commands on one line; in this case a semicolon is necessary to separate commands:» x=1:10; y=(x-5).^2; plot(x,y);

• Assignment using indices – remember that you can index into matrices to either look up values or to assign value:» x=rand(50,1); inds=find(x<0.1); y=x(inds); x(inds)=-x(inds); x(inds)=3;

Outline

(1) Functions (2) Flow Control(3) Line Plots(4) Image/Surface Plots(5) Vectorization

User-defined Functions• Functions look exactly like scripts, but for ONE difference

Functions must have a function declaration

Help file

Function declaration

InputsOutputs


User-defined Functions

• Some comments about the function declaration

• No need for return: MATLAB 'returns' the variables whose names match those in the function declaration

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

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

Must have the reserved word: function

Function name should match MATLAB file nameIf more than one output,

must be in brackets

Inputs must be specified

Functions: overloading

• We're familiar with» zeros

» 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: overloading

• 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); %n-dimensional matrices are OK

» D=size(a)

» [m,n]=size(a)

» [x,y,z]=size(a)

» m2=size(a,2)

• You can overload your own functions by having variable input and output arguments (see varargin, nargin, varargout, nargout)

Functions: Excercise

• Write a function with the following declaration:function plotSin(f1)

• In the function, plot a sin wave with frequency f1, on the range [0,2π]:

• To get good sampling, use 16 points per period.

( )1sin f x

0 1 2 3 4 5 6 7-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Functions: Excercise

• Write a function with the following declaration:function plotSin(f1)

• In the function, plot a sin wave with frequency f1, on the range [0,2π]:

• To get good sampling, use 16 points per period.

• In an MATLAB file saved as plotSin.m, write the following:» function plotSin(f1)

x=linspace(0,2*pi,f1*16+1);figureplot(x,sin(f1*x))

( )1sin f x

Outline


Relational Operators

• MATLAB uses mostly standard relational operatorsequal ==not equal ~=greater than >less than <greater or equal >=less or equal <=

• Logical operators elementwise short-circuit (scalars)And & &&Or | ||Not ~Xor xorAll true allAny true any

• Boolean values: zero is false, nonzero is true• See help . for a detailed list of operators

if/else/elseif

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

IF

if cond

commands

end

ELSE

if cond

commands1

else

commands2

end

ELSEIF

if cond1

commands1

elseif cond2

commands2

else

commands3

end

• No need for parentheses: command blocks are between reserved words

Conditional statement: evaluates to true or false

for

• for loops: use for a known number of iterations• MATLAB syntax:

for n=1:100commands

end

• The loop variableIs defined as a vectorIs a scalar within the command blockDoes not have to have consecutive values (but it's usually cleaner if they're consecutive)

• The command blockAnything between the for line and the end

Loop variable

Command block

while

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

• The command block will execute while the conditional expression is true

• Beware of infinite loops!

WHILE

while condcommands

end

Exercise: Conditionals

• Modify your plotSin(f1) function to take two inputs: plotSin(f1,f2)

• If the number of input arguments is 1, execute the plot command you wrote before. Otherwise, display the line 'Two inputs were given'

• Hint: the number of input arguments are in the built-in variable nargin

Exercise: Conditionals

• Modify your plotSin(f1) function to take two inputs: plotSin(f1,f2)

• If the number of input arguments is 1, execute the plot command you wrote before. Otherwise, display the line 'Two inputs were given'

• Hint: the number of input arguments are in the built-in variable nargin

» function plotSin(f1,f2)

x=linspace(0,2*pi,f1*16+1);figure

if nargin == 1plot(x,sin(f1*x));

elseif nargin == 2disp('Two inputs were given');

end

Outline


Plot Options

• Can change the line color, marker style, and line style by adding a string argument» plot(x,y,’k.-’);

• Can plot without connecting the dots by omitting line style argument» plot(x,y,’.’)

• Look at help plot for a full list of colors, markers, and linestyles

color marker line-style

Playing with the Plot

to select lines and delete or change properties

to zoom in/outto slide the plot around

to see all plot tools at once


Line and Marker Options

• Everything on a line can be customized» plot(x,y,'--s','LineWidth',2,...

'Color', [1 0 0], ...'MarkerEdgeColor','k',...'MarkerFaceColor','g',...'MarkerSize',10)

• See doc line_props for a full list of properties that can be specified

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You can set colors by using a vector of [R G B] values or a predefined color character like 'g', 'k', etc.

Cartesian Plots

• We have already seen the plot function» x=-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

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3D Line Plots

• We can plot in 3 dimensions just as easily as in 2» time=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

Axis Modes

• Built-in axis modes

» axis square

makes the current axis look like a box» axis tight

fits axes to data» axis equal

makes x and y scales the same» axis xy

puts the origin in the bottom left corner (default for plots)» axis ij

puts the origin in the top left corner (default for matrices/images)

Multiple Plots in one Figure

• To have multiple axes in one figure» subplot(2,3,1)

makes a figure with 2 rows and three columns of axes, and activates the first axis for plottingeach axis can have labels, a legend, and a title

» subplot(2,3,4:6)activating a range of axes fuses them into one

• To close existing figures» close([1 3])

closes figures 1 and 3» close all

closes all figures (useful in scripts/functions)

Copy/Paste Figures• Figures can be pasted into other apps (word, ppt, etc)• Edit copy options figure copy template

Change font sizes, line properties; presets for word and ppt

• Edit copy figure to copy figure• Paste into document of interest


Saving Figures• Figures can be saved in many formats. The common ones

are:

.fig preserves all information

.bmp uncompressed image

.eps high-quality scaleable format

.pdf compressed image


Advanced Plotting: Exercise

• Modify the plot command in your plotSin function to use squares as markers and a dashed red line of thickness 2 as the line. Set the marker face color to be black(properties are LineWidth, MarkerFaceColor)

• If there are 2 inputs, open a new figure with 2 axes, one on top of the other (not side by side), and activate the top one (subplot)

0 1 2 3 4 5 6 7-1

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plotSin(6) plotSin(1,2)

Advanced Plotting: Exercise

• Modify the plot command in your plotSin function to use squares as markers and a dashed red line of thickness 2 as the line. Set the marker face color to be black(properties are LineWidth, MarkerFaceColor)

• If there are 2 inputs, open a new figure with 2 axes, one on top of the other (not side by side), and activate the top one (subplot)

» if nargin == 1plot(x,sin(f1*x),'rs--',...'LineWidth',2,'MarkerFaceColor','k');

elseif nargin == 2subplot(2,1,1);

end

Outline


Visualizing matrices

• Any matrix can be visualized as an image» mat=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• You can change the colormap:

» imagesc(mat)

default map is jet» colormap(gray)

» colormap(cool)

» colormap(hot(256))

• See help hot for a list

• Can define custom colormap» map=zeros(256,3);

» map(:,2)=(0:255)/255;

» colormap(map);

Surface Plots

• It is more common to visualize surfaces in 3D

• Example:

• surf puts vertices at specified points in space x,y,z, andconnects all the vertices to make a surface

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

• How can we make these matricesloop (DUMB)built-in function: meshgrid

( ) ( ) ( )[ ] [ ]

f x, y sin x cos y

x , ; y ,π π π π=

∈ − ∈ −

2 4 6 8 10 12 14 16 18 20

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-2

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surf

• Make the x and y vectors» x=-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 matrices » Z =sin(X).*cos(Y);

• Plot the surface» surf(X,Y,Z)

» surf(x,y,Z);

surf Options

• See help surf for more options• There are three types of surface shading

» shading faceted

» shading flat

» shading interp

• You can change colormaps» colormap(gray)

contour

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

takes same arguments as surfcolor indicates heightcan modify linestyle propertiescan set colormap

» hold on

» mesh(X,Y,Z)

Exercise: 3-D Plots

• Modify plotSin to do the following:

• If two inputs are given, evaluate the following function:

• y should be just like x, but using f2. (use meshgrid to get the X and Y matrices)

• In the top axis of your subplot, display an image of the Z matrix. Display the colorbar and use a hot colormap. Set the axis to xy (imagesc, colormap, colorbar, axis)

• In the bottom axis of the subplot, plot the 3-D surface of Z (surf)

( ) ( )1 2sin sinZ f x f y= +

Exercise: 3-D Plots

» function plotSin(f1,f2)

x=linspace(0,2*pi,round(16*f1)+1);figure

if nargin == 1plot(x,sin(f1*x),'rs--',...'LineWidth',2,'MarkerFaceColor','k');

elseif nargin == 2y=linspace(0,2*pi,round(16*f2)+1);[X,Y]=meshgrid(x,y);Z=sin(f1*X)+sin(f2*Y);subplot(2,1,1); imagesc(x,y,Z); colorbar;axis xy; colormap hotsubplot(2,1,2); surf(X,Y,Z);

end

Exercise: 3-D Plots

plotSin(3,4) generates this figure

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1

2

3

4

5

6

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-1

0

1

2

01

23

45

67

0

24

68-2

0

2

Specialized Plotting Functions

• 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 graphs» bar(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 functions» stairs(1:10,rand(1,10));

• fill-draws and fills a polygon with specified vertices» fill([0 1 0.5],[0 0 1],'r');

• see help on these functions for syntax• doc specgraph – for a complete list

Outline


Revisiting find

• 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 & x<0.6);

• inds will contain the indices at which x has values between 0.4 and 0.6. This is what happens:

x>0.4 returns a vector with 1 where true and 0 where falsex<0.6 returns a similar vector The & combines the two vectors using an andThe find returns the indices of the 1's

Example: Avoiding Loops

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

Using a loop and if/else

count=0;

for n=1:length(x)

if x(n)>0

count=count+1;

end

end

Being more clever

count=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!• Built-in functions will make it faster to write and execute

Efficient Code

• Avoid loopsThis is referred to as vectorization

• Vectorized code is more efficient for MATLAB• Use indexing and matrix operations to avoid loops• For example, to sum up every two consecutive terms:

» 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

» end

Slow and complicated

» a=rand(1,100);

» b=[0 a(1:end-1)]+a;

Efficient and clean. Can also do this using conv

End of Lecture 2

(1) Functions (2) Flow Control (3) Line Plots(4) Image/Surface Plots(5) Vectorization

Vectorization makes coding fun!





http://ocw.mit.edu



IAP 2008

Lecture 3 : Solving Equations and Curve Fitting

Homework 2 Recap

• How long did it take?

• Using min with matrices:» a=[3 7 5;1 9 10; 30 -1 2];

» b=min(a); % returns the min of each column

» m=min(b); % returns min of entire a matrix

» m=min(min(a)); % same as above

» m=min(a(:)); % makes a a vector, then gets min

• Common mistake:» [m,n]=find(min(a)); % think about what happens

• How to make and run a function: save the file, then call it from the command window like any other function. No need to 'compile' or make it official in any other way

Outline

(1) Linear Algebra(2) Polynomials(3) Optimization(4) Differentiation/Integration(5) Differential Equations

Systems of Linear Equations

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

• Construct matrices so the system is described by Ax=b» A=[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. Solution

depends on whether the system is over or underdetermined.

MATLAB makes linear algebra fun!

More Linear Algebra

• Given a matrix» mat=[1 2 -3;-3 -1 1;1 -1 1];

• Calculate the rank of a matrix» r=rank(mat);

the number of linearly independent rows or columns

• Calculate the determinant» d=det(mat);

mat must be squareif determinant is nonzero, matrix is invertible

• Get the matrix inverse» E=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

• 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:

System 1:

System 2:

4 343 2x y

x y+ =

− + =

2 2 43

3 4 2

x yx y

x y

− =− + =

+ =

Exercise: Linear Algebra

• Solve the following systems of equations:

System 1:

System 2:

4 343 2x y

x y+ =

− + =

2 2 43

3 4 2

x yx y

x y

− =− + =

+ =

» 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 matrix» x1=A\b;

gives least squares solution» error=abs(A*x1-b)

Outline


Polynomials

• Many functions can be well described by a high-order polynomial

• MATLAB represents a polynomials by a vector of coefficientsif 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 polynomial• To get the roots of a polynomial

» r=roots(P)

r is a vector of length N

• Can also get the polynomial from the roots» P=poly(r)

r is a vector length N

• To evaluate a polynomial at a point» y0=polyval(P,x0)

x0 is a single value; y0 is a single value

• To evaluate a polynomial at many points» y=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

• 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 = -3:.01:3;

» plot(x,polyval(p2,x), ‘r--’);

Exercise: Polynomial Fitting

• Evaluate for x=-4:0.1:4.

• Add random noise to these samples. Use randn. Plot the noisy signal with . markers

• Fit a 2nd degree polynomial to the noisy data

• Plot the fitted polynomial on the same plot, using the same x values and a red line

2y x=

Exercise: Polynomial Fitting

• Evaluate for x=-4:0.1:4.» x=-4:0.1:4;

» y=x.^2;

• Add random noise to these samples. Use randn. Plot the noisy signal with . markers» y=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 same x values and a red line» hold on;

» plot(x,polyval(p,x),’r’)

2y x=

Outline


Nonlinear Root Finding

• Many real-world problems require us to solve f(x)=0• 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 solving

equations.• Make a separate function file

» x=fzero('myfun',1)

» x=fzero(@myfun,1)

1 specifies apoint close to whereyou think the root is


Minimizing a Function

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

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

• fminsearch: unconstrained interval» x=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» x=fzero(@myfun,1)

What if myfun is really simple?

• 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 the optimization toolbox

• Useful for larger, more structured optimization problems

• Sample functions (see help for more info)» linprog

linear programming using interior point methods» quadprog

quadratic programming solver» fmincon

constrained nonlinear optimization

Exercise: Min-Finding

• Find the minimum of the function over the range –π to π. Use fminbnd.

• Plot the function on this range to check that this is the minimum.

( ) ( ) ( )cos 4 sin 10 xf x x x e−=

Exercise: Min-Finding

• Find the minimum of the function over the range –π to π. Use fminbnd.

• Plot the function on this range to check that this is the minimum.

• Make the following function:» function y=myFun(x)

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

• Find the minimum in the command window:» x0=fminbnd('myFun',-pi,pi);

• Plot to check if it's right» figure; x=-pi:.01:pi; plot(x,myFun(x));

( ) ( ) ( )cos 4 sin 10 xf x x x e−=

Outline


Numerical Differentiation

• MATLAB can 'differentiate' numerically» x=0:0.01:2*pi;

» y=sin(x);

» dydx=diff(y)./diff(x);

diff computes the first difference

• Can also operate on matrices» mat=[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 detailsThe opposite of diff is the cumulative sum cumsum

• 2D gradient» [dx,dy]=gradient(mat);

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Numerical Integration

• MATLAB contains common integration methods

• Adaptive Simpson's quadrature (input is a function)» q=quad('myFun',0,10);

q is the integral of the function myFun from 0 to 10

» q2=quad(@(x) sin(x)*x,0,pi)

q2 is the integral of sin(x)*x 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 pi» z2=trapz(x,sqrt(exp(x))./x)

z2 is the integral of from 0 to pixe x

Outline


ODE Solvers: Method

• Given a differential equation, the solution can be found by integration:

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

ODE Solvers: MATLAB

• MATLAB contains implementations of common ODE solvers

• Using the correct ODE solver can save you lots of time and give more accurate results» ode23

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

» ode45

High order (Runge-Kutta) solver. High accuracy and reasonable speed. Most commonly used.

» ode15s

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

ODE Solvers: Standard Syntax

• To use standard options and variable time step» [t,y]=ode45('myODE',[0,10],[1;0])

• Inputs:ODE function name (or anonymous function). This function takes inputs (t,y), and returns dy/dtTime interval: 2-element vector specifying initial and final timeInitial conditions: column vector with an initial condition for each ODE. This is the first input to the ODE function

• Outputs:t contains the time pointsy contains the corresponding values of the integrated variables.

ODE integrator: 23, 45, 15s

ODE function Time range

Initial conditions

ODE Function• The ODE function must return the value of the derivative at

a given time and function value

• Example: chemical reactionTwo equations

ODE file:

– y has [A;B]– dydt has

[dA/dt;dB/dt]

A B

10

5010 50

10 50

dA A BdtdB A Bdt

= − +

= −


ODE Function: viewing results

• To solve and plot the ODEs on the previous slide:» [t,y]=ode45('chem',[0 0.5],[0 1]);

assumes that only chemical B exists initially» plot(t,y(:,1),'k','LineWidth',1.5);

» hold on;

» plot(t,y(:,2),'r','LineWidth',1.5);

» legend('A','B');

» xlabel('Time (s)');

» ylabel('Amount of chemical (g)');

» title('Chem reaction');

ODE Function: viewing results

• The code on the previous slide produces this figure

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1

Time (s)

Am

ount

of c

hem

ical

(g)

Chem reaction

AB

Higher Order Equations• Must make into a system of first-order equations to use

ODE solvers• Nonlinear is OK!• Pendulum example:

( )

( )

( )

0g sinL

g sinL

let g sinL

x

dxdt

θ θ

θ θ

θ γ

γ θ

θγ

θγ

+ =

= −

=

= −

⎡ ⎤= ⎢ ⎥⎣ ⎦⎡ ⎤

= ⎢ ⎥⎣ ⎦

&&

&&

&

&

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]);

assume pendulum is almost horizontal» 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

8PositionVelocity

Position in terms ofangle (rad)

Velocity (m/s)

Plotting the Output

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

» xlabel('Position');

» yLabel('Velocity');

• The phase plane is just a plot of one variable versus the other:

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

-6

-4

-2

0

2

4

6

8

Position

Vel

ocity

Velocity is greatest when theta=0

Velocity=0 when theta 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]);

Specify the timestep by giving a vector of timesThe function value will be returned at the specified pointsFixed timestep is usually slower because function values are interpolated to give values at the desired timepoints

• You can customize the error tolerances using odeset» options=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 than RelTol times the value at that step, and less than AbsTol

Decreasing error tolerance can considerably slow the solverSee doc odeset for a list of options you can customize

Exercise: ODE

• Use ode45 to solve for on the range t=[0 10], with initial condition and

• Plot the result.10dy dt t y= −

( )y t( )0 10y =

Exercise: ODE

• Use ode45 to solve for on the range t=[0 10], with initial condition and

• Plot the result.

• Make the following function» function dydt=odefun(t,y)

» dydt=-t*y/10;

• Integrate the ODE function and plot the result» [t,y]=ode45(‘odefun’,[0 10],10);

• Alternatively, use an anonymous function» [t,y]=ode45(@(t,y) –t*y/10,[0 10],10);

• Plot the result» plot(t,y);xlabel('Time');ylabel('y(t)');

10dy dt t y= −( )y t

( )0 10y =

Exercise: ODE

• The integrated function looks like this:

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Time

y(t)

Function y(t), integrated by ode45

End of Lecture 3


We're almost done!





http://ocw.mit.edu



IAP 2010

Lecture 4: Advanced Methods

Homework 3 Recap

• How long did it take?

• Common issues:• The ODE file should be separate from the command that

solves it. ie. you should not be calling ode45 from within your ODE file

• The structure of the output of an ode solver is to have time running down the columns, so each column of y is a variable, and the last row of y are the last values

• HW 4 was updated today, so download it again if you already started. Show a juliaAnimation

• Today is the last required class: make sure the sign-in sheet is accurate regarding your credit/listener status

Outline

(1) Probability and Statistics(2) Data Structures(3) Images and Animation(4) Debugging(5) Online Resources

Statistics

• Whenever analyzing data, you have to compute statistics» scores = 100*rand(1,100);

• 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, 25…95» N=histc(scores,0:10:100);

returns the number of occurrences between the specified bin edges 0 to <10, 10 to <20…90 to <100. you can plot these manually:

» bar(0:10:100,N,'r')

Random Numbers

• Many probabilistic processes rely on random numbers

• MATLAB contains the common distributions built in» rand

draws from the uniform distribution from 0 to 1» randn

draws from the standard normal distribution (Gaussian)» random

can give random numbers from many more distributionssee doc random for helpthe docs also list other specific functions

• You can also seed the random number generators» rand('state',0); rand(1); rand(1); rand('state',0); rand(1);

Changing Mean and Variance

• We can alter the given distributions» y=rand(1,100)*10+5;

gives 100 uniformly distributed numbers between 5 and 15» y=floor(rand(1,100)*10+6);

gives 100 uniformly distributed integers between 10 and 15. 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 script‘brown’

• Make a 10,000 element vector of zeros• Write a loop to keep track of the particle’s position at each time• Start at 0. To get the new position, pick a random number, and if

it’s <0.5, go left; if it’s >0.5, go right. Store each new position in the 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 script‘brown’

• Make a 10,000 element vector of zeros• Write a loop to keep track of the particle’s position at each time• Start at 0. To get the new position, pick a random number, and if

it’s <0.5, go left; if it’s >0.5, go right. Store each new position in the kth position in the vector

• Plot a 50 bin histogram of the positions.

» x=zeros(10000,1);» for n=2:10000» if rand<0.5» x(n)=x(n-1)-1;» else» x(n)=x(n-1)+1;» end» end» figure;» hist(x,50);

Outline


Advanced Data Structures

• We have used 2D matricesCan have n-dimensionsEvery element must be the same type (ex. integers, doubles, characters…)Matrices are space-efficient and convenient for calculationLarge matrices with many zeros can be made sparse:

» a=zeros(100); a(1,3)=10;a(21,5)=pi; b=sparse(a);

• Sometimes, more complex data structures are more appropriate

Cell array: it's like an array, but elements don't have to be the same typeStructs: can bundle variable names and values into one structure

– Like object oriented programming in MATLAB

Cells: organization

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

• One cell can contain people's names, ages, and the ages of their 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

[ ]

Cells: initialization

• To initialize a cell, specify the size» a=cell(3,10);

a will be a cell with 3 rows and 10 columns

• or do it manually, with curly braces {}» c={'hello world',[1 5 6 2],rand(3,2)};

c is a cell with 1 row and 3 columns

• Each element of a cell can be anything

• To access a cell element, use curly braces {}» a{1,1}=[1 3 4 -10];

» a{2,1}='hello world 2';

» a{1,2}=c{3};

Structs

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

• To initialize an empty struct:» s=struct([]);

size(s) will be 1x1initialization is optional but is recommended when using large structs

• To add fields» s.name = 'Jack Bauer';

» s.scores = [95 98 67];

» s.year = 'G3';

Fields can be anything: matrix, cell, even structUseful for keeping variables together

• For more information, see doc struct

Struct Arrays

• To initialize a struct array, give field, values pairs» ppl=struct('name',{'John','Mary','Leo'},...'age',{32,27,18},'childAge',{[2;4],1,[]});

size(s2)=1x3every cell must have the same size

» person=ppl(2);

person is now a struct with fields name, age, childrenthe values of the fields are the second index into each cell

» person.name

returns 'Mary'» ppl(1).age

returns 32

ppl ppl(1) ppl(2) ppl(3)

name: 'John' 'Mary' 'Leo'

age: 32 27 18

childAge: [2;4] 1 []

Structs: access

• To access 1x1 struct fields, give name of the field» stu=s.name;

» scor=s.scores;

1x1 structs are useful when passing many variables to a function. put them all in a struct, and pass the struct

• To access nx1 struct arrays, use indices » person=ppl(2);

person is a struct with name, age, and child age» personName=ppl(2).name;

personName is 'Mary'» a=[ppl.age];

a is a 1x3 vector of the ages; this may not always work, the vectors must be able to be concatenated.

Exercise: Cells

• Write a script called sentGen

• Make a 3x2 cell, and put three names into the first column, and adjectives into the second column

• Pick two random integers (values 1 to 3)• 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 three names into the first column, and adjectives into the second column

• Pick two random integers (values 1 to 3)• 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


Figure Handles

• Every graphics object has a handle» L=plot(1:10,rand(1,10));

gets the handle for the plotted line» A=gca;

gets the handle for the current axis» F=gcf;

gets the handle for the current figure

• To see the current property values, use get» get(L);

» yVals=get(L,'YData');

• To change the properties, use set» set(A,'FontName','Arial','XScale','log');

» set(L,'LineWidth',1.5,'Marker','*');

• Everything you see in a figure is completely customizable through handles

Reading/Writing Images

• Images can be imported into matlab» im=imread('myPic.jpg');

• MATLAB supports almost all image formatsjpeg, tiff, gif, bmp, png, hdf, pcx, xwd, ico, cur, ras, pbm, pgm, ppmsee help imread for a full list and details

• To write an image, give an rgb matrix or indices and colormap» imwrite(rand(300,300,3),'test1.jpg');

» imwrite(ceil(rand(200)*256),jet(256),...'test2.jpg');

see help imwrite for more options

Animations

• MATLAB makes it easy to capture movie frames and play them back automatically

• The most common movie formats are:avianimated gif

• Avigood when you have ‘natural’ frames with lots of colors and few clearly defined edges

• Animated gifGood for making movies of plots or text where only a few colors exist (limited to 256) and there are well-defined lines

Making Animations

• Plot frame by frame, and pause in between» close all

» for t=1:30

» imagesc(rand(200));

» colormap(gray);

» pause(.5);

» end

• Can also use drawnow instead of pause• When plotting lines or points, it's faster to change the

xdata and ydata properties rather than plotting each time» h=plot(1:10,1:10);

» set(h,'ydata',10:1);

Saving Animations as Movies

• A movie is a series of captured frames» close all

» for n=1:30

» imagesc(rand(200));

» colormap(gray);

» M(n)=getframe;

» end

• To play a movie in a figure window» movie(M,2,30);

Loops the movie 2 times at 30 frames per second

• To save as an .avi file on your hard drive» movie2avi(M,'testMovie.avi','FPS',30, ... 'compression', 'cinepak');

• See doc movie2avi for more information

Making Animated GIFs

• You can use imwrite to save an animated GIF. Below is a trivial example

» temp=ceil(rand(300,300,1,10)*256);

» imwrite(temp,jet(256),'testGif.gif',...'delaytime',0.1,'loopcount',100);

• Alternatively, you can use getframe, frame2im, and rgb2ind to convert any plotted figure to an indexed image and then stack these indexed images into a 4-D matrix and pass them to imwrite. Read the doc on imwrite and these other functions to figure out how to do this.

Outline


display

• When debugging functions, use disp to print messages» disp('starting loop')

» disp('loop is over')

disp prints the given string to the command window

• 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 breakpoints

Click on – next to line numbers in MATLAB filesEach red dot that appears is a breakpointRun the programThe 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:


Performance Measures

• It can be useful to know how long your code takes to runTo predict how long a loop will takeTo pinpoint inefficient code

• You can time operations using tic/toc:» tic

» CommandBlock1

» a=toc;

» CommandBlock2

» b=toc;

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

Performance Measures

• For more complicated programs, use the profiler» profile on

Turns on the profiler. Follow this with function calls» profile viewer

Displays gui with stats on how long each subfunction took


Outline


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/

End of Lecture 4


THE END





http://ocw.mit.edu

6.094Introduction to Programming in MATLAB

Lecture 5: Symbolics, Simulink®,

File I/O, Building GUIs

Instructor:Danilo Šćepanović

IAP 2010

Outline

(1) Symbolic Math(2) Simulink(3) File I/O(4) Graphical User Interfaces

Symbolic Math Toolbox

• Don’t do nasty calculations by hand!• Symbolics vs. Numerics

Advantages Disadvantages

Symbolic •Analytical solutions•Lets you intuit things about solution form

•Sometimes can't be solved•Can be overly complicated

Numeric •Always get a solution•Can make solutions accurate•Easy to code

•Hard to extract a deeper understanding•Num. methods sometimes fail•Can take a while to compute

Symbolic Variables

• Symbolic variables are a type, like double or char

• To make symbolic variables, use sym» a=sym('1/3');

» b=sym('4/5');

» mat=sym([1 2;3 4]);

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

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

• Or use syms» syms x y real

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

Symbolic Expressions

• Multiply, add, divide expressions» d=a*b

does 1/3*4/5=4/15;

» expand((a-c)^2);

multiplies out

» factor(ans)

factors the expression

» matInv=inv(mat)

Computes inverse symbolically

Cleaning up Symbolic Statements

» pretty(ans)

makes it look nicer

» collect(3*x+4*y-1/3*x^2-x+3/2*y)

collects terms

» simplify(cos(x)^2+sin(x)^2)

simplifies expressions

» subs(‘c^2’,c,5)

Replaces variables with numbersor expressions. To do multiple substitutionspass a cell of variable names followed by a cell of values

» subs(‘c^2’,c,x/7)

ans= 25

ans= 1/49*x^2

More Symbolic Operations

• We can do symbolics with matrices too» mat=sym('[a b;c d]');

» mat2=mat*[1 3;4 -2];

compute the product» d=det(mat)

compute the determinant» i=inv(mat)

find the inverse

• You can access symbolic matrix elements as before» i(1,2)

Exercise: Symbolics

• The equation of a circle of radius r centered at (a,b) is given by:

• Use solve to solve this equation for x and then for y

• It’s always annoying to integrate by parts. Use int to do the following integral symbolically and then compute the value by substituting 0 for a and 2 for b:

( ) ( )2 2 2x a y b r− + − =

bx

a

xe dx∫

Exercise: Symbolics

• The equation of a circle of radius r centered at (a,b) is given by:

• Use solve to solve this equation for x and then for y

» syms a b r x y

» solve('(x-a)^2+(y-b)^2=r^2','x')

» solve('(x-a)^2+(y-b)^2=r^2','y')

• It’s always annoying to integrate by parts. Use int to do the following integral symbolically and then compute the value by substituting 0 for a and 2 for b:

» Q=int(x*exp(x),a,b)

» subs(Q,{a,b},{0,2})

( ) ( )2 2 2x a y b r− + − =

bx

a

xe dx∫

Outline


SIMULINK• Interactive graphical environment • Block diagram based MATLAB add-on environment• Design, simulate, implement, and test control, signal

processing, communications, and other time-varying systems


Getting Started

• In MATLAB,Start Simulink

• Create a new Simulink file,similar to howyou make a newscript



Simulink Library Browser• The Library Browser contains various blocks that you can

put into your model• Examine some blocks:

Click on a library: “Sources”– Drag a block into Simulink: “Band limited white noise”

Visualize the block by going into “Sinks”– Drag a “Scope” into Simulink


Connections

• Click on the carat/arrow on the right of the band limited white noise box

• Drag the line to the scopeYou’ll 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


Connections, Block Specification• To split connections, hold down ‘Ctrl’ when clicking on a

connection, and drag it to the target block; or drag backwards from the target block

• To modify properties of a block, double-click it and fill in the property values.


Behind the curtain• Go to “Simulation”->”Configuration Parameters”

at the top menuSee ode45? Change the solver type here


Exercise: Simulink

• Take your white noise signal, and split it into high frequency and low frequency components. Use the Transfer Function block from Continuous and use these transfer functions:

• Hook up scopes to the input and the two outputs• Send the two outputs to the workspace by using the to

Workspace block from Sink

10.1 1

LPs

=+

0.10.1 1

sHPs

=+

Exercise: Simulink• The diagram should look like this. To change the transfer

function parameters, double click the blocks and specify the numerator and denominator as polynomials in s (remember how we defined polynomial vectors before)


Exercise: Simulink

• After running the simulation, double-clicking the scopes will show:

InputLow pass

High Pass


Toolboxes

• MathTakes the signal and performs a math operation

» Add, subtract, round, multiply, gain, angle

• ContinuousAdds differential equations to the system

» Integrals, Derivatives, Transfer Functions, State Space

• DiscontinuitiesAdds nonlinearities to your system

• DiscreteSimulates discrete difference equationsUseful for digital systems

Building systems

• Sources» Step input, white noise, custom input, sine wave, ramp input,

Provides input to your system

• Sinks » Scope: Outputs to plot

» simout: Outputs to a MATLAB vector on workspace

» MATLAB mat file

Outline


Importing Data

• MATLAB is a great environment for processing data. If you have a text file with some data:

• To import data from files on your hard drive, use importdata» a=importdata('textFile.txt');

a is a struct with data, textdata, and colheaders fields

» x=a.data;

» names=a.colheaders;

Importing Data

• With importdata, you can also specify delimiters. For example, for comma separated values, use:» a=importdata('filename', ', ');

The second argument tells matlab that the tokens of interest are separated by commas or spaces

• importdata is very robust, but sometimes it can have trouble. To read files with more control, use fscanf (similar to C/Java), textread, textscan. See help or doc for information on how to use these functions

Writing Excel Files

• MATLAB contains specific functions for reading and writing Microsoft Excel files

• To write a matrix to an Excel file, use xlswrite» [s,m]=xlswrite('randomNumbers',rand(10,4),... 'Sheet1'); % we specify the sheet name

• You can also write a cell array if you have mixed data:» C={'hello','goodbye';10,-2;-3,4};

» [s,m]=xlswrite('randomNumbers',C,'mixedData');

• s and m contain the 'success' and 'message' output of the write command

• See doc xlswrite for more usage options

Reading Excel Files

• Reading excel files is equally easy• To read from an Excel file, use xlsread

» [num,txt,raw]=xlsread('randomNumbers.xls');

Reads the first sheetnum contains numbers, txt contains strings, raw is the entire cell array containing everything

» [num,txt,raw]=xlsread('randomNumbers.xls',... 'mixedData');

Reads the mixedData sheet» [num,txt,raw]=xlsread('randomNumbers.xls',-1);

Opens the file in an Excel window and lets you click on the data you want!

• See doc xlsread for even more fancy options

Outline


Making GUIs• It's really easy to make a graphical user interface in

MATLAB• To open the graphical user interface development

environment, type guide» guide

Select Blank GUI


Draw the GUI• Select objects from the left, and draw them where you

want them


Change Object Settings• Double-click on objects to open the Inspector. Here you can

change all the object's properties.


Save the GUI• When you have modified all the properties, you can save

the GUI• MATLAB saves the GUI as a .fig file, and generates an

MATLAB file!


Add Functionality to MATLAB file• To add functionality to your buttons, add commands to the

'Callback' functions in the MATLAB file. For example, when the user clicks the Draw Image button, the drawimage_Callback function will be called and executed

• All the data for the GUI is stored in the handles, so use setand get to get data and change it if necessary

• Any time you change the handles, save it using guidata» guidata(handles.Figure1,handles);


Running the GUI• To run the GUI, just type its name in the command window

and the GUI will pop up. The debugger is really helpful for writing GUIs because it lets you see inside the GUI


Outline


Now you know EVERYTHING!


6.094 Introduction to MATLAB®January (IAP) 2010



http://ocw.mit.edu

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