MATLAB Programming

First Edition, 2009

ISBN 978 93 80168 38 8

© All rights reserved.

Published by: Global Media 1819, Bhagirath Palace, Chandni Chowk, Delhi-110 006 Email: [email protected]

Table of Contents 1. Basic MATLAB Concepts 2. Data Storage and Manipulation 3. Graphics 4. M File Programming 5. Mathematical Manipulations 6. More advanced I-O & Examples 7. Object-Oriented Programming 8. Comparing Octave and MATLAB 9. Toolboxes

Basic MATLAB Concepts The Current Directory and Defined Path It is necessary to declare a current directory before saving a file, loading a file, or running an M-file. By default, unless you edit the MATLAB shortcut, the current directory will be .../MATLAB/work. After you start MATLAB, change the current directory by either using the toolbar at the left-hand side of the screen, or entering the path in the bar at the top. The current directory is the directory MATLAB will look in first for a function you try to call. Therefore if you have multiple folders and each of them has an M-file of the same name, there will not be a discrepancy if you set the current directory beforehand. The current directory is also the directory in which MATLAB will first look for a data file. If you still want to call a function but it is not part of the current directory, you must define it using MATLAB's 'set path' utility. To access this utility, follow the path: file > set path... > add folder... You could also go to "add folder with subfolders...", if you're adding an entire group ,as you would if you were installing a toolbox. Then look for and select the folder you want. If you forget to do this and attempt to access a file that is not part of your defined path list, you will get an 'undefined function' error.

Saving Files There are many ways to save to files in MATLAB. • • • •

save - saves data to files, *.mat by default uisave - includes user interface hgsave - saves figures to files, *.fig by default diary [filename] - saves all the text input in the command window to a text file.

All of them use the syntax: save filename.ext

or similar for the other functions. The files are saved in your current directory, as seen on the top of the window. By default the current directory is .../MATLAB/work.

Loading Files

Likewise, there are many ways to load files into the workspace. One way is to use the "file" menu. To open a .m file click "open", whereas to import data from a data file select "import data..." and follow the wizard's instructions. An alternative way to load a saved .mat file (within a function, for example) is to type: >> load filename.ext

The file must be in a recognized directory (usually your current directory, but at least one for which the path has been set). The data in the .mat file is stored with the same name as the variable originally had when it was saved. To get the name of this and all other environment variables, type "who". To open an .m file, you can use file -> open, or type >>open filename.ext

File Naming Constraints You can name files whatever you want (usually simpler is better though), with a few exceptions: •

"

MATLAB for Windows retains the file naming constraints set by DOS. The following characters cannot be used in filenames: / : * < > | ?

•

You're not allowed to use the name of a reserved word as the name of a file. For example, while.m is not a valid file name because while is one of MATLAB's reserved words.

•

When you declare an m-file function, the m-file must be the same name as the function or MATLAB will not be able to run it. For example, if you declare a function called 'factorial':

function Y = factorial(X)

You must save it as "factorial.m" in order to use it. MATLAB will name it for you if you save it after typing the function declaration, but if you :change the name of the function you must change the name of the file manually, and vice versa.

Introduction MATLAB is interesting in that it is dynamically compiled. In other words, when you're using it, you won't run all your code through a compiler, generate an executable, and then

run the executable file to obtain a result. Instead, MATLAB simply goes line by line and performs the calculations without the need for an executable. Partly because of this, it is possible to do calculations one line at a time at the command line using the same syntax as would be used in a file. It's even possible to write loops and branches at the command line if you want to! Of course this would lead to a lot of wasted effort a lot of the time, so doing anything beyond very simple calculations, testing to see if a certain function, syntax, etc. works, or calling a function you put into an .m file should be done within an .m file.

Calculator MATLAB, among other things, can perform the functions of a simple calculator from the command line. Let us try to solve a simple problem: Sam's car's odometer reading was 3215 when he last filled the fuel tank. Yesterday he checked his odometer and it read 3503. He filled the tank and noticed that it took 10 gallons to do that. If his car's gas tank holds 15.4 gallons, how long can he drive before he is going to run out of gas, assuming the gas mileage is the same as before? First let us compute the distance Sam's car has travelled in between the two gas fillings >> 3503-3215 ans = 288

Gas mileage of Sam's car is >> 288/10 ans = 28.8

With this, he can drive >> 28.8 * 15.4 ans = 443.5200

443.52 miles before he runs out of gas. Let us do the same example, now by creating named variables >> distance = 3503-3215 distance = 288 >> mileage = distance/10 mileage = 28.8000 >> projected_distance = mileage * 15.4 projected_distance =

443.5200

To prevent the result from printing out in the command window, use a semicolon after the statement. The result will be stored in memory. You can then access the variable by calling its name. Example: >>projected_distance = mileage * 15.4; >> >>projected_distance projected_distance = 443.5200

Using the command line to call functions The command line can be used to call any function that's in a defined path. To call a function, the following general syntax is used: >> [Output variable 1, output variable 2, ...] = functionname(input 1, input 2, input 3,...)

MATLAB will look for a file called functionname.m and will execute all of the code inside it until it either encounters an error or finishes the file. If the former is true, you'll hear an annoying noise and will get an error message in red. In the latter case, MATLAB will relinquish control to you, which you can see when the >> symbol is visible on the bottom of the workspace and the text next to "start" button on the bottom-left of the screen says "ready". Use this in order to call homemade functions as well as those built into MATLAB. MATLAB has a large array of functions, and the help file as well as this book are good places to look for help on what you need to provide as inputs and what you will get back. BE CAREFUL because the syntax for functions and for indexing arrays is the same! To avoid confusion, just make sure you don't name a variable the same as any function. To ensure this, type the name of the variable you want to define in the command prompt. If it tells you: Error using ==> (functionname) Not enough input arguments.

then you'll have a conflict with an existing function. If it tells you: ??? Undefined function or variable '(functionname)'

you'll be OK.

Reading and Writing from a file: Command Line Reading and Writing data from/to a .mat file The quickest mean of saving and retrieving data is through the binary .mat file format MATLAB provides. This is the native format for MATLAB. •

Note: This author has had some problems with certain classes not being saved correctly when saving data using version 7 for use in version 6. Most data items will work just fine. Of particular interest was an issue with State-Space objects that were saved using version 7 to a version 6 compatible file. When the file was opend in MATLAB version 6+ the State-Space objects did not load.Spradlig (talk) 04:52, 31 March 2008 (UTC)

Saving Data The save command is the used to save workspace data to a file. •

Save all workspace data to the file mySave.mat in the current directory.

>> save('mySave.mat') >> save(fullfile(pwd, 'mySave.mat')) •

Save just the variables myData1 and myData2 to mySave.mat.

>> save('mySave.mat', 'myData1', 'myData2') •

Save all myData variables to mySave.mat.

>> save('mySave.mat', 'myData*') •

Save all myData variables to a mySave.mat file compatible with version 6 of MATLAB.

>> save('mySave.mat', 'myData*', '-v6') •

Save all myData variables to an ASCII file.

>> save('mySave.txt', 'myData*', '-ASCII') •

Append new variables to the data file.

>> save('mySave.mat', 'newData*', '-append')

Loading Data The load command is used to load data from a file into the current workspace. •

Load all variables from the file mySave.mat into the current workspace.

>> load('mySave.mat') >> load(fullfile(pwd, 'mySave.mat')) •

Load just the variables myData1 and myData2.

>> load('mySave.mat', 'myData1', 'myData2') •

Load all myData variables.

>> load('mySave.mat', 'myData*') •

Get a cell array of variables in saved file.

>> whos('-file', 'mySave.mat')

Reading and Writing from an Excel spreadsheet Since analyzing data is one of the more common motivations for using input output I will start with reading and writing from a spreadsheet. I cover the command line first since it is often necessary to import the data while an m-function is being evaluated. MATLAB makes it easy to read from an Excel spreadsheet. It has the built in command "xlsread". To use the xlsread function use the syntax: >>g=xlsread('filename');

This line of code reads filename.xls (from the current directory) and places it in an identical array inside MATLAB called g. You can then manipulate the array g any way you want. Make sure that the file you choose is in the same directory were you save your M-files (usually the work directory) otherwise you get an error. You can specify the path to a file but, this can get messy. To write data to an .xls the procedure is very similar. The xlswrite command below creates a spreadsheet called filename.xls in the current directory from the variable g: >> xlswrite('filename',g);

NOTE: if you are using MATLAB 6.5 there is no "xlswrite" command (that I'm aware of). There are several ways to write to a file. The simplest way I have found is fid=fopen('newFile.xls', 'w');

fprintf(fid,'%6.3f %6.3f %10.3f\n', g); fclose(fid);

You can substitute newFile.xls with .txt. Also, there might be some issues with formatting in Excel. The formatting issues can usually be handled inside Excel but if they can't you might have to play around with the fopen command parameters. This is pretty similar (if not the same) way you would write to a file in C.

Reading and Writing from and to other text files If a file is not an excel spreadsheet, it can still be read using "load" function: >> load newfile.txt

This works only if the text is entirely numerical, without special formatting. Otherwise you get a 'unrecognized character' error. The easiest way to write to a non-excel file, or using MATLAB 6.5 or less, is to use the same code as that for writing excel files but change the extension. Usually there are no formatting difficulties with plain text files.

Reading and Writing from a data file: GUI MATLAB contains a nice GUI application that will guide you through importing data from any recognized data file (usually .mat, .txt, or .xls on a Windows system). To use it, go to file > import data, and select the file you want. Then, choose what column separators are present (by selecting the appropriate radio button). Finally, click "next". MATLAB saves the variable under a name similar to that of the file, but with modifications to make it conform with MATLAB syntax. Spaces are omitted, plusses and minuses are turned into other characters. To see the name MATLAB generated (and probably change it) type "who" in the command prompt.

Data Storage and Manipulation Data Types and Operations on Point Values

Introduction A large number of MATLAB's functions are operations on two types of numbers: rational numbers and boolean numbers. Rational numbers are what we usually think of when we think of what a number is. 1, 3, and -4.5 are all rational numbers. MATLAB stores rational numbers as doubles by default, which is a measure of the number of decimal places that are stored in each variable and thus of how accurate the values are. Note that MATLAB represents irrational numbers such as pi with rational approximations, except when using the symbolic math toolbox. See that section for details. Boolean numbers are either "TRUE" or "FALSE", represented in MATLAB by a 1 and a 0 respectively. Boolean variables in MATLAB are actually interchangable with doubles, in that boolean operators can be performed with arrays of doubles and vice versa. Any non-zero number in this case is considered "TRUE". Most of the rational operators also work with complex numbers. Complex numbers, however, cannot be interchanged with boolean values like the real rationals can. See the complex numbers section for details on how to use them.

Rational Operators on Single Values MATLAB has all the standard rational operators. It is important to note, however, that Unless told otherwise, all rational operations are done on entire arrays, and use the matrix definitions. Thus, even though for now we're only talking about operations on a single value, when we get into arrays, it will be important to distinguish between matrix and componentwise multiplication, for example. Add, Subtract, multiply, divide, exponent operators: %addition a = 1 + 2 %subtraction b = 2 - 1 %matrix multiplication c = a * b %matrix division (pseudoinverse) d = a / b %exponentiation e = a ^ b

The modulo function returns the remainder when the arguments are divided together, so a modulo b means the remainder when a is divided by b. %modulo remainder = mod(a,b)

All of these functions except for the modulus work for complex numbers as well.

Boolean Operators on Single Values The boolean operators are & (boolean AND) | (boolean OR) and ~ (boolean NOT /negation). A value of zero means false, any non-zero value (usually 1) is considered true. Here's what they do: >>%boolean >> y = 1 & y = 0 >> y = 1 & y = 1 >>%boolean >> y = 1 | y = 1 >> y = 1 | y = 1

AND 0 1 OR 0 1

The negation operation in MATLAB is given by the symbol ~, which turns any FALSE values into TRUE and vice versa: >> c = (a == b) c = 1 >> ~c ans = 0

This is necessary because conditionals (IF/SWITCH/TRY) and loops (FOR/WHILE) always look for statements that are TRUE, so if you want it to do something only when the statement is FALSE you need to use the negation to change it into a true statement. The NOT operator has precedence over the AND and OR operators in MATLAB unless the AND or OR statements are in parenthesis: >> y = ~1 & 0 y = 0 >> y = ~(1&0) y = 1

Relational Operators

Equality '==' returns the value "TRUE" (1) if both arguments are equal. This must not be confused with the assignment operator '=' which assigns a value to a variable. >> %relational >>a=5;b=5; >>a==b ans = 1 %Assignment >>a=5;b=3; >>a=b a = 3

Note that in the first case, a value of 1 (true) is returned, however for the second case a gets assigned the value of b. Greater than, less than and greater than or equal to, less than or equal to are given by >, =, >a=3;b=5; >>a>b> fstring = 'hello' fstring = hello

If you would like to include a single quote this is one way to do it: >> fstring = '''' fstring = ' >> fstring = 'you''re' fstring = you're

An important thing to remember about strings is that MATLAB treats them as array of characters. To see this, try executing the following code:

>> fstring = 'hello'; >> class(fstring) ans = char

Therefore, many of the array manipulation functions will work the same with these arrays as any other, such as the 'size' function, transpose, and so on. You can also access specific parts of it by using standard indexing syntax. Attempting to perform arithmetic operations on character arrays converts them into doubles. >> fstring2 = 'world'; >> fstring + fstring2 ans = 223 212 222

216

211

These numbers come from the standard numbers for each character in the array. These values are obtained by using the 'double' function to turn the array into an array of doubles. >> double(fstring) ans = 104 101 108

108

111

The 'char' function can turn an array of integer-valued doubles back into characters. Attempting to turn a decimal into a character causes MATLAB to round down: >> char(104.6) ans = h >> char(104) ans = h

Since the MATLAB strings are treated as character arrays, they have some special functions if you wish to compare the entire string at once rather than just its components: findstr(bigstring, smallstring) looks to see if a small string is contained in a bigger string, and if it is returns the index of where the smaller string starts. Otherwise it returns []. strrep(string1, replaced, replacement) replaces all instances of replaced in string1 with replacement

Displaying values of string variables If all you want to do is display the value of a string, you can omit the semicolon as is standard in MATLAB. If you want to display a string in the command window in combination with other text you must use array notation combined with either the 'display' or the 'disp' function: >> fstring = 'hello';

>> display( [ fstring 'world'] ) helloworld

MATLAB doesn't put the space in between the two strings. If you want one there you must put it in yourself. This syntax is also used to concatenate two or more strings into one variable, which allows insertion of unusual characters into strings: >> fstring = ['you' char(39) 're'] fstring = you're

Any other function that returns a string can also be used in the array.

Comparing strings Unlike with rational arrays, strings will not be compared correctly with the relational operator, because this will treat the string as an array of characters. To get the comparison you probably intended, use the strcmp function as follows: >> string1 = 'a'; >> strcmp(string1, 'a') ans = 1 >> strcmp(string1, 'A') ans = 0

Note that MATLAB strings are case sensitive so that 'a' and 'A' are not the same. In addition the strcmp function does not discard whitespace: >> strcmp(string1, ' a') ans = 0

The strings must be exactly the same in every respect. If the inputs are numeric arrays then the strcmp function will return 0 even if the values are the same. Thus it's only useful for strings. Use the == operator for numeric values. >> strcmp(1,1) ans = 0.

This section discusses inline functions, anonymous functions, and function handles. Each of these is portable in that rather than having to write an equation multiple times in a program, you can just define it once and then call it whenever you want to use it. In addition, function handles in particular allow you to pass an equation to another function for direct evaluation as needed. Inline and anonymous functions are useful for commandline evaluation or for multiple evaluations in the same m-file.

Inline functions An inline function is one way that a function can be created at the command or in a script: >>f = inline('2*x^2-3*x+4','x'); >>f(3) ans = 13

To create a multi-variable inline function, send more inputs to the 'inline' function: >> f = inline('2*x*y', 'x', 'y'); >> f(2,2) ans = 8

You cannot make an array of inline functions, MATLAB will tell you that you have too many inputs or do something weird like that.

Anonymous functions A function can also be created at the command or in a script: >>f = @(x) 2*x^2-3*x+4; >>f(3) ans = 13

To make an anonymous function of multiple variables, use a comma-separated list to declare the variables: >>f = @(x,y) 2*x*y; >>f(2,2) ans = 8

It is possible to make an array of anonymous functions in MATLAB 7.1 but this will become outdated soon so using this construct in a distributed program is not recommended. To pass anonymous functions to other functions, just use the name of the anonymous function in your call: >> f = @(t,x) x; >> ode45(f, [0:15],1)

Function Handles

A function handle passes an m-file function into another function. This of course lets you have more control over what's passed there, and makes your program more general as it lets you pass any m-file (as long as it meets other requirements like having the right number of input arguments and so on). To pass an m-file to a function, you must first write the m-file, say something like this: function xprime = f(t,x) xprime = x;

Save it as myfunc.m. To pass this to another function, say an ODE integrator, use the @ symbol as follows: >> ode45(@myfunc, [0:15], 1)

One advantage of using function handles over anonymous functions is that you can evaluate more than one equation in the m-file, thus allowing you to do things like solve systems of ODEs rather than only one. Anonymous functions limit you to one equation.

Declaring a complex number in MATLAB Complex numbers in MATLAB are doubles with a real part and an imaginary part. The imaginary part is declared by using the 'i' or 'j' character. For example, to declare a variable as '1 + i' just type: >> compnum = 1 + i compnum = 1.000 + 1.000i >> compnum = 1 + j compnum = 1.000 + 1.000i

Note that if you use j MATLAB still displays i on the screen. Since i is used as the complex number indicator it is not recommended to use it as a variable, since it will assume it's a variable if given a choice. >> i = 3; %bad idea >> a = 1 + i a = 4

However, since implicit multiplication is not normally allowed in MATLAB, it is still possible to declare a complex number like this: >> i = 3; >> a = 1i + 1 a = 1.000 + 1.000i

It's best still not to declare i as a variable, but if you already have a long program with i as a variable and need to use complex numbers this is probably the best way to get around it. If you want to do arithmetic operations with complex numbers make sure you put the whole number in parenthesis, or else it likely will not give the intended results.

Arithmetic operations that create complex numbers There are several operations that create complex numbers in MATLAB. One of them is taking an even root of a negative number, by definition. >> (-1)^0.5 ans = 0.000 + 1.000i >> (-3)^0.25 ans = 0.9306 + 0.9306i

As a consequence of the Euler formula, taking the logarithm of a negative number also results in imaginary answers. >> log(-1) ans = 0 + 3.1416i

In addition, the roots of functions found with the 'roots' function (for polynomials) or some other rootfinding function will often return complex answers.

MATLAB functions to manipulate complex values First of all, it is helpful to tell whether a given matrix is real or complex when programming, since certain operations can only be done on real numbers. Since complex numbers don't have their own class, MATLAB comes with another function called 'isreal' to determine if a given matrix is real or not. It returns 0 if any of the inputs are complex. >> A = [1 + i, 3]; >> isreal(A) ans = 0 >> isreal(A(2)) ans = 1

Notice that it is possible to have real and complex numbers in the same array, since both are of class double. The function is set up this way so that you can use this as part of a conditional, so that a block only is executed if all elements of array A are real. To extract just the real part of a complex variable use the 'real' function. To extract just the complex part use the 'imag' function. >> real(A) ans = 1 3 >> imag(A)

ans = 1

0

One thing you may need to do is perform an operation on the real values of an array but not the complex values. MATLAB does not have a function to directly do this, but the following pair of commands lets you put only the real values into another array: >> RealIndex = (imag(A) == 0); %if imaginary part is zero then the number is real) >> RealOnly = A(RealIndex) RealOnly = 3

Arrays and Matrices

Introduction to Arrays Arrays are the fundamental data type of MATLAB. Indeed, the former data types presented here, strings and number, are particular cases of arrays. As in many traditional languages, arrays in MATLAB are a collection of several values of the same type (by default, the type is equivalent to the C type double on the same architecture. On x86 and powerpc, it is a floating point value of 64 bits). They are indexed through the use of a single integer or, to get more than one value, an array of integers..

Declaring Arrays Row and Column Arrays A simple way to create a row array is to give a comma separated list of values inside brackets: >> array = [0, 1, 4, 5] array = 0 1 4 5

The commas can be omitted for a row array because MATLAB will assume you want a row array if you don't give it any separators. However, the commas make it easier to read and can help with larger arrays. The commas indicate that the array will be a horizontal array. To make a column array you can use semicolons to separate the values. >> column = [1; 2; 3] column = 1 2 3

All elements of an array must be the same data type, so for example you cannot put a function handle and a double in the same array.

Declaring multi-dimensional arrays Arrays can be multi-dimensional. To create a 2 dimensional array (a matrix in Linear Algebra terms), we have to give a list of comma separated values, and each row should be separated by a semi colon: >> matrix = [1, 2, 3; 4, 5, 6] matrix = 1 2 3 4 5 6

In MATLAB the term array is synonymous with matrix and will more often than not be referred to as a matrix. It should be noted that a matrix, as its mathematical equivalent, requires all its rows and all its columns to be the same size: >> matrix = [1, 2, 3; 4, 5] ??? Error using ==> vertcat All rows in the bracketed expression must have the same number of columns.

Properties of MATLAB arrays and matrices Contrary to low level languages such as C, an array in MATLAB is a more high level type of data: it contains various information about its size, its data type, and so on. >> array = [0,1,4,5]; >> length(array) ans = 4 >> class(array) ans = double

The number of rows and columns of the matrix can be known through the built-in size function. Following the standard mathematical convention, the first number is the number of rows and the second is the number of columns: >> matrix = [1, 2, 3; 4, 5, 6]; >> size(matrix) ans = 2 3

The goal of MATLAB arrays is to have a type similar to mathematical vectors and matrices. As such, row and column arrays are not equivalent. Mono-dimensional arrays are actually a special case of multi-dimensional arrays, and the 'size' function can be used for them as well. >> size(array)

ans = 1

4

Row and column do not have the same size, so they are not equivalent: >> size(column) ans = 3 1 >> size(row) ans = 1 3

Why Use Arrays? A major advantage of using arrays and matrices is that it lets you avoid using loops to perform the same operation on multiple elements of the array. For example, suppose you wanted to add 3 to each element of the array [1,2,3]. If MATLAB didn't use arrays you would have to do this using a FOR loop: >> array = [1,2,3]; >> for ii = 1:3 array(ii) = array(ii) + 3; >> end >> array array = [4,5,6]

Doing this is not efficient in MATLAB, and it will make your programs run very slowly. Instead, you can create another array of 3s and add the two arrays directly. MATLAB automatically separates the elements: >> array = [1,2,3]; >> arrayofthrees = [3,3,3]; >> array = array + arrayofthrees array = [4,5,6];

If all you are doing is adding a constant, you can also omit the declaration of 'arrayofthrees', as MATLAB will assume that the constant will be added to all elements of the array. This is very useful, for example if you use an array with variable size: >> array = [1,2,3]; >> array + 3 ans = [4,5,6]

The same rule applies to scalar multiplication. See Introduction to array operations for more information on the operations MATLAB can perform on arrays.

Arrays are a fundamental principle of MATLAB, and almost everything in MATLAB is done through a massive use of arrays. To have a deeper explanation of arrays and their operations, see Arrays and matrices.

Introduction to array operations As arrays are the basic data structure in MATLAB, it is important to understand how to use them effectively. See the previous section for that. Arrays in MATLAB obey the same rule as their mathematical counterpart: by default, the matrix definitions of operations are used, unless a special operator called the dot operator is applied. Because arrays operations are so similar to the equivalent mathematical operations, a basic knowledge of linear algebra is mandatory to use matlab effectively. However, we won't be as precise as in mathematics when using the terms vector and matrix. In MATLAB, both are arrays of doubles (thus being a matrix in the real mathematical meaning), and MATLAB considers vectors as a matrics with only one row or only one column. However, there are special functions just for vectors; see the vector module for an explanation of how to use these.

Basics How to input an array The common way to input an array from the matlab command line is to put the input figures into list into square brackets: >> [1, 2, 3] ans = 1

2

3

Comma is used to separate columns elements, and semicolon is used to separate rows. So [1, 2, 3] is a row vector, and [1; 2; 3] is a column vector >> [1; 2; 3] ans = 1 2 3

If a blankspace is used to seperate elements, the default separator is comma, thus making the vector a row vector.

Logically, inputing a matrix is done by using a comma separated list of column vectors, or a semicolon seperated list of row vectors: >> [1, 2, 3; 4, 5, 6] ans = 1 4

2 5

3 6

Variable assignment To reuse an array in subsequent operations, one should assign the array to a variable. Variable assignement is done through the equal symbol: >> a = [1, 2, 3] a = 1

2

3

Notice that instead of ans =, the name of the variable is displayed by matlab. If you forget to assign the last statement to a variable, the variable ans always point to the last non assigned: >> [1, 2, 3] ans = 1 2 >> a = ans

3

a = 1

2

3

But: >> [1, 2, 3] ans = 1

2

3

>> b = [4, 5, 6] b = 4

5

>> a = ans a =

6

1

2

3

Ie ans is really the last non assigned result, and not the result of the last statement. As it is the case for most interpreted languages, you do not need to declare a variable before using it, and reusing a variable name in an assignement will overwrite the previous content. To avoid cluttering the command line of matlab, you can postfix any command with a semicolon: >> a = [1, 2, 3];

Accessing elements of a matrix Now that you know how to define a simple array, you should know how to access its elements. Accessing the content of an array is done through the operator (), with the index inside the parenthesis; the indexing of the first element is 1: >> a = [1, 2, 3]; >> a(1) ans = 1 >> a(3) ans = 3

Accessing an element outside the bounds will result in an error: >> a(5) ??? Index exceeds matrix dimensions.

To access a single matrix element, you can use the (i,j) subscript, where i is the index in the row, and j in the column: >> a= [1, 2; 3, 4]; >> a(1, 2) ans = 2 >> a(2, 1) ans =

3

You can also access a matrix element through a unique index; in this case, the order is column major, meaning you first go through all elements of the first column, then the 2d column, etc... The column major mode is the same than in Fortran, and the contrary of the order in the C language. >> a = [1, 2, 3; 4, 5, 6]; >> a(3) ans = 2

It is also possible to access blocks of matrices using the colon (:) operator. This operator is like a wildcard; it tells MATLAB that you want all elements of a given dimension or with indices between two given values. For example, say you want to access the entire first row of matrix a above, but not the second row. Then you can write: >> a = [1, 2, 3; 4, 5, 6]; >> a(1,:) %row 1, every column ans = 1 2 3

Now say you only want the first two elements in the first row. To do this, use the following syntax: >> a = [1, 2, 3; 4, 5, 6]; >> a(1, 1:2) ans = 1 2

The syntax a(:) changes a into a column vector (column major): >> a = [1, 2, 3; 4, 5, 6] >> a(:) ans = 1 4 2 5 3 6

Finally, if you do not know the size of an array but wish to access all elements from a certain index until the end of the array, use the end operator, as in >> a = [1, 2, 3; 4, 5, 6]

>> a(1, 2:end) %row 1, columns from 2 until end of the array ans = 2 3

Logical Addressing In addition to index addressing, you can also access only elements of an array that satisfy some logical criterion. For example, suppose a = [1.1, 2.1, 3.2, 4.5] and you only want the values between 2 and 4. Then you can achieve this in two ways. The first is to use the find function to find the indices of all numbers between 2 and 4 in the array, and then address the array with those indices: >> a = [1.1, 2.1, 3.2, 4.5]; >> INDICES = find(a >= 2 & a > a(INDICES) ans = 2.1 3.2

This does not work in Matlab 2006b The second method is to use logical addressing, which first changes a into a logical array, with value 1 if the logical expression is true and 0 if it is false. It then finds and returns all values in the a which are true. The syntax for this is as follows: >> a = [1.1, 2.1, 3.2, 4.5]; >> a(a >= 2 & a > [1, 2, 3] - [1, 2, 1] ans = 0

0

2

The multiplication by a scalar also works as expected: >> 2 * [1, 2, 3] ans =

[2, 4, 6]

Multiplication and division are more problematic: multiplying two vectors in does not make sense. It makes sense only in the matrix context. Using the symbol * in matlab computes the matrix product, which is only defined when the number of columns of the left operand matches the number of rows of the right operand: >> a = [1, 2; 3, 4]; >> a * a ans = 7 10 15 22 >> a = [1, 2, 3]; b = [1; 2; 3]; >> a * a ??? Error using ==> * Inner matrix dimensions must agree. >> a * b ans = 14

Using the division symbol / has even more constraints, as it imposes the right operand to be invertible. For square matrices, a / b is equivalent to a * b − 1. For example : >> a = [1, 2; 3, 4]; b = [1, 2; 1, 2] >> b / a ans = 1 0 1 0 >> a / b Warning: Matrix is singular to working precision. ans = Inf Inf

Inf Inf

If you desire to multiply or divide two matrices or vectors componentwise, or to raise all components of one matrix to the same power, rather than using matrix definitions of these operators, you can use the dot (.) operator. The two matrices must have the same dimensions. For example, for multiplication, >> a = [1, 2, 3]; >> b = [0, 1, 2]; >> a .* b ans =

0

2

6

The other two componentwise operators are ./ and .^. As matlab is a numerical computing language, you should keep in mind that a matrix which is theoritically invertible may lead to precision problems and thus giving imprecise results or even totally wrong results. The message above "matrix is singular to working precision" should appear in those cases, meaning the results cannot be trusted. Non-square matrices can also be used as the right operand of /; in this case, it computes the pseudoinverse. This is especially useful in least square problems.

Boolean Operators on Arrays The same boolean operators that can be used for point values can also be used to compare arrays. To do this, MATLAB compares the elements componentwise and returns them in a logical array of the same size as the two arrays being compared. The two arrays must have the same size. For example, >> A = [2,4], B = [1,5]; >> A < B ans = [0

1]

You must be careful when using comparisons between arrays as loop conditions, since they clearly do not return single values and therefore can cause ambiguous results. The loop condition should be reducable to a single boolean value, T or F, not an array. Two common ways of doing this are the "any" and the "all" functions. A function call any(array) will return true if array contains any nonzero values and false if all values are zero. It does the comparisons in one direction first then the other, so to reduce a matrix you must call the any function twice. The function all, similarly, returns true if and only if all elements in a given row or column are nonzero.

Solving Linear Systems To solve a linear system in the form Ax = b use the "\" operator. Example: >>A = [4 5 ; 2 8]; b = [23 28]'; x = A\b x =

2 3

A vector in MATLAB is defined as an array which has only one dimension with a size greater than one. For example, the array [1,2,3] counts as a vector. There are several operations you can perform with vectors which don't make a lot of sense with other arrays such as matrices. However, since a vector is a special case of a matrix, any matrix functions can also be performed on vectors as well provided that the operation makes sense mathematically (for instance, you can matrix-multiply a vertical and a horizontal vector). This section focuses on the operations that can only be performed with vectors.

Declaring a vector Declare vectors as if they were normal arrays, except all dimensions except for one must have length 1. It does not matter if the array is vertical or horizontal. For instance, both of the following are vectors: >> Horiz = [1,2,3]; >> Vert = [4;5;6];

You can use the isvector function to determine in the midst of a program if a variable is a vector or not before attempting to use it for a vector operation. This is useful for error checking. >> isvector(Horiz) ans = 1 >> isvector(Vert) ans = 1

Another way to create a vector is to assign a single row or column of a matrix to another variable: >> A = [1,2,3;4,5,6]; >> Vec = A(1,:) Vec = 1 2 3

This is a useful way to store multiple vectors and then extract them when you need to use them. For example, gradients can be stored in the form of the Jacobian (which is how the symbolic math toolbox will return the derivative of a multiple variable function) and extracted as needed to find the magnitude of the derivative of a specific function in a system.

Declaring a vector with linear or logarithmic spacing Suppose you wish to declare a vector which varies linearly between two endpoints. For example, the vector [1,2,3] varies linearly between 1 and 3, and the vector [1,1.1,1.2,1.3,...,2.9,3] also varies linearly between 1 and 3. To avoid having to type out

all those terms, MATLAB comes with a convenient function called linspace to declare such vectors automatically: >> LinVector = linspace(1,3,21) Lin Vector = Columns 1 through 9 1.0000 1.1000 1.2000 1.3000 1.7000 1.8000 Columns 10 through 18 1.9000 2.0000 2.1000 2.2000 2.6000 2.7000 Columns 19 through 21 2.8000 2.9000 3.0000

1.4000

1.5000

1.6000

2.3000

2.4000

2.5000

Note that linspace produces a row vector, not a column vector. To get a column vector use the transpose operator (') on LinVector. The third argument to the function is the total size of the vector you want, which will include the first two arguments as endpoints and n - 2 other points in between. If you omit the third argument, MATLAB assumes you want the array to have 100 elements. If, instead, you want the spacing to be logarithmic, use the logspace function. This function, unlike the linspace function, does not find n - 2 points between the first two arguments a and b. Instead it finds n-2 points between 10^a and 10^b as follows: >> LinVector = logspace(1,3,21) logvector = 1.0e+003 * Columns 1 through 9 0.0100 0.0126 0.0158 0.0501 0.0631 Columns 10 through 18 0.0794 0.1000 0.1259 0.3981 0.5012 Columns 19 through 21 0.6310 0.7943 1.0000

0.0200

0.0251

0.0316

0.0398

0.1585

0.1995

0.2512

0.3162

Both of these functions are useful for generating points that you wish to evaluate another function at, for plotting purposes on rectangular and logarithmic axes respectively.

Vector Magnitude The magnitude of a vector can be found using the norm function: >> Magnitude = norm(inputvector,2);

For example: >> magHoriz = norm(Horiz) magHoriz = 3.7417 >> magVert = norm(Vert) magVert = 8.7750

The input vector can be either horizontal or vertical.

Dot product The dot product of two vectors of the same size (vertical or horizontal, it doesn't matter as long as the long axis is the same length) is found using the dot function as follows: >> DP = dot(Horiz, Vert) DP = 32

The dot product produces a scalar value, which can be used to find the angle if used in combination with the magnitudes of the two vectors as follows: >> theta = acos(DP/(magHoriz*magVert)); >> theta = 0.2257

Note that this angle is in radians, not degrees.

Cross Product The cross product of two vectors of size 3 is computed using the 'cross' function: >> CP = cross(Horiz, Vert) CP = -3 6 -3

Note that the cross product is a vector. Analogous to the dot product, the angle between two vectors can also be found using the cross product's magnitude: >> CPMag = norm(CP); >> theta = asin(CPMag/(magHoriz*magVert)) theta = 0.2257

The cross product itself is always perpendicular to both of the two initial vectors. If the cross product is zero then the two original vectors were parallel to each other.

Introduction to Structures MATLAB provides a means for structure data elements. Structures are created and accessed in a manner familiar for those accustomed to programming in C. MATLAB has multiple ways of defining and accessing structure fields. See Declaring Structures for more details. Follow this link for MATLAB structure help.

Note: Structure field names must begin with a letter, and are case-sensitive. The rest of the name may contain letters, numerals, and underscore characters. Use the namelengthmax function to determine the maximum length of a field name.

Declaring Structures Structures can be declared using the struct command. >> a = struct('b', 0, 'c', 'test') a = b: 0 c: 'test'

In MATLAB, variables do not require explicit declaration before their use. As a result structures can be declared with the '.' operator. >> b.c = 'test' b = b: 0 c: 'test'

Structures can be declared as needed and so can the fields.

Arrays of Structures Structures can also be arrays. Below is an example >> >> array >>

a = struct('b', 0, 'c', 'test'); a(2).b = 1; by creating another element a(2).c = 'testing' a = 1x2 struct array with fields: b c >> a(1) ans = b: 0 c: 'test' >> a(2) ans = b: 1 c: 'testing'

% Create structure % Turn it into an

% Initial structure

% The second element

Accessing Fields When the field name is known the field value can be accessed directly. >> a.c

ans = test ans = testing

In some cases you may need to access the field dynamically which can be done as follows. >> str = 'c'; >> a(1).(str) ans = test >> a(1).c ans = test

Accessing Array Elements Any given element in a structure array can be accessed through an array index like this >> a(1).c ans = test

To access all elements in a structure array use the syntax {structure}.{field}. In order to get all values in a vector or array use the brackets ([]) as seen below. >> [a.('c')] ans = testtesting >> [a.('b')] ans = 0 1

Graphics 2D Graphics Plot Plots a function in Cartesian Coordinates, x and y. Example: x=0:0.1:2; % creates a line vector from 0 to 2 fx=(x+2)./x.^2; % creates fx plot(x,fx,'-ok') % plots 2d graphics of the function fx

To plot 2 or more graphs in one Figure, then simply append the second (x,y) pair to the first: >>>x1 = [1,2,3,4] >>>y1 = [1,2,3,4] >>>y2 = [4,3,2,1] >>>plot(x1,y1,x1,y2)

This will plot y1 and y2 on the same x-axis in the output.

Polar Plot Plots a function using θ and r(θ) t = 0:.01:2*pi; polar(t,sin(2*t).^2)

3D Graphics plot3 The "plot3" command is very helpful and easy to see three dimensional images. It follows the the same syntax as the "plot" command. If you search the MATlab help (not at the command prompt. Go to the HELP tab at the top of the main bar then type plot3 in the search) you will find all the instruction you need. Example: l=[-98.0556 ; 1187.074]; f=[ -33.5448 ; -240.402]; d=[ 1298 ; 1305.5] plot3(l,f,d); grid on;

This example plots a line in 3d. I created this code in an M-file. If you do the same you change the values and hit the run button in the menu bar to see the effect.

Mesh Creates a 3D plot using vectors x and y, and a matrix z. If x is n elements long, and y is m elements long, z must be an m by n matrix. Example: x=[0:pi/90:2*pi]'; y=x'; z=sin(x*y); mesh(x,y,z);

Contour Creates a 2D plot of a 3D projection, using vectors x and y, and a matrix z. If x is n elements long, and y is m elements long, z must be an m by n matrix. Example: x=[0:pi/90:2*pi]'; y=x'; z=sin(x*y); contour(x,y,z);

Contourf Same as contour, but fills color between contour lines

Surface Basically the same as mesh MATLAB offers incomparable control over the way you can add details to your plot. From inserting text at the right positions to labelling the axes, MATLAB from the command line offers you an easy way to create publication style graphics. With support for Encapsulated postscript and Illustrator output. Complex figures with several axes and conveying large amounts of information can be created.

Concept of a handle Most operations on figures generate objects with a set of properties. Users familiar with Object oriented programming would realize that the functions and the data are encapsulated into the object. A typical figure would contain at least half a dozen objects. These objects are called handles. A very tacky analogy would be like handles to several different refrigerators with several different contents. To provide an intuitive feel. I have listed out the properties from a text handle.

Finding a handle Various commands provide required handles, for example: h = gcf; % Get current figure h = gca; % Get current axis

Examples Axis Label xlabel labels the x-axis of the current plot >>xlabel('string')

You can display text on two lines or insert the value of variables >>xlabel({['First Line or line n° ',int2str(a)],['Second Line or line n°',int2str(b)]})

ylabel labels the y-axis of the current plot. It works in same way of xlabel but the output is vertical in 2D plots.

Documenting a Maximum Value % Previous code set the x value of the peak data point into x_peak plot(lags(1:1000:end),abs_cs(1:1000:end)); ptitle = 'UUT and Source Correlation Score Magnitude'; xlabel('Lag'); ylabel('Correlation Magnitude'); title(ptitle); yloc = max(get(gca,'YLim')); % Put at top of plot text(lags(x_peak),yloc,[' \leftarrow ' num2str(x_peak) 'ns']); lstr{1} = sprintf(' Test %d', TESTNUM); lstr{2} = sprintf(' Unit %d%s', UNITNUM, comparestr); text(lags(1),mean(get(gca,'YLim')),lstr);

M File Programming M-files There are 2 types of m-file • •

Scripts Functions

Scripts are a type of m-file that runs in the current workspace. So if you call a script from the command line (base workspace) the script will use and manipulate the variables of the base workspace. This can get very messy and lead to all sorts of strange errors when loops are involved and the coder is lazy about about naming their loop variables (i.e. for i = 1:10, if every loop uses i, j, or k then it's likely that any script called from a loop will alter the loop variable). Functions are wholly contained in themselves. They possess their own workspace keeping workspaces separate. This means that all variables necessary for a particular function must be passed or defined in some way. This can get tedious for complex algorithms requiring lots of variables. However, any manipulations of variables are discarded when the function is exited. Only those output arguments provided by the function are available to the calling workspace. This means that loops can use i, j, or k all they want because the function's workspace and the calling workspace do not mix. Any command valid at the command line is valid in any m-file so long as the necessary variables are present in the m-files operating workspace. Using functions properly any change can be affected to any algorithm or plotting tool. This allows for automation of repetitive tasks. It is optional to end the M-file with 'end'; doing so, however, can lead to complications if you have conditionals or loops in your code, or if you're planning on using multiple functions in the same file (see nested functions for details on this).

Requirements for a function Custom functions follow this syntax in their most basic form: function [output1, output2, ...]= function_name(input_arg1,input_arg2) statements return;

In current versions of MATLAB the return; line is not required. The function_name can be anything you like but it is best if the m-file name is function_name.m. Calling the

function from the command line or another m-file is done by invoking the m-file name of the function with the necessary input and output arguments. Within the function itself, there must be a statement that defines each of the output arguments (output1, output2, etc.). Without some declaration the variable for the output argument doesn't exist in the function's workspace. This will cause an error about "one or more output arguments". It is good practice to initialize the output arguments at the beginning of the function. Typically output arguments are initialized to empty ([]) or 0 or -1 or something equivalent for other data types. The reason is that if the function encounters an error you've anticipated then the function can return (via the return command) with those default values. If the initialization value is an invalid value then it can easily be checked by the calling function for any errors which may not throw a MATLAB error. Path In order to invoke a function that function's m-file must be in the current path. There is a default path that can be setup through the File menu or the addpath command. The order of the path is important as MATLAB searches the path in order and stops searching after it finds the 1st instance of that m-file name. The current path is • •

the current directory (which can be seen at the top of the MATLAB window or by typing pwd at the command prompt the default path

Note that MATLAB will always search the current directory before searching any of the rest of the path.

nargin & nargout The nargin and nargout commands are only valid inside functions since scripts are not passed any arguments. The nargin command returns the number of passed input arguments. This is useful in conjunction with nargchk nargchk(min, max, nargin)

where min is the minimum number of arguments necesary for the function to operate and max is the maximum number of valid input arguments. The nargout command is useful for determining which output arguments to return. Typically, the outputs are the end results of some algorithm and they are easily calculated. However, in some instances secondary output arguments can be time consuming to calculate or require more input arguments than the primary output

arguments do. So the function can check the number of output arguments being requested through the nargout command. If the caller isn't saving the secondary output arguments then they do not need to be calculated.

varargin & varargout When using MATLAB objects and functions they often allow the user to set properties. The functions and objects come with default values for these properties but the user is allowed to override these defaults. This is accomplished through the use of varargin. varargin is a cell array that is usually parsed where varargin{i} is a property and varargin{i+1} is the value the user wishes for that property. The parsing is done with a for or while loop and a switch statement. function [out] = myFunc(in, varargin)

The varargout output argument option allows for a variable number of output arguments just as varargin allows for a variable number of input arguments. From the MATLAB site function [s,varargout] = mysize(x) nout = max(nargout,1)-1; s = size(x); for k=1:nout, varargout(k) = {s(k)}; end returns the size vector and, optionally, individual sizes. So [s,rows,cols] = mysize(rand(4,5)); returns s = [4 5], rows = 4, cols = 5.

Useful syntax guidelines Placing the semicolon symbol after every line tells the compiler not to place that line of code in the command prompt and then execute. This can make your programs run a lot faster. Also, placing a semicolon after every line helps with the debugging process. syms x y z; w=[x y z]; e=[1 2 3]; t=jacobian(e,w);

Placing comments in your code can help other people (and yourself) understand your code as it gets more complex. syms x y z; w=[x y z]; e=[1 2 3]; t=jacobian(e,w);

%syms command makes x y and z symbolic

Comments can also Identify who wrote the code and when they wrote it. %Some code writer

%mm/dd/yyyy

Nested functions Placing comments Comment lines begin with the character '%', and anything after a '%' character is ignored by the interpreter. The % character itself only tells the interpreter to ignore the remainder of the same line. In the Matlab editor, commented areas are printed in green by default, so they should be easy to identify. There are two useful keyboard shortcuts for adding and removing chunks of comments. Select the code you wish to comment or uncomment, and then press Ctrl-R to place one '%' symbol at the beginning of each line and Ctrl-T to do the opposite. Matlab also supports multi-line comments, akin to /* ... */ in languages like C or C++, via the %{ and %} delimiters.

Common uses Comments are useful for explaining what function a certain piece of code performs especially if the code relies on implicit or subtle assumptions or otherwise perform subtle actions. Doing this is a good idea both for yourself and for others who try to read your code. For example, % Calculate average velocity, assuming acceleration is constant % and a frictionless environment. force = mass * acceleration

It is common and highly recommended to include as the first lines of text a block of comments explaining what an M file does and how to use it. Matlab will output the comments leading up to the function definition or the first block of comments inside a function definition when you type: >> help functionname

All of Matlab's own functions written in Matlab are documented this way as well.

The input() function lets your scripts process data entered at the command line. All input is converted into a numerical value or array. The argument for the input() function is the message or prompt you want it to display. Inputting strings require an additional 's' argument. Example: %test.m %let's ask a user for x

x = input('Please enter a value for x:')

Then running the script would produce the output: Please enter a value for x:3 x = 3 >>

Control Flow IF statement An IF statement can be used to execute code once when the logical test (expression) returns a true value (anything but 0). An "else" statement following an if statement is executed if the same expression is false (0). Syntax: if expression statements elseif expression2 statements end

SWITCH statement Switch statements are used to perform one of several possible sets of operations, depending on the value of a single variable. They are intended to replace nested "if" statements depending on the same variable, which can become very cumbersome. The syntax is as follows: switch variable case value1 statements case value2 statements ... otherwise statements end

The end is only necessary after the entire switch block, not after each case. If you terminate the switch statement and follow it with a "case" statement you will get an error saying the use of the "case" keyword is invalid. If this happens it is probably because you deleted a loop or an "if" statement but forgot to delete the "end" that went with it, thus

leaving you with surplus "end"s. Thus MATLAB thinks you ended the switch statement before you intended to. The otherwise keyword executes a certain block of code (often an error message) for any value of variable other than those specified by the "case" statements.

TRY/CATCH statement The TRY/CATCH statement executes a certain block of code in the "try" block. If it fails with an error or a warning, the execution of this code is terminated and the code in the "catch" block is executed rather than simply reporting an error to the screen and terminating the entire program. This is useful for debugging and also for filtering out erroneous calculations, like if you accidentally try to find the inverse of a singular matrix, when you don't wish to end the program entirely. Syntax: try statements catch statements end

Note that unlike the other control flow statements, the TRY/CATCH block does not rely on any conditions. Therefore the code in the TRY block will always be at least partially executed. Not all of the TRY block code will always be executed, since execution of the TRY ends when an error occurs.

FOR statement The FOR statement executes code a specified number of times using an iterator. Syntax: for iterator = startvalue:increment:endvalue statements end

The iterator variable is initialized to startvalue and is increased by the amount in increment every time it goes through the loop, until it reaches the value endvalue. If increment is omitted, it is assumed to be 1, as in the following code: for ii = 1:3 statements end

This would execute statements three times.

WHILE statement

The while statement executes code until a certain condition evaluates to false or zero. Example: while condition statements end

Forgetting to change the condition within a while loop is a common cause of infinite loops.

BREAK and CONTINUE MATLAB includes the "break" and "continue" keywords to allow tighter loop control. The "break" keyword will cause the program to leave the loop it is currently in and continue from the next line after the loop ends, regardless of the loop's controlling conditions. If the code is in a nested loop it only breaks from the loop it's in, not all of them. The syntax is simply to write the word "break" within the loop where you desire it to break. In contrast to "break", "continue" causes the program to return back to the beginning of the loop it is presently in, and to recheck the condition to see if it should continue executing loop code or not. The code in the loop after the "continue" statement is not executed in the same pass.

Program Flow The idea of program flow is simple. However, implementing and using flow techniques effectivly takes practice. MATLAB flow control is almost identical to flow control in C. There is a tremendous amount of text on the subject of flow in C. If you do a little homework in about an hour you can know all you need to from one of numerous C tutorials. To be good at flow control all you have to do is practice. Here are a few concepts that you can practice using flow control to implement: • • •

Calculate compounding interest using a while loop (don't cheat by using the algebraic form). Create a moving average filter using a for loop Make a counter that keeps track of keystrokes:How many times a typist hits a certain letter.

As far as I've seen there is little help out there to help people decipher MATLAB's error messages. Most of the syntax errors are not difficult to fix once you know what is causing them so this is intended to be a guide to identifying and fixing errors in MATLAB code. Warnings are also shown here as these often lead to errors later.

Arithmetic errors Usually these are self-explanitory. As a reminder, here are some common functions that cannot be performed and what MATLAB returns (along with a warning for each one): a/0 = Inf if a > 0, -Inf if a < 0, and NaN if a = 0. log(0) = -Inf MATLAB defines 0^0 to be 1.

NaN will very often result in errors or useless results unless measures are taken to avoid propogating them. ???Error using ==> minus Matrix dimensions must agree.

So check the dimensions of all the terms in your expression. Often it is an indexing mistake that causes the terms to be of different size. If you are using power function you might add a single dot after the parameter. i.e. y=x.^2 instead of y=x^2 Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second. Otherwise, you get the message: ??? Error using ==> mtimes Inner matrix dimensions must agree.

Note the difference between this error and the previous one. This error often occurs because of indexing issues OR because you meant to use componentwise multiplication but forgot the dot. Attempting to take the inverse of a singular matrix will result in a warning and a matrix of Infs. It is wise to calculate the determinant before attempting to take the inverse or, better, to use a method that does not require you to take the inverse since its not numerically stable. Attempting to take a power of a nonsquare matrix results in the error ??? Error using ==> mpower Matrix must be square.

This is usually because you meant to use componentwise exponentiation and forgot the dot.

Indexing errors Indexing is a pain in MATLAB, it is probably one of the hardest things to get down, especially since the syntax for an index is the same as the syntax for a function. One annoying fact is that the names of variables are case sensitive, but the names of functions

are NOT. So if you make an array called Abs and you try to index abs(1), it will return 1 no matter what the first value in the array Abs is. Unfortunately, MATLAB will not return an error for this, so a good rule of thumb is never ever name your variables the same as a function. This clears up some indexing problems. Some things are rather obvious but take some practice in avoiding: You cannot try to access part of an array that does not exist yet. >> A = [1,3]; >> A(3) ??? Index exceeds matrix dimensions.

Unfortunately, MATLAB doesnt tell you which variable you exceeded the dimensions on if there's more than one so you'll have to check that. This often occurs if, for example, you are using a loop to change which part of an array is accessed, but the loop doesn't stop before you reach the end of the array. This also happens if you end up with an empty matrix as a result of some operation and then try to access an element inside it. You cannot try to access a negative, complex, noninteger, or zero part of an array; if you do you get this message: >> A(-1) >> A(i) >> A(1.5) >> A(0) ??? Subscript indices must either be real positive integers or logicals.

Note that MATLAB starts counting at 1, not 0 like C++. And again, it doesn't tell you which index is not real or logical. Also note that if 0 was a logical 0 (false) then the statement A(0) would be valid and would return all 0 values in the array. Attempting to use non-standard MATLAB syntax in your indexing will often result in the error: >> A(1::, 2) ??? A(1::, 2) | Error: Unexpected MATLAB operator.

The "operator" :: is one of several possible operators that MATLAB does not accept. This could be an example of someone trying to access all rows of A after the first one and the second column, in which case you should use the "end" syntax, as in: >> A(1:end, 2) ans = 3

Make sure you are careful when using the colon operator, because it does many different things depending on where you put it and misuse often results in these errors. Try putting in one piece of your code at a time and see what it is doing, it may surpise you.

Assignment errors Ah, assignment, that is using the = sign to give a variable, or certain elements of an array, a particular value. Let's start with a classic mistake: >> a = 2; >> if a = 3 ??? if a = 3 | Error: The expression to the left of the equals sign is not a valid target for an assignment.

This error occurs because you meant to see if "a" equaled 3, but instead you told MATLAB to assign "a" a value of 3. You cannot do that on the same line that the if/while statement is on. The correct syntax is >> if a == 3 >> end

This creates no errors (and you can put anything inside the conditional you want). You cannot have a normal array with two different classes of data inside it. For example, >> A = @(T) (1+T) A = @(T) (1+T) >> A(2) = 3 ??? Conversion to function_handle from double is not possible.

For such a purpose you should use cell arrays or struct arrays. Here's the tricky one. Take a look at the following code: >> A = [1,2,3;4,5,6;7,8,9]; >> A(2,:) = [3,5]; ??? Subscripted assignment dimension mismatch. >> A(2,:) = [1,4,5,6]; ??? Subscripted assignment dimension mismatch. >> A(1:2, 1:2) = [1,2,3,4]; ??? Subscripted assignment dimension mismatch.

What is happening here? In all three cases, take a look at the dimensions of the left and the right hand sides. In the first example, the left hand side is a 1x3 array but the right

side is a 1x2 array. In the second, the Left hand side is 1x3 while the right is 1x4. Finally, in the third, the left hand side is 2x2 while the right is 1x4. In all three cases, the dimensions do not match. They must match if you want to replace a specific portion of an existing variable. It doesn't matter if they have the same number of data points or not (as the third example shows); the dimensions must also be the same, with the exception that if you have a 1xn array on one side and an nx1 on the other MATLAB will automatically transpose and replace for you: >> A(2,:) = [1;2;3] A = 1 2 3 1 2 3 7 8 9

If you do not want this be aware of it!

Struct array errors Struct arrays are rather complex, and they have a rigid set of rules of what you can and can not do with them. Let us first deal with indexing within struct arrays. Suppose you define the variable "cube" and want to store the volume and the length of one side of two different cubes in a struct array. This can be done as follows: >> >> >> >>

cube(1).side = cube(1).volume cube(2).side = cube(2).volume

1; = 1; 2; = 8;

This seems like a good way of storing data and it is for some purposes. However, suppose you wanted to abstract the volumes from the struct and store them in one array. You cannot do it this way: >> volumes = cube.volume ??? Illegal right hand side in assignment. Too many elements.

You'll notice that if you tell MATLAB to display cube.volume, it will display both values, but reassign the variable ans each time, because it is treated as two separate variables. In order to avoid the error, you must format 'cube.volume' as an array upon assignment. >> volumes = {cube.volume}

You can also write in a separate assignment for each cube but this is more adaptable to larger numbers of cubes. Just like extracting data, you must input the data one at a time, even if it is the same for all instances of the root (cube). >> cube.volForm = @(S) (S^3)

??? Incorrect number of right hand side elements in dot name assignment. Missing [] around left hand side is a likely cause. >> cube(:).volForm = @(S) (S^3) ??? Insufficient outputs from right hand side to satisfy comma separated list expansion on left hand side. Missing [] are the most likely cause.

Unfortunately missing [] is not the cause, since adding them causes more errors. The cause is that you cannot assign the same value to all fields of the same name at once, you must do it one at a time, as in the following code: >> for ii = 1:2 >> cube(ii).volForm = @(S) (S^3); >> end >> cube ans = 1x2 struct array with fields: volume side volForm

The same volume formula is then found in both cubes. This problem can be alleviated if you do not split the root, which is highly recommended. For example, you can use a struct like this: >> shapes.cubeVol = @(S) (S^3); >> shapes.cube(1).vol = 1; >> shapes.cube(2).vol = 8;

This avoids having to use a loop to put in the formula common to all cubes.

Syntax errors Parenthesis errors Unlike in C++, you are not required to terminate every line with anything but a line break of some sort. However, there are still syntax rules you have to follow. In MATLAB you have to be especially careful with where you put your parenthesis so that MATLAB will do what you want it to. A very common error is illustrated in the following: >> A(1 ??? A(1 | Error: Expression or statement is incorrect--possibly unbalanced (, {, or [.

This error is simple enough, it means you're missing a parenthesis, or you have too many. Another closely related error is the following:

>> A(1)) ??? A(1)) | Error: Unbalanced or misused parentheses or brackets.

MATLAB tries to tell you where the missing parenthesis should go but it isn't always right. Thus for a complex expression you have to go through it very carefully to find your typo. A useful trick is to try to set a breakpoint a line after the offending line. It won't turn red until the error is corrected, so keep trying to correct it and saving the file until that breakpoint turns red. Of course, after this you have to make sure the parenthesis placement makes sense, otherwise you'll probably get another error related to invalid indecies or invalid function calls.

String errors There are two ways that you can create a string; use the ' string ' syntax, or type two words separated by only whitespace (not including line breaks), as in >> save file.txt variable

In this line, file.txt and variable are passed to the save function as strings. It is an occasional mistake to forget a parenthesis and accidentally try and pass a string to a function that does not accept strings as input: >> eye 5 ??? Error using ==> eye Only input must be numeric or a valid numeric class name.

These should not be hard to spot because the string is color-coded purple. Things like this occur if you uncomment a line of text and forget to change it. Forgetting the closing ' in the other syntax for a string results in an obvious error: >> A = 'hi ??? A = 'hi | Error: A MATLAB string constant is not terminated properly.

The unterminated string is color-coded red to let you know that it is not terminated, since it's otherwise easy to forget.

Other miscellaneous errors You cannot leave trailing functions, and if you do MATLAB gives you an error that is similar but not exactly the same as that for a missing parenthesis, since it doesn't want to venture a guess: >> A = 1+3+ ??? A = 1+3+

| Error: Expression or statement is incomplete or incorrect.

These usually are not hard to spot, and often result from forgetting the "..." necessary to split a line. The double colon is not the only "unexpected MATLAB operator", there is also "..", "....", and several other typos that generate this error. If you accidentally type the ` character you get the error: >> ??? ` | Error: The input character is not valid in MATLAB statements or expressions.

This usually occurs because you intended to put a "1" in the equation but missed the key.

Function Calling errors It is necessary to know the nature of the input and output arguments of a given function in order to call it. For MATLAB's built-in functions, this information is found in the documentation, or by typing >> help functionname

Most functions (not all however) require at least one input argument, and calling it with too few will result in an error: >> A = ode45() ??? Error using ==> ode45 Not enough input arguments.

See ODE45.

You cannot call the function with too many input arguments either: >> A = plus(1,2,3) ??? Error using ==> plus Too many input arguments.

You can choose how many of the output arguments you want out of those available by using the bracket notation, but you cannot assign more than the function can output. >> A = [1,2;3,4] D = eig(A); %one output argument [V,D] = eig(A); %two output arguments [V,D,Mistake] = eig(A); ??? Error using ==> eig Too many output arguments.

Finally, you must make sure that all arguments passed TO the function are the right class, and all those returned are also the right class if you are replacing parts of an array that already exists (see the section on assignment for more on this).

Control Flow errors The most common one by far is if you forget the 'END', which is an issue in M-file functions. It will tell you 'at least one END is missing' and try to tell you where the loop or conditional statement starts. If you have too many END statements and more than one function in an M-file, MATLAB may give you a cryptic message about not formatting the functions correctly. This is because all functions in the same M-file must either end with an END statement or not. It doesn't matter which, but if you have too many END statements in one of the functions, MATLAB will think your function is ending early and will get confused when the next function in line does not have an END statement at the end of it. So if you get this confusing message, look for extra END statements and it should fix your problem. Having an extra END in a 'switch' statement gives a message that you used the 'case' keyword illegaly, because MATLAB thinks you ended the switch statement early, and 'case' has no meaning outside a 'switch' statement.

Other errors There are numerous types of errors that do not generate errors from the MATLAB compiler, which have to do with calling the wrong function, using the wrong operation, using the wrong variable, introducing an infinite loop, and so on. These will be the hardest to fix, but with the help of the MATLAB debugger, they will be easier to find. See Debugging M Files for details on how to use the debugger.

1. In MATlab 6.x (not sure exactly which builds this problem occurs in) the random number generator will generate the same sequence the first time you execute the command.

This section explains things you can do if you fix all the syntax errors (the ones that give you actual error messages), the program runs... but it gives you some result you don't want. Maybe it goes into an infinite loop, maybe it goes through the loop one too few or one too many times, maybe one of your "if" statements doesn't work, maybe the program is giving you "infinity" or "NaN" as an answer (which usually isn't very useful!)... there's

as many things that can go wrong as there are lines in the code. Thankfully there are techniques for both fixing and improving on working MATLAB code.

Using MATLAB's Debugging tool Using the Debugging Tool will let you stop your program in mid-execution to examine the contents of variables and other things which can help you find mistakes in your program. M-file programs are stopped at "breakpoints". To create a breakpoint, simply press F12 and a red dot will appear next to the line where your cursor is. You can also click on the dash next to the line number on the left side of the M-file window to achieve the same result. Then press F5 or Debug->Run from the menu to run the program. It will stop at the breakpoint with a green arrow next to it. You can then examine the contents of variables in the workspace, step, continue or stop your program using the Debug menu. To examine contents of a variable, simply type its name into the workspace, but be warned: you can only look at the values of variables in the file you stop in, so this means that you'll probably need multiple breakpoints to find the source of your problem. There are several different ways you can move through the program from a breakpoint. One way is to go through the whole program, line by line, entering every function that is called. This is effective if you don't know where the problem is, but since it enters every function (including MATLAB functions like ode45), you may not desire to use it all the time. Thankfully, there's also a way to simply step through the function you're currently stopped in, one line at a time, and instead of going through the child functions line by line MATLAB will simply give you the results of those functions. Finally, note that you cannot set a breakpoint until you save the M-file. If you change something, you must save before the breakpoint "notices" your changes. This situation is depicted in MATLAB by changing the dots from red to gray. Sometimes, you'll save but the dots will still be gray; this occurs when you have more than one breakpoint in multiple files. To get around this (which is really annoying), you have to keep going to "exit debug mode" until it turns gray. Once you're completely out of debug mode, your file will save and you'll be ready to start another round of debugging.

Using comments to help you debug code If you want to test the effects of leaving out certain lines of code (to see, for example, if the program still returns Inf if you take them out), you can comment out the code. To do this, highlight it and then go to:

Text -> Comment

Or press CRTL+R. This will simply put a '%' in front of every line; if the line is already commented out it will put another '%' there so when you uncomment them the pattern of comment lines will not change. Commented lines will be ignored by the compiler, so the effect will be that the program is run without them. To uncomment a line go to Text -> Uncomment

Or press CTRL+T. Another use of commenting is to test the difference between two different possible sets of code to do something (for example, you may want to test the effect of using ODE113 as opposed to ODE45 to solve a differential equation, so you'd have one line calling each). You can test the difference by commenting one out and running the program, then uncommenting that one and commenting the other one out, and calling the program again.

How to escape infinite loops If your program is doing nothing for a long time, it may just be slow (MATLAB creates a lot of overhead and if you don't use arrays wisely it will go very, very slow) but if you are testing a small module, it is more likely that you have an infinite loop. Though MATLAB can't directly tell you you have an infinite loop, it does attempt to give you some hints. The first comes when you terminate the program. Terminate it by pressing CTRL+C and MATLAB will give you a message telling you exactly what line you stopped on. If your program is running a long time, it is likely the line you stopped in is in the middle of an infinite loop (though be warned, if the loop calls a sub-function, it is likely that you will stop in the sub-function and not the parent. Nevertheless, MATLAB also will tell you the lines of the parents too so you can track down the loop easily enough). However, sometimes MATLAB won't even let you return to the main window to press CTRL-C. In this case you probably have to kill the whole MATLAB process. After this, add a "pause (0.001)" or a similarly small value in the loop you suspect to be the infinite one. Whenever MATLAB passes this instruction you will be able to interact with MATLAB for a (very) short time, e.g. go to the main window and press STRG-C with MATLAB being able to respond to your command.

Mathematical Manipulations Linear Algebra

Operations Squaring a matrix a=[1 2;3 4]; a^2;

a^2 is the equivalent of a*a. To square each element: a.^2

The period before the operator tells MATLAB to perform the operation element by element.

Determinant Getting the determinant of a matrix, requires that you first define your matrix, then run the function "det()" on that matrix, as follows: a = [1 2; 3 4]; det(a) ans = -2

Symbolic Determinant You can get the symbolic version of the determinant matrix by declaring the values within the matrix as symbolic as follows: m00 = sym('m00'); m01 = sym('m01'); m10 = sym('m10'); m11 = sym('m11');

or syms m00 m01 m10 m11;

Then construct your matrix out of the symbolic values: m = [m00 m01; m10 m11];

Now ask for the determinant:

det(m) ans = m00*m11-m01*m10

Transpose To find the transpose of a matrix all you do is place a apostrophe after the bracket. Transpose- switch the rows and columns of a matrix. Example: a=[1 2 3] aTranspose=[1 2 3]'

or b=a' %this will make b the transpose of a

when a is complex, the apostrophe means transpose and conjugate. Example a=[1 2i;3i 4]; a'=[1 -3i;-2i 4]; For a pure transpose, use .' instead of apostrophe.

Systems of linear equations Ther are lots of ways to solve these equations.

Homogeneous Solutions Particular Solutions State Space Equations

Special Matrices Often in MATLAB it is necessary to use different types of unique matrices to solve problems.

Identity matrix To create an identity matrix (ones along the diagonal and zeroes elsewhere) use the MATLAB command "eye": >>a = eye(4,3) a =

1 0 0 0

0 1 0 0

0 0 1 0

Ones Matrix To create a matrix of all ones use the MATLAB command "ones" a=ones(4,3)

Produces: a = 1 1 1 1

1 1 1 1

1 1 1 1

Zero matrix The "zeros" function produces an array of zeros of a given size. For example, a=zeros(4,3)

Produces: a = 0 0 0 0

0 0 0 0

0 0 0 0

This type of matrix, like the ones matrix, is often useful as a "background", on which to place other values, so that all values in the matrix except for those at certain indecies are zero.

Row reduced echelon form To find the Row reduced echelon form of a matrix just use the MATLAB command rref Example: a=[1 2 3; 4 5 6]; b=rref(a);

It's that simple. (I believe that MATLAB uses the Gauss-Jordan elimination method to make this computation; don't quote me on that (I'm not even sure if there are other methods)).

Inverse To find the inverse of a matrix use the MATLAB command inv. (note the matrix must be square) Example: a=[1 2 3;4 5 6;7 8 9]; b=inv(a);

Coffactor, minor The Jacobian t=jacobian(e,w);

e is a scalar vector, w is a vector of functions. Also, this does not solve equations symbolicaly unless you define the w vector functions as symbolics prior to executing this statment. Example: syms x y z; w=[x y z]; e=[1 2 3]; t=jacobian(e,w);

Differential Equations

Differential equation solver syntax All of the differential equations have the same syntax that you must use, and the same input and output arguments. All of the solvers require three input arguments: a function handle to the differential equation you want to solve, a time interval over which to integrate, and a set of initial conditions. Let us suppose we want to solve the simple differential equation y' = y, y(0) = 1, which has the true solution y(t) = e^t. Suppose we want the solution between t = 0 and t = 1. To use function handles, you must first create an M-file with the function in it like so: function Yprime = func(t, y)

Yprime = 0; Yprime = y;

Note that you must include the time argument even if it is not used in the differential equation. The initialization of Yprime to an array of zeros will save you grief if you try to solve more than one function; Yprime must be returned as a VERTICAL array but, if you don't initialize it as a verical array (or transpose at the end), it will return a HORIZONTAL array by default. Also note that, with the exception of ode15i, the function must be solved explicitly for y'. Once this file is created, call the ODE function with the arguments in the order (function, timeinterval, initialcond): >> ode45(@func, [0,1], 1)

The other method is to use anonymous functions, which is only useful if you have one function (otherwise you must use function handles). You again must declare the anonymous function in terms of both the dependent variable(s) and time: >> func = @(t,y) y; >> ode45(func, [0,1], 1)

Calling the ODE function without input arguments gives a graph of the solution. Calling it with one output argument returns a struct array: >> Struct = ode45(func, [0,1], 1) Struct = solver: 'ode45' extdata: [1x1 struct] x: [1x11 double] y: [1x11 double] stats: [1x1 struct] idata: [1x1 struct]

This data is mostly used to, in the future, call the 'deval' function to get the answer at any time you want: >> Solution = deval(Struct, 0.5) Solution = 1.6487

Deval can also return the derivative at the point of interest by including a second output argument. >> [Solution, Derivative] = deval(Struct, 0.5) Solution = 1.6487 Derivative = 1.6487

Since the derivative of e^x is itself it makes sense that the derivative and solution are the same here.

Calling ode45 with two output arguments returns two lists of data; t first, then the independent variables in an appropriately-sized matrix.

ODE Options The are a rather large number of options that MATLAB gives you to modify how it solves the differential equations. The help file does a pretty good job describing all of them so they won't be described in detail here. To get a list use the help function: >>help odeset

To get a list of the different options' names and what you have to pass to it, just type 'odeset' into the command prompt. It returns either a data type or a finite list of options. It also lists, in parenthesis, the default values of all the options. To set a specific option or list of options, type the name of the option first and then the value of the option you want. For example, suppose you want to tighten the default relative tolerance from 10^-3 to 10^-4. You would call 'odeset' as follows: >> option = odeset('RelTol', 10^-4);

Note that the option name must be passed as a string, or else you'll get an 'undefined variable' error most likely. More than one option can be passed at a time by just putting them all in a list: >> option = odeset('RelTol', 10^-4, 'AbsTol', 10^-7, 'MassSingular', 'no');

The options structure can then be passed to the ode function as a forth (optional) input argument: >> [T,X] = ode45(@func, [0,1], 1, option);

This will return more accurate values than default because the error tolerances are tighter. It should also compute faster because MATLAB is not checking to see if this is a differential-algebraic equation (this is what the MassSinglular option does; it is usually set to 'maybe' so MATLAB checks by itself).

The ODE solvers MATLAB doesn't just have one ODE solver, it has eight as of the MATLAB 7.1 release. Some are more suited for certain problems than others, which is why all of them are included. The MATLAB help has a list of what functions each one can do, but here is a

quick summary, in roughly the order you should try them unless you already know the classification of the problem: Most problems can be solved using ode45, and since this is the best tradeoff of speed and accuracy it should be the first one you use. If you need a really tight error tolerance or a lot of data points, use ode113. If you have a relatively loose error tolerance or the problem is slow with ode45, try ode23. If the problem is truly stiff and ode45 fails, use ode23tb for loose tolerances, ode23s for slightly tighter tolerances with a constant mass matrix, or ode15s for tigheter tolerances or nonconstant mass matrices. If you need a solution without numerical damping on a stiff problem, use ode23t. The s indicates that the algorithm is intended for stiff problems. There is one other ODE solver, which is special: Implicit problems can only be solved using ode15i. Since ode15i has some differences in its syntax, it is discussed next.

Implicit ODEs: ODE15i Since ode15i is the only ODE solver that solves implicit equations, it must have some special syntactical rules on how to input the function. If you are using ode15i declare the function as follows: function Z = func(t,y,Yprime) Z = 0; Z = y - Yprime;

Note that with ode15i, you must put the function into normal form (solve it for 0), whereas for all other ODE functions you must solve explicitly for y'. Also notice that you must declare three input arguments instead of the usual two. When you call ode15i, you must not only include initial conditions for y but also for Yprime. The initial conditions for Yprime go into the fourth argument: >> [t,x] = ode15i(@func, [0,1], 1, 1);

This will return similar results to ode45 when used for the explicit equation, but has less data points. The options structure is passed to ode15i as the optional fifth argument. All output options from ode15i are the same as for the other ODE solvers. MATLAB Programming/Partial Differential Equations

More advanced I/O Now that we've covered basic file input output (I'll get into more detailed file manipulation later) we can move on to the next step. In the section Basic Reading and Writing data we already had the data in the form of a spreadsheet. How did it get to that spreadsheet and how do we send it some where else? This question will attempt to be answered in the following section. (there are many ways to transmitt and recieve data. I've even heard of this newfangled thing called the easternet or internet or whatever, but for now we will assume we are in a University lab with no ethernet connection (dont laugh I've been in this situation more times than I like to remember)). MATLAB provides us with there style of reading and writing to the serial port. 1. You must create a serial port object. object=serial(port,'Property name',propertyValue);

Example: I will now create a serial object called serialOne serialOne=serial('COM1', 'BaudRate', 9600);

(for those of you more familiar with C this should look very similar to creating a handle: In fact it is the MATLAB way of making handles) Most computers call the serial ports COM and then some number usually 1-4 (depending on how many there are). This also includes DB-9, DB-25 and other various DB type connectors.

2. Now that the computer knows there is a serial port object you have to tell it to open the object. Use the fopen command 3. Write to the port 4. Close object. Example: serialOne=serial('COM1', 'BaudRate', 9600); fopen(serialOne); fprintf(serialOne,'textFile.txt'); fclose(serialOne);

The Moving Average Filter Formula:

MATLAB implementation(All the code here was intended to be put in an M-file): clc; clear;

% clear all

v=.01 f=100; fs=5000; t=0:1/fs:.03 x=sin(2*pi*f*t); r=sqrt(v)*randn(1,length(t)); Xw=x+r; input)

%original signal %noise %signal plus noise (filter % I have chosen h=3

for n= 3:length(Xw), y(n)=(Xw(n)+(Xw(n-1))+(Xw(n-2)))/3; signal end

%y[n] is the filtered

plot(y); hold; plot(x,'r'); over top the

%plot the original signal %filtered signal to see

the difference

The moving average filter is simple and effective. One of the things that is a problem is the lag associated with the moving average filter. The more samples used the longer the lag experienced(All filters have lag). How much lag can be tolerated is up to the individual.

The Alpha Beta filter The Kalman Filter The Kalman filter is a recursive method of combining to estimates to determine the truth. A few parameters that are widely used are the initial conditions or current value and measured data.

Equation:

Example: n=100; sigma=(20/6076); R=100; Rm=R+sigma*randn; Rs(1)=Rm(1); Cs=sigma^2 for i=2:n Rm(i)=R+sigma*randn; alpha=Cs/(Cs+sigma^2); Rs(i)=Rs(i-1)+alpha*(Rm(i)-Rs(i-1)); Cs=(1-alpha)*Cs; end

All this code does is take a constant value R and adds noise to it. Then it filters the new signal in an effort to separate the noise from the original signal.

The discrete Fourier transform DFT quick reference What is it?

DFT element

matlab example and comments fsample=100;

How often do you want to sampling frequency fs sample?

sample at 100Hz psample=1/fsample;

sample period

For how long do you want time range to sample?

tmax=10;

run from 0 to 10 sec. inclusive

nsamples=tmax/psample+1;

+1 because we include t=0 and t=tmax. nsamples=2^ceil(log2(nsamples));

How many samples does N that give you?

round up to a power of 2 in length so FFT will work. times=(0:psample:(nsamples1)*psample)';

Create a column vector of sample times. delf=fsample/nsamples;

How far apart are each of the frequencydomain result points?

frequencies=(0:delf:(nsamples1)*delf)';

Column vector of result frequencies

What signal do you want input x(t) to sample?

x=sin(2*pi*10*times)+sin(2*pi*3*times);

Make a 10Hz sine wave plus a 3Hz sine wave

Fourier transform What are the results?

fft_x=fft(x, length(x));

What frequencies Xm(f) = | X(f) | does the signal have?

fft_x_mag=abs(fft_x);

What phase relationships?

fft_x_phase=unwrap(angle(fft_x));

plot(frequencies, fft_x_mag);

How do you view the results?

Or, to match the amplitude of the magnitude peak to the amplitude of the sine wave, plot(frequencies, (2/nsamples)*fft_x_mag);

What about the power Xp(f) = | X(f) | 2 spectrum?

fft_x_power=fft_x_mag.^2;

plot(frequencies, fft_x_power);

Object-Oriented Programming A struct as defined and used in Octave A structure in Octave groups different data types called fields in a single object. Fields are accessed by their names.

Declaring a structure A structure is declared by assigning values to its fields. A period (.) separates the name of the field and the name of the structure: >> >> >> >>

city.name = 'Liege'; city.country = 'Belgium'; city.longitude = 50.6333; city.latitude = 5.5666;

The fields of a structure and their value can be displayed by simply entering the name of the struct: >> city city = { name = Liege country = Belgium longitude = 50.633 latitude = 5.5666 }

Manipulating structures A structure can be copied as any objects: >> city_copy = city;

In most circumstance, the fields of a structure can be manipulated with the period operator. The value of a field can be overwritten by: >> city.name = 'Outremeuse';

In the same way, the value of a field can be retrieved by: >> city.name ans = Outremeuse

The function isstruct can be used to test if object is a structure or not. With the function fieldnames all field names are returned as a cell array: >> fieldnames(city) ans = { [1,1] = name [2,1] = country [3,1] = longitude [4,1] = latitude }

To test if a structure contains the a given field named, the function isfield can be used: >> isfield(city,'name') ans = 1

The value of a field can be extract with getfield: >> getfield(city,'name') ans = Liege

In a similar way, the value of a field can be set with setfield: >> setfield(city,'name','Outremeuse')

The functions isfield, getfield and setfield are useful when the names of a structure are determined during execution of the program. MATlab stores methods (seperate M-file) in class directories not on the standard search path. The two minimum things needed in order to create a class are the constructor and display M-files.

Comparing Octave and MATLAB Octave is a free computer program for performing numerical computations which is mostly compatible with MATLAB.

History The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Real development was started by John W. Eaton in 1992. The first alpha release dates back to January 4, 1993 and on February 17, 1994 version 1.0 was released. The name has nothing to do with music. It was the name of a former professor of one of the authors of Octave who was known for his ability to quickly come up with good approximations to numerical problems.

Technical details • • • •

Octave is written in C++ using STL libraries. Octave has an interpreter that interprets the Octave language. Octave itself is extensible using dynamically loadable modules. Octave interpreter works in tandem with gnuplot and Grace software to create plots, graphs, and charts, and to save or print them.

Octave, the language The Octave language is an interpreted programming language. It is a structured programming language (an example of which is the C language) and supports many common C standard library constructs, and can be extended to support UNIX system calls and functions. However, it does not support passing argument by reference. Octave programs consist of a list of function calls or script. The language is matrix-based and provides various functions for matrix operation. It is not object-oriented, but supports data structures. Its syntax is very similar to MATLAB, and carefully programming a script will allow it to run on both Octave and MATLAB.

Notable features •

Command and variable name completion

Typing a TAB character on the command line causes Octave to attempt to complete variable, function, and file names. Octave uses the text before the cursor as the initial portion of the name to complete. •

Command history

When running interactively, Octave saves the commands typed in an internal buffer so that they can be recalled and edited. •

Data structures

Octave includes a limited amount of support for organizing data in structures. For instance: octave:1> x.a = 1; x.b = [1, 2; 3, 4]; x.c = "string"; octave:2> x.a x.a = 1 octave:3> x.b x.b = 1 3

2 4

octave:4> x.c x.c = string •

Short-circuit boolean operators

Octave's `&&' and `||' logical operators are evaluated in a short-circuit fashion (like the corresponding operators in the C language) and work differently than the element by element operators `&' and `|'. •

Increment and decrement operators

Octave includes the C-like increment and decrement operators `++' and `--' in both their prefix and postfix forms. •

Unwind-protect

Octave supports a limited form of exception handling modelled after the unwind-protect form of Lisp. The general form of an unwind_protect block looks like this: unwind_protect body unwind_protect_cleanup cleanup end_unwind_protect •

Variable-length argument lists

Octave has a real mechanism for handling functions that take an unspecified number of arguments without explicit upper limit. Here is an example of a function that uses the new syntax to print a header followed by an unspecified number of values: function foo (heading, ...) disp (heading); va_start (); while (--nargin) disp (va_arg ()); endwhile endfunction •

Variable-length return lists

Octave also has a real mechanism for handling functions that return an unspecified number of values. For example: function [...] = foo (n) for i = 1:n vr_val (i); endfor endfunction

Octave has been mainly built with MATLAB compatibility in mind. It essentially shares a lot of features in common with MATLAB: 1. 2. 3. 4.

Matrices as fundamental data type. Built-in support for complex numbers. Powerful built-in math functions and extensive function libraries. Extensibility in the form of user-defined functions.

Some of the differences that do exist between Octave and MATLAB can be worked around using "user preference variables." GNU Octave is mostly compatible with Matlab. However, Octave's parser allows some (often very useful) syntax that Matlab's does not, so programs written for Octave might not run in Matlab. For example, Octave supports the use of both single and double quotes. Matlab only supports single quotes, which means parsing errors will occur if you try to use double quotes (e.g. in an Octave script when run on Matlab). Octave and Matlab users who must collaborate with each other need to take note of these issues and program accordingly. Note: Octave can be run in "traditional mode" (by including the --traditional flag when starting Octave) which makes it behave in a slightly more Matlabcompatible way.

This chapter documents instances where Matlab's parser will fail to run code that will run in Octave, and instances where Octave's parser will fail to run code that will run in Matlab. This page also contains notes on differences between things that are different between Octave (in traditional mode) and Matlab. Tip for sharing .mat files between Octave and Matlab: always use save -V6 if you are using Matlab 7.X. Octave version 2.1.x cannot read Matlab 7.X .mat files. Octave 2.9.x can read Matlab 7.X .mat files.

load For compatablility, it is best to specify absolute paths of files for LOAD. Matlab (7.0) vs Octave (2.1.71): paths are not searched for .mat files in the same way as .m files: Matlab a = 1; save /tmp/a.mat a ; addpath('/tmp'); load a.mat % OK

Octave a = 1; save /tmp/a.mat a ; LOADPATH=['/tmp:',LOADPATH]; load a.mat % error: load: nonexistent file: `a.mat'

For any other purpose, don't use absolute paths. It is bad programming style. Don't do it. It causes many problems.

C-Style Autoincrement and Assignment operators Octave (3.0.1) supports C-style autoincrement and assignment operators: i++; ++i; i+=1; etc.

MatLab (7.0) does not.

product of booleans Matlab (7.0) and Octave (3.0.2) responds differently when computing the product of boolean values: X = ones(2,2) ; prod(size(X)==1) Matlab: ??? Function 'prod' is not defined for values of class 'logical'. Octave: ans = 0

nargin

Matlab (7.0) will not allow the following; Octave (2.1.71) will. function a = testfun(c) if (nargin == 1) nargin = 2; end

startup.m Matlab will execute a file named startup.m in the directory it was called from on the command line. Octave does not. It will, however, execute a file named .octaverc.

['abc ';'abc'] ['abc ';'abc'] is allowed in Octave; Matlab returns: ?? Error using ==> vertcat In Octave the result will be a 2 by 4 matrix where the last element of the last row is a space.

Calling Shells the "! STRING" syntax calls a shell with command STRING in Matlab. Octave does not recognize !. Always use system(STRING) for compatibility. If you really miss the one-character shortcut, for convenience on the command line you can create a similar shortcut by defining the following in your 2.9.x octave startup file: function S(a), system(a); end mark_as_rawcommand('S')

Now "S STRING" will evaluate the string in the shell.

hist hist.m in Octave has a normalization input, Matlab does not.

Getting Version Information Octave (2.1.7X) uses "OCTAVE_VERSION", Matlab uses "ver" for version informaton. (Octave 2.9.5 and later has "ver", so is compatible in this sense.)

Format of printing to screen

Cell arrays and structures print to screen in different format. There may be switches to make them the same, for example, struct_levels_to_print. None are set in --traditional mode, however.

Attempting to load empty files MATLAB lets you load empty files, OCTAVE does not. system('touch emptyfile') ; A = load('emptyfile') Matlab 6.5 : A=[] Octave 2.1.71 : error: load: file `emptyfile' seems to be empty! error: load: unable to extract matrix size from file `emptyfile'

fprintf and printf Matlab doesn't support `printf' as a command for printing to the screen. foo = 5; printf('My result is: %d\n', foo)

works in Octave, but not Matlab. If using Matlab, `fprintf' covers writing both to the screen and to a file: foo = 5; fprintf('My result is: %d\n', foo)

In Matlab, the omission of the optional file-handle argument to fprintf (or using the special value 1 for standard output or 2 for standard error) causes the text to be printed to the screen rather than to a file.

Whitespace Matlab doesn't allow whitespace before the transpose operator. [0 1]'

works in Matlab, but [0 1] '

doesn't. Octave allows both cases.

Line continuation Matlab always requires `...' for line continuation. rand (1, ... 2)

while both rand (1, 2)

and rand (1, \ 2)

work in Octave, in addition to `...'.

Logical operators (And, Or, Not) Octave allows users to use two different group of logical operators: the ones used in Matlab, or the ones familiar to C/Java/etc programmers. If you use the latter, however, you'll be writing code that Matlab will not accept, so try to use the Matlab-compatible ones: •

For not-equal comparison, Octave can use '~=' or '!='. Matlab requires '~=' .

•

For a logical-and, Octave can use `&' or `&&'; Matlab requires `&' . (note: Matlab supports `&&' and `||' as short-circuit logical operators since version 6.5.)

•

For a logical-or, Octave can use `|' or `||'; Matlab requires `|' . (note: Octave's '||' and '&&' return a scalar, '|' and '&' return matrices)

Some other differences •

There used to be a difference in datestr between Octave-forge and Matlab. Has this been changed?

•

MATLAB uses the percent sign to begin a comment. Octave uses both the pound sign and the percent sign interchangeably.

•

MATLAB has no fputs function. Call fprintf instead.

•

load in Matlab = load --force in Octave (should --force be default in --traditional?)

•

page_output_immediately = 1 should this be default in --traditional?

•

For exponentiation, Octave can use `^' or `**'; Matlab requires `^'.

•

For string delimiters, Octave can use ` or "; Matlab requires '.

•

For ends, Octave can use `end{if,for, ...}'; Matlab requires `end'.

•

For "dbstep in" use "dbstep"; for "dbstep" use "dbnext"

•

For "eig(A,B)" use "qz(A,B)"

•

For gallery, compan, and hadamard install http://www.ma.man.ac.uk/~higham/testmat.html

•

For subspace try http://www.math.com/support/ftp/linalgv5.shtml

•

For datetick use gnuplot commands:

__gnuplot_set xdata time __gnuplot_set timefmt "%d/%m" __gnuplot_set format x "%b %d" •

If something (like Netlab) need a function called fcnchk you can just put the following into a file called fcnchk.m and put it somewhere Octave can find it:

function f=fcnchk(x, n) f = x; end

Toolboxes Introduction to the Symbolic Math Toolbox The symbolic toolbox is a bit difficult to use but it is of great utility in applications in which symbolic expressions are necessary for reasons of accuracy in calculations. The toolbox simply calls the MAPLE kernel with whatever symbolic expressions you have declared, and then returns a (usually symbolic) expression back to MATLAB. It is important to remember that MAPLE is not a numeric engine, which means that there are certain things it doesn't let you do that MATLAB can do. Rather, it is useful as a supplement to provide functions which MATLAB, as a numerical engine, has difficulty with. The symbolic math toolbox takes some time to initialize, so if nothing happens for a few seconds after you declare your first symbolic variable of the session, it doesn't mean you did anything wrong. The MATLAB student version comes with a copy of the symbolic math toolbox.

Symbolic Variables You can declare a single symbolic variable using the 'sym' function as follows. >> a = sym('a1') a = a1

You can create arrays of symbolic expressions like everything else:

>> a2 = sym('a2'); >> a = [a1, a2] a = [ a1, a2]

Symbolic variables can also be declared many at a time using the 'syms' function. By default, the symbolic variables created have the same names as the arguments of the 'syms' function. The following creates three symbolic variables, a b and c. >> syms a b c >> a a = a

Symbolic Numbers

Symbolic numbers allow exact representations of fractions, intended to help avoid rounding errors and representation errors. This section helps explain how to declare them. If you try to add a number into a symbolic array it will automatically turn it into a symbolic number. >> syms a1, a2; >> a = [a1, a2]; >> a(3) = 1; %would normally be class 'double' >> class(a(3)) ans = sym

Symbolic numbers can also be declared using the syntax a(3) = sym('number'). The difference between symbolic numbers and normal MATLAB numbers is that, if possible, MAPLE will keep the symbolic number as a fraction, which is an exact representation of the answer. For example, to represent the number 0.5 as a fraction, you can use: >> sym(0.5) ans = 1/2

Here, of course, MATLAB would normally return 0.5. To make MATLAB change this back into a 'double', type: >> double(ans) ans = 0.5000

Other class conversions are possible as well; for instance, to change it into a string use the 'char' function. There is no function to directly change a symbolic variable into a function handle, unfortunately. A caveat: Making a symbolic variable of negative exponentials can create problems if you don't use the correct syntax. You cannot do this: >> sym('2^-5') ??? Error using ==> sym.sym>char2sym Not a valid symbolic expression.

Instead, you must do this: >> sym('2^(-5)') ans = 2^(-5)

MAPLE is thus more picky about what operators you can use than MATLAB.

Symbolic Functions You can create functions of symbolic variables, not just the variables themselves. This is probably the most intuitive way to do it:

>> syms a b c %declare variables >> f = a + b + c ans = a + b + c

If you do it this way, you can then subsequently perform substitutions, differentiations, and so on with respect to any one of these variables.

Substituting Values into Symbolic Variables Substitutions can be made into functions of symbolic variables. Suppose you defined the function f = a + b + c and wish to substitute a = 3 into f. You can do this with the following syntax: >> syms a b c %declare variables >> f = a + b + c; >> subs(f, a, 3) ans = 3+b+c

Notice the form of this function call. The first argument is the name of the function you wish to substitute into. The second can be either the name of the symbolic variable you want to plug in for or its present value, but if you want to avoid confusion they should be the same anyway. The third argument is the value you want to plug in for that variable. The value you're plugging in need not be a number. You can also plug in other variables (including those already present in the function) by using strings. Using the same f: >> subs(f, a, 'x') ans = x+b+c >> subs(f, a, 'b') ans = 2*b + c

If x is already a symbolic variable you can omit the quotes (but if it's not you'll get an undefined variable error): >> syms x >> subs(f,a,x) ans = x+b+c

Multiple substitutions are allowed; to do it, just declare each of them as an array. For example, to plug in 1 for a and 2 for b use: >> subs(f, [a,b], [1,2]) ans = 3+c

Finally, if you substitute for all of the symbolic values in a function MATLAB automatically changes the value back into a double so that you can manipulate it in the MATLAB workspace. >> subs(f, [a,b,c], [1,2,3])

ans = 6 >> class(ans) ans = double

Algebraic Function Manipulations The symbolic math toolbox allows you several different ways to manipulate functions. First off you can factor a function using 'factor' and multiply it out using 'expand': >> syms a b >> f = a^2 - 2*a*b + b^2; >> factor(f); ans = (a - b)^2 >> expand(ans) ans = a^2 - 2*a*b + b^2

The 'collect' function does the same thing as the 'expand' function but only effects polynomial terms. 'Expand' can also be used to expand trigonometric and logarithmic/exponential functions with the appropriate identities. The Horner (nested) representation for a function is given by 'horner': >> horner(f) ans = b^2+(-2*b+a)*a

This representation has a relatively low number of operations required for its evaluation compared to the expanded version and is therefore helpful in making calculations more efficient. A common problem with symbolic calculations is that the answer returned is often not in its simplest form. MATLAB's function 'simple' will perform all of the possible function manipulations and then return the one that is the shortest. To do this do something like: >> Y = simple(f) Y = (a - b)^2

Algebraic Equations The symbolic math toolbox is able to solve an algebraic expression for any variable, provided that it is mathematically possible to do so. It can also solve both single equations and algebraic systems.

Solving Algebraic Equations With a Single Variable MATLAB uses the 'solve' function to solve an algebraic equation. The syntax is solve(f, var) where f is the function you wish to solve and var is the variable to solve for. If f is a function of a single variable you will get a number, while if it is multiple variables you will get a symbolic expression.

First, let us say we want to solve the quadratic equation x^2 = 16 for x. The solutions are x = -4 and x = 4. To do this, you can put the function into 'solve' directly, or you can define a function in terms of x to solve and pass that into the 'solve' function. The first method is rather intuitive: >> solve('x^2 = 16', x) ans = -4 4 >> solve(x^2 - 16, x) ans = -4 4

Either of these two syntax works. The first must be in quotes or you get an 'invalid assignment' error. In the second, x must be defined as a symbolic variable beforehand or you get an 'undefined variable' error. For the second method you assign a dummy variable to the equation you want to solve like this: >> syms x >> y = x^2 - 16; >> solve(y, x);

Note that since MATLAB assumes that y = 0 when you're solving the equation, you must subtract 16 from both sides to put the equation into normal form.

Solving Symbolic Functions for Particular Variables The format for doing this is similar to that for solving for a single variable, but you will get a symbolic function rather than a number as output. There are a couple of things to look out for though. As an example, suppose that you want to solve the equation y = 2x + 4 for x. The expected solution is x = (y-4)/2. Lets see how we can get MATLAB to do this. First let's look at how NOT to do it: >> syms x >> y = 2*x + 4; >> solve(y, x) ans = -2

What has happened here? The MATLAB toolbox assumes that the 'y' you declared is 0 for the purposes of solving the equation! So it solved the equation 2x + 4 = 0 for x. In order to do what you intended to do you have to put your original equation, y = 2x + 4, into normal form, which is 2x + 4 - y = 0. Once this is done, you need to assign a 'dummy' variable like this: >> syms x y >> S = 2*x + 4 - y; %S is the 'dummy'

>> solve(S, x) ans = -2 + 1/2*y

This is, of course, the same thing as what we expected. You could also just pass the function into the 'solve' function like this, as done with functions of a single variable: >> solve('y = 2*x + 4', x);

The first method is preferable, because once it is set up it is much more flexible to changes in the function. In the second method you would have to change every call to 'solve' if you changed the function at all, whereas in the first you only need to change the original definition of S.

Solving Algebraic Systems The 'solve' command also allows you to solve systems of algebraic equations, and will attempt to return all solutions to these systems. As a first example, let us consider the linear system a + b = 3 a + 2*b = 6,

which has the solution (a,b) = (0,3). You can tell MATLAB to solve it as follows: >> syms a b >> f(1) = a + b - 3; >> f(2) = a + 2*b - 6; >> [A,B] = solve(f(1), f(2)) A = 0 B = 3

If only one output variable is specified but there are multiple equations, MATLAB will return the solutions in a struct array: >> SOLUTION = solve(f(1), f(2)) SOLUTION = a: [1x1 sym] b: [1x1 sym] >> SOLUTION.a a = 0 >> SOLUTION.b b = 0

The good thing about this is that the fields have the same names as the original variables, whereas in the other form it is easy to get confused which variable is going into which spot in the array. In addition, the struct array is more convenient for large systems. Now let us look at a slightly more complex example: a^2 + b^2 = 1

a + b = 1

This has solutions (a,b) = (0,1) and (a,b) = (1,0). Now putting this into MATLAB gives: >> f = [a^2 + b^2 - 1, a + b - 1]; SOLUTION = solve(f(1), f(2)); >> SOLUTION.a ans = 1 0 >> SOLUTION.b ans = 0 1

Here both solutions are given, a(1) corresponds to b(1) and a(2) corresponds to b(2). To get one of the solutions into a single array together you can use normal array indexing, as in: >> Solution1 = [SOLUTION.a(1), SOLUTION.b(1)] Solution1 = [1, 0]

Analytic Calculus MATLAB's symbolic toolbox, in addition to its algebraic capabilities, can also perform many common calculus tasks, including analytical integration, differentiation, partial differentiation, integral transforms, and solving ordinary differential equations, provided the given tasks are mathematically possible.

Differentiation and Integration with One Variable Differentiation of functions with one or more variables is achieved using the 'diff' function. As usual, you can either define the function before the differentiation (recommended for M files) or you can manually write it in as an argument (recommended for command-line work). If there is only one symbolic variable in the expression, MATLAB assumes that is the variable you are differentiating with respect to. The syntax is simply: >> syms x >> f = x^2 - 3*x + 4; >> diff(f) % or diff('x^2 - 3*x + 4') ans = 2*x - 3

Integration, similarly, is achieved using the 'int' function. Only specifying the function results in an indefinite integral, or the antiderivative of the function. >> int(f) ans = 1/3*x^3-3/2*x^2

Note that if you only specify one output argument (or none at all), the 'int' function omits the integration constant. You just have to know it's there.

To do a definite integral on a one-variable function, simply specify the beginning and end points. >> int(f, 0,1) ans = -7/6

Differentiation and Integration of Multivariable Functions A convenient way of representing the derivatives of multivariate functions (partial derivatives) is with the Jacobian, which is performed by the 'jacobian' function. To use it, you should define an array of symbolic functions and then just pass it to the function (note the difference between the use of this function, which requires all equations to be in the same array, and the use of 'solve', which requires you to separately pass each equation): >> syms a b >> f = [a^2 + b^2 - 1, a + b - 1]; >> Jac = jacobian(f) Jac = [ 2*a, 2*b] [ 1, 1]

Note that the first row is the gradient of f(1) and the second the gradient of f(2). If you only want a specific partial derivative, not the entire Jacobian, you can call the 'diff' function with the function you want to differentiate and the variable you wish to differentiate with respect to. If none is specified, differentiation occurs with respect to the variable closest to 'x' in the alphabet. >> diff(f(1), a) ans = 2*a

It is worth noting that to complete an implicit differentiation, one can explicitly state the implicit assumption by multiplying the differentiated function by variable in a multivariable equation.

, where xi is the ith

Indefinite integration of multivariate functions works the same as for single functions; pass the function and MATLAB will return the indefinite integral with respect to the variable closest to x: >> int(f(1)) ans = a^2*b+1/3*b^3-b

This is the integral with respect to b. To avoid confusion, you can specify the variable of integration with a second argument, as with differentiation. >> int(f(1), a) ans = 1/3*a^3+b^2*a-a

Definite integration (as far as I can tell) can only be done with respect to one variable at a time, and this is done by specifying the variable, then the bounds: >> int(f(1), a, 1, 2) %integrate a from 1 to 2, holding b constant ans = 4/3 + b^2

Analytic Solutions to ODEs MATLAB can solve some simple forms of ODEs. Unlike with the integration and algebraic solving techniques, the syntax for the differential equation solver requires that you put the function in manually in a specific manner. The derivatives must be specified using the symbol 'DNV', where N is the order of the derivative and V is the variable that is changing. For example, suppose you seek the solution to the equation , the solutions to which are of the form x(t) = A*cos(t) + B*sin(t). You would put this equation into the 'dsolve' function as follows: >> syms x >> dsolve('D2x = -x') ans = C1*sin(t)+C2*cos(t)

Unlike the 'int' function, dsolve includes the integration constants. To specify initial conditions, just pass extra arguments to the 'dsolve' function as strings. If x'(0) = 2 and x(0) = 4 these are inserted as follows: >> dsolve('D2x = -x', 'Dx(0) = 2', 'x(0) = 4') ans = 2*sin(t)+4*cos(t)

Note that the initial conditions must also be passed as strings. MATLAB can also solve systems of differential equations. An acceptable syntax is to pass each equation as a separate string, and then pass each initial condition as a separate string: >> SOLUTION = dsolve('Df=3*f+4*g', 'Dg =-4*f+3*g', 'f(0) = 0', 'g(0) = 1') SOLUTION = f: [1x1 sym] g: [1x1 sym] >> SOLUTION.f SOLUTION.f = exp(3*t)*sin(4*t) >> SOLUTION.g SOLUTION.g = exp(3*t)*cos(4*t)

The 'dsolve' function, like 'solve', thus returns the solution as a structure array, with field names the same as the variables you used. Also like 'solve', you can place the variables in separate arrays by specifying more than one output variable.

Integral Transforms

MATLAB's symbolic math toolbox lets you find integral transforms (in particular, the Laplace, Fourier, and Z-transform) and their inverses when they exist. The syntax is similar to the other symbolic math functions: declare a function and pass it to the appropriate functions to obtain the transform (or inverse). Computer Science > Programming > MATLAB Programming/Print Version The GUIDE Toolbox provided by MATLAB allows advanced MATLAB programmers to provide Graphical User Interfaces to their programs. GUIs are useful because they remove end users from the command line interface of MATLAB and provide an easy way to share code across nonprogrammers. In addition by using special compilers the mathematical ability of MATLAB seamlessly blends in with the GUI functionality provided. Just to provide an example, assume you are writing a nonlinear fitting system based on the levenburg marquardt algorithm. Implementing a same GUI in VC++ would take at least a month of effort. But in MATLAB with the existing nlinfit function the time for such an endeavor would be hours instead of days. The figure shows an example of a simple GUI created with the GUIDE toolbox, it takes as input two numbers adds them and displays them in the third textbox, very simple but it helps illustrate the fact that such a GUI was created in minutes. The first section we need to understand is the concept of a callback

CallBack A callback is a functions executed whenever the user initiates an action by clicking for example on a button or pressing a key on the keyboard. Programming a callback is therefore the most important part of writing a GUI. For example in the GUI illustrated above, we would need to program a callback for the the button Add. This is provided as a callback in the code. The code is provided below and illustrates how to write a simple callback. % --- Executes on button press in pushbutton1. function pushbutton1_Callback(hObject, eventdata, handles) % hObject handle to pushbutton1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) n1 = get(handles.edit2,'string'); n2 = get(handles.edit1,'string'); S = str2double(n1) + str2double(n2); set(handles.edit3,'String',num2str(S))

In this piece of code I get the numbers as a strings from the edit boxes and then convert them into numbers using the str2double function provided in MATLAB. I then set the string for the other edit box as the sum of these two numbers. This completes the simple example of a GUI needed to add two numbers. To illustrate a more complex example I

show how a simple exponential function can be plotted and you change the function's parameters, with a little bit of imagination you could make it plot any arbitrary function you enter. To make the example even more complex I have two GUIS, one is the control gui and the other is the plotting GUI, this allows the user to program some of the more complicated functionality expected out of the modern GUI systems.

Sample Time Colors By selecting Format->Sample Time Colors, you can get Simulink to color code signal lines according to their sample times. Colors are only updated when the model is updated or simulated.

The most common colors are: summary="This table gives the color, sample time pairings for Simulink's sample time colors. Magenta Constant Black

Continuous

Red

Fastest Discrete Sample Time

Yellow Hybrid/Mixed Sample Time For the rest of the colors and other information, see Enabling Sample Time Colors (the Mathworks website) Note: Constant sample time will only be displayed if Inline Parameters is checked in the Advanced Tab of Simulation->Simulation Parameters. This is because constant blocks can have variables as their arguments.Click here to go back up to the MATLAB programming book.