Introduction to MATLAB

Transcription

1 Introduction to MATLAB Getting MATLAB to Run Programming The Command Prompt Simple Expressions Variables Referencing Matrix Elements Accessing Matrix Elements Assigning into Submatrices Matrix Concatenations More Expressions Plotting Logical Constructs Formatting Text "if" Statement "for" Loops "while" Statements Variable Scope Multiple Input Functions More on Logic

2 Getting Matlab to run To start the Matlab, type: >> matlab Or >> _matlab ver Programming The most awful truth about computers is that they are all dumb. Everything must be explicitly told. Spelling out the most minor and obvious detail. Otherwise, you will either get a syntax error (computer not understanding what you are saying) or a programming error (the computer understanding what you are saying but it is different from what you mean). There are many programing languages. Why use Matlab? As a first programming language, it is great, it has simple syntax, it is easy to get started with and easy to do graphics. While other programming languages are faster running they are slower to learn and to use. In addition, Matlab, as an interpreted language, is interactive, which allows for more exploratory programming. Once the exploration is done, one should consider rewriting the program in a different language if intensive use is required. The Command Prompt In the interactive terminal of Matlab you will see the command prompt (>>). This is Matlab's way of telling you that it is ready to receive instructions from you. To command Matlab, type your request and hit Enter. Matlab then evaluates your command and displays the output (if any). To get help use the help, and lookfor commands. help should be used to find out information about a known command, e.g. help zeros. lookfor can be used to find a command to perform a task e.g. lookfor roots. If you do not get the command prompt for some reason, you may have typed a syntax error, or have put Matlab into an infinite loop. Don't panic! press Ctrl-C and you'll get the prompt back.

3 Simple Expressions Matlab's most basic building blocks are expressions: 1, 1+1,4*3, 4^2, 5/6. But Matlab's real strength is in handling matrices and vectors. To enter a vector we use brackets. Try the following examples to see what happens: [1 2], [1:10], [1, 2 3 4; 5 6, 7 8 ; ] Notice that the colon is used to create an arithmetic sequence. We can also create a sequence with a step-size different from 1: [4:0.1:5], [5:-2:-5]. To transpose a matrix (or vector) use the apostrophe ('): >> [1:10]' Variables Asking Matlab to calculate things for us is great, but most of the times a calculation is only one step out of many and the result of a calculation need to be saved for the (very near) future. To do this we use 'variables', which are simply a way to name a bit of Matlab data. This is best explained by example: >> a=1 puts the value 1 into a variable called a. typing >> a results back in the value 1. We can also assign a calculation into a variable: >> b=1+1 and now b will contain the answer, 2. Variable names can (and should) be longer than one letter. A word or a couple of words strung together ts to work the best.

4 Referencing matrix elements We can access the elements of a matrix in the following manner: >> a=[ ] >> a(4) >> b=[1 2 ; 3 4] >> b(2,2) Where the numbers in the parenthesis are (row, column). Accessing Matrix Elements Submatrices We already say in the previous lecture that you can access the (2,3)th element of a matrix a with the notation >> a(2,3) This gives the entry of a in the second row and the third column. We can also gain access to several elements at the same time. Try for example: >> a(2, [1 2]) >> a([3 4], 3) >> a([2 3], [3 4]) These commands extract a submatrix of a with the corresponding set of rows and columns. It is often useful to extract a whole row or column of a given matrix. Assuming that B has 10 rows, we can extract the second column of B with the command >> B(2,1:10) Using the column notation instead of spelling out [ ]. In fact, there is more shorthand to be learned here. First, we can replace the 10 with the

5 keyword: >> B(2,1:) This keyword gets replaced by the largest number allowed for that place. So if it is used in the rows it will be replaced wit the number of rows, if in the columns, with the number of columns. It is useful to know that one can manipulate it as one would any other variable: >> B(2,1:-1) >> B(2,1:/2) Of course, if by the manipulation you get an illegal expression, matlab will complain: >> B(2,1:+1) >> B(2,sin():) The 1: construction is so useful that Matlab has a further shorthand for it: >> B(2,:) Means the same as B(2,1:). So, one should think of the bare colon (:) as meaning "everything". Non-contiguous element access The elements of the matrix that we access do not have to be contiguous: >> a(3,[2 4 6]) >> a([2 4], [1 3]) They don't have to be in increasing order: >> a(3,[2 6 4]) >> a([4 1], [3 2]) Nor do they have to only appear once: >> a(3,[ ]) >> a(3*ones(1,3),2*ones(1,10))

6 Assigning into submatrices Just as in the reference and assignment of single numbers into matrices, we can also assign new values into complete submatrices. The only thing that needs to be checked is that the matrix on the right of the equal sign has the same dimension as the matrix referenced on the left: >> a([2 3],[1 4]) =[1 2 ;3 4] >> a(1,1:4)=2:5 Scalar expansion Matlab has a nice time-saving notation. If an operator requires a matrix of a known size and instead is given a scalar (i.e a number). It "expands" the scalar into a full matrix of the required size. This is quite cumbersome to say...much better to show examples: >>1:10.^2 >>2.^1:10 >>[1 2; 3 4] + 5 In these three cases, the relevant operator is looking for a matrix of the same size as the one it already has on the other side, but instead it finds a number. So Matlab prets as if the matrix of the expected size is there with each of its entries equal to the number. Matrix concatenations Using the brackets we can create more and more complicated matrixs: >> [ones(2,2) zeros(2,2)] >> [ones(2,2) ; zeros(2,2)]

7 More Expressions All Matlab expressions are made out of basic building blocks put together: Constants Variables Operators Keywords Functions Constants We have met some constants already: 1, 3, 1.56, -5, pi. But there are more that we haven't yet met: i, j, (root of -1), 2.4e12 (2.4 times 10^12), 'a' (the character a),'hello' (a vector consisting of the letters 'h','e','l','l','o'). Note that i and jcan be used in a way different from variables: you can write 3i or -15j, but if you define a=10, writing 4a is an error... Also note that Matlab allow using the colon notation with letters:'a':'z'. Note that you can access letters in a string just elements in any other vector: >> a='hello' >> a(2) >> a(4:5)='p!' Variables There isn't much to say about variables. Variable names must start with a letter, and the variable name length must be no more than 32 characters. Names may use letters, numbers and the underscore '_'. Note that matlab will make matrixs and vectors grow as needed (filling the empty spaces with zeros): >> a = [1 2 ; 3 4] >> a(5,5) = 1

8 Operators We have met many operators: =,+,-,/,*,^,.,',[,],(,),;,:. There are a few others, as we shall see shortly. Keywords We have not yet seen many keywords...the only exception is. Functions We have seen: ones, zeros, diag, size,...there are others. For example: sin, cos, exp, log, sqrt. You can also define new functions as we shall learn in the next lecture or two. Plotting Let's learn how to make nice pictures. plotting is one of the basic tools of Matlab. Most of the plotting happens through plot and its variants. Best is to learn by example: >> x = 0:0.01:1 ; >> y = sin(x) ; >> plot(x,y) Notice several things. First, the semicolon ';' at the of the expression suppresses the output of the result. However, it does not inhibit the evaluation of the expression. So x and y are still as they would be if the semicolon was missing but the screen is blissfully clean (try it without the semicolon). Second, notice that the plot function created a nice figure for us, with the first "wave" of the sin. try it put with other functions. Logical Constructs Sometimes it is important to compare numbers. For example we might want to act differently if a given number is positive or negative. For this we need logical operators and comparisons.

9 Testing First, comparisons. we use the operators ==, ~=, < >, <=, >= (the ~ key is usually at the top left of the keyboard, and it needs the SHIFT key) to compare numbers e.g: >> 1==2 >> 3~=5 >> 5<=6 >> 7<=10 Notice that these expressions return 0 when they are false and 1 when the are true. This is the convention. 0 is false and 1 is true. While this notation looks similar to math, there are a few pitfalls 1<x<5 does NOT return what it does in math...in fact this will alway be true regardless of the value of x. Can you see why? == is not =. Regardless of how many time I will say this, people will still make this mistake many times...but perhaps by saying it lots, it will happen less...maybe. remember...you have been warned...= is assignment, and == is a test for equality. Boolean operations What if we want to check two things? A and B? or perhaps A and (B or C)...for this we need Boolean operators: & (AND), (OR), and ~(NOT) (The is usually found at the middle right of the keyboard, and it needs a SHIFT key.) 1~=2 3<4 3~=4 & 8==(4+4) ~(2<=8) Formatting Text Once in a while we would like to display some text that is nicely formatted and not just the output as Matlab wants to display. To do this we use the sprintf command. In its simplest form it is quite straightforward...the first arguemnt is a format string, and the rest are parameters for the string. examples: sprintf('hello, here is a number %d',17)

10 sprintf('my name is %s and I am %g years old','meysam',32) Read up on sprintf to get the whole picture Flow Control Up to now, we saw how to make Matlab execute a list of commands without any decision making. Now we will learn to have Matlab do different things deping on the results of previous expressions. if Statement We might want have Matlab evaluate a bunch of expressions if some condition is satisfied, and another bunch of expressions otherwise. The syntax for this is (don't type this into Matlab..) if condition Bunch Of expressions else Other expressions Note that you must replace the condition with an expression that Matlab will evaluate first. Example: >> if 3<4 '3 is less than 4' else '3 is not less than 4' >> Since the condition is true, the expression which is evaluated is '3 is less than 4', and this is then displayed on the screen. We can try something more interesting. Type the following into a file (say, if_example.m)

11 x = rand; s = 'The number %g is %s than 0.5\n'; if x<0.5 fprintf(s,x,'less') else fprintf(s,x,'more') Save the file and run it (either from the menu, or by typing the name of the file). You might be asked to change directory, do that. As you know, rand returned a (pseudo-)random number (to initialize it, you need to call something like rand('twister', sum(100*clock)), but read more about rand to find out why.) fprintf is similar to sprintf except that it writes the result instead of returning it. If you want to understand this a little further..read the help files and look at the difference in the output from the following commands: >> a=sprintf( Hello! ) >> b=fprintf( Hello! ) The '\n' at the of the string s is to make sure that Matlab actually goes down a line before returning us the command prompt...try removing it and see what happens. What is Truth? I'm not trying to get all philosophical or anything, but we need to know what matlab considers to be "true" and what to be "false", if we want to understand how the execution of an if statement works. For this I want you to create another file (oracle.m) with the following if x 'True' else 'False' Then, from the command line, assign various values to x and run your file. See if each x value is true or false. Things you can try out. x=(1==4) x=4 x=0

12 x=[0 0] x=[] x=[0 1] x=[1 2 3] x='hello' x='' x=' ' x=-1 x=i for Loops Loops are good when we want to do stuff over and over again. For example, if we didn't know how to sum up the elements of a vector using sum or linear algebra commands, we could use a loop to do it (a bad idea...much slower than the alternative, about 100 times slower!). But we might still need them...the syntax is as follows: for variable = vector a bunch of expressions here variable is an ELEMENT of vector The way it works is that for every element in vector, Matlab runs through all the commands in the block. during each such execution, the value of variable is set to that element...let see a simple example: for v = 1:10 v v^2 Notice that for every number from 1 to 10 we have two printouts (since there were two commands in the block) one simply the number and the other is the number squared...just as we asked. One can have any legal variable name instead of the variable, but note that it gets overwritten. Also the vector, can be any vector... for v = 'hello' v

13 One can "break out" of a loop before it is done by issuing a break command: for v = 'hello and goodbye' v if v=='d' break OK, enough new material..lets exercise! while Statements Sometime, you need to iterate many times but don't really want to create a vector, or you don't know in advance how many time you need to iterate...for this you can use a while statement. The syntax for a while statement is while expression a bunch of statements At the beginning of each iteration, Matlab evaluates the expression at the top of the loop. if it is true, Matlab executes the statements to the. If it is false, Matlab jumps to the and continues after it. So, for example: >> x=10; >> while x>0 x=x-1 >>

14 will print out all the numbers from 9 to 0 (including both) Take note of two things: Matlab does not check the expression all the time. Only before starting a new iteration If the expression at the top is false when the first iteration should start, it will not. Defining your own function While Matlab has a vast collection of functions, sometimes you want to add some of your own. A function should be thought of as a black box, with a slot in the top (for the input) and a drawer at the bottom for the output. When you give the function input ("call the function"), it calculates the output and puts it in the drawer ("returns it"). Once the function is written, you do not need to care about how it works. Functions must be defined in their own file, and the name of the file must match the name of the function. So, for example, to define a function called example_function_1, create a file named example_function_1.m. That file must start with the keyword function, so function y = example_function_1(x) is the first line of the file. let's dissect this first line: function As I said, the first word must be "function". y = This declares the local name of a variable whose value will be returned when the execution of the function is over. example_function_1 The name of the function. This must match the name of the file. (x) The local name of the variable that will contain the input value. So the rest of the file could be

15 y=2*x; admittedly, a very boring function: It takes the input value (locally called x), doubles it, and assigns that result to y. Since y is declared to be the return value and the function has ed, the return value is simply twice the input. we call the function by giving it a value: >> example_function_1(17) or >> example_function_1(rand) The return value can then be assigned to another variable: >> z= example_function_1(18); >> z Variable Scope An important difference between a function and a script (the files that don't have "function" as their first word are called scripts) is that of "variable scope". When executing a script from the command-line, the scripts has access to all the variables that are defined on the command line. For instance, if you have a script called script1.m whose contents are x+1 y=y/2 and you run this script from the commandline like so: >> x=2 >> y = 6 >> script1 >> y

16 You will see that the script knows that x=2 and that y=6. Not only this, note that y is changed in the scripts, the new value of y, 3 is also the value of y when check on the command line. This seems natural, perhaps, but this is not how functions behave. A function with the same content function1.m function function1 x+1 y=y/2 will fail to run, since within the function there are no such variables x and y. The variables of the function, are different from the variables outside the function, even if they have the same name. change your prime checking program so that the input is the number, and the output is 1 if the number is prime and 0 if not. change your n-changing program, so that the input is the initial value of n and the output is the number of steps it took to reach 1. use your primechecking function in another program (can be a script) that finds (a) the 1000th prime number and (b) the first prime number greater than use your n-changing function in another program (can be a script) that finds the number of iterations needed for the first 100 numbers, and plots then nicely. Multiple Input Functions You can define a function to have more than one input. This is the syntax: function r = func1(x,y) Remember that the name of the function must match the name of the file (with a.m added at the ). Therefore, no spaces, operators etc. in the name of the function. More on Logic

17 x = [ ], y = [ ].Evaluate and understand the following: o x>y o y>x o x==y o x<=y o x y o x&y o x&~y o ((x>y) (y>x)) o x<5 o x(y<5) x = [1:10], y = [ ]. Evaluate and understand the following: o (x>3)&(y<5) o x(x>5) o x(y>5) o y(x<=4) o x( (x<2) (x>=8) ) o y( (x<2) (x>=8) ) (remember, all nonzero values are true)

18 Exercises Learn about the commands ones, zeros, sum, and diag. Learn about magic and verify what you learned with sum. Create the vector consisting of the whole even numbers between 15 and 27. Create the vector consisting of the whole odd numbers between 15 and 27. Create the vector of the previous question in decreasing order. Verify that the sum along the anti-diagonal of the magic matrix is also the same as all the other rows, cols and the diagonal. (Hint: you can first change the order of the rows or columns before taking the diag) If A is a matrix, how can you get the sub-matrix of elements whose both coordinates are odd? Let x = [ ] and y = [ ]. Think of y as a specific reordering of the numbers 1:4. Use y to reorder x in the same fashion. Given a ``permutation'' vector as y is in the previous item. Find the vector which gives the inverse of the permutation! (Tricky!) Given two permutations p and q, find the permutations that is the same as p(q(s)) (that is it permutes the set s as if you would first permute with q and then with p) Write an expression for the sum of the integers from 1 to 100 Write an expression for the sum of the squares of the numbers from 1 to 10 Write an expression for the sum of the powers of 0.5 to the numbers from 1 to 10 Let x = [ ]. Add 3 to just the odd-index elements (resulting in a 2-vector), adds 3 to them and puts the result in the even positions of the original vector (overwriting the 2 and the 1). Create the matrix whose first column is the numbers 1:10, the next column contains the first 10 squares 1, 4, and the third column contains the first 10 powers of 2: Create the string of letters 'zyx...cba' Intertwine the two strings 'Hello!!' and 'Goodbye' (create the string 'HGeolold...!e') Plot the sin over a different interval Plot a more squiggily sin over the same interval plot the function x^2 between the numbers -5 and 5 plot the function log(x) from 1 to 10 (be sure that your plots are nice and smooth) plot a circle (think parametric plot) plot a Lissajou curve Plot a "heart" r = sin(theta/2)

19 Read more about plot and its optional arguments! Note that using plot with only one input argument does something a little strange (but useful at times...). Evaluate the sum of -1^(n+1) / (2n-1) for n from 1 to N where N=100. Find a value of N so that the sum is close to pi/4 (with difference < 10^-6) Find the sum of a vector x=[ ] without using the old tricks...that is, use afor loop and don't use sum. Write a little program that checks if x=73 is prime. Do not use isprime. But you might find mod or rem useful. Modify your program to find the first 20 primes Write a program that find the error in the calculation from the top of the page for N = 1, 10, 100,...10^6 and plots the results in a meaningful way. (you might want to consider using a log-plot, with loglog, or manually, taking the log from both axes before plotting) Find the value of the 100th Fibonacci number, using a while loop, and no big vectors. Write your prime checking program using a while loop instead of a for loop. Write a while loop that modifies n. if n is even it changes n to n/2, otherwise, it changes it to 3n+1. if n is 1 the loop stops. Try this out for several different starting values. Count how many steps you need to get to 1. Make a function for which the input is x and y and the output is x^y. From the command line, check the following inputs: o o >powerfunc(2,3); powerfunc(3,2); x=2; y=3; powerfunc(x,y); powerfunc(y,x) You can define more than one function inside a function file: func2.m: function y = func2(x) y = helper(x) function r = helper(q) r = q^2;

20 In this way, the functions inside the file are visible to all the other functions in the file. However, only the first can be called from outside the file. For x = [ ]: o Set to 0 the positive elements of x. o Set to 3 the multiples of 3. with 0 without 0 o Multiply by 5 the even elements of x o Create y from the elements of x that are > 10. o Set to 0 the elements less than the average. o Set to the distance from the average the elements greater than the average. Create r and c such that: ( ) ( ) r = ( ) ( ) ( ) c = r' Using r and c, create ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ) ( )