MATLAB Tutorial The Basics

Transcription

1 MATLAB Tutorial The Basics Chienmin Chuang School of Mathematics, University of Birmingham May 24, 2011

2 WHAT IS MATLAB? MATLAB, coined from MATrix LABoratory, is a mathematical computing software developed by MathWorks. Simply speaking, it can be thought as a calculator which is powerful in matrix operations. To a further extent, it is a powerful programming language equipped with various built-in functions and (optional) tool boxes and can be used to build a customized program.

3 HOW TO START MATLAB? Simply double click the MATLAB icon on your desktop or execute the program from the start menu. You will see the Current Directory on the top row of the interface where you create, save and connect all your programs and files. In addition, there are three main windows: Workspace: All the variables currently used are displayed here. Command: Used for executing all commands, built-in functions and your own programs. History: Show what you have done on MATLAB.

4 HOW TO START MATLAB? Your best friend when using MatLab help function. Simply type Help, you can get instruction of how to use MATLAB. To know how to use a certain built-in function, simply type help <function-name>. Another choice: Help>Using the Command Window.

5 NUMERICAL VARIABLE Type any number in the command window and then press Enter. MATLAB always echoes back with ans=the typed number. MATLAB generates the ans variable automatically when a user specifies no output argument. A user can define a numerical variable using the syntax: variable-name = numerical value. An semicolon (;) can be put in the end for no echo.

6 NUMERICAL VARIABLE Some preset constants are i and j both refer to 1. pi is the ratio of circle s circumference to its diameter. Nan refers to Not a Number and has the form 0/0. Inf refers to Infinity and has the form 1/0. Technical points: 1. Preset names can be reloaded with new values or data. 2. Inf-Inf and Inf/Inf both produce NaN. 3. The exponential base is not preset as e. To define the number, we can use e=exp(1).

7 DISPLAY FORMATS Numerical values or variables can be displayed in different formats. Below are widely used types: format short/long displays values with 4/14-15 digits after the decimal point. format short/long eng displays values in the engineering format with 4/16 digits after the point. format rat displays values in terms of an rational number which may be exact or approximate to the original one. Short is he default format type. Simply type help format for more details.

8 ARITHMETIC OPERATIONS Five basic operations: + (add), -(subtract), * (multiply), (divide) and ˆ(exponentiate or raise to a power). Tips: 1. Priority: ˆ, / +,. 2. Use space and bracket () for clear expression. 3. A comma, is used to separate two expressions. 4. If a semicolon ; ends an equation, the result will not be shown on the screen. It is also used to end a line of command and separate it from the next one. 5. A variable can be defined by simply assigning a value to the variable name. To clear a variable x from memory, simply type clear x. clear all clears all variables. Note that MATLAB are case-sensitive to variable names. 6. To check what variables are currently used, type whos.

9 ARITHMETIC OPERATIONS The command clc is used to to clear command window.

10 ELEMENTARY FUNCTIONS There are a variety of elementary mathematical functions built in MATLAB. Type help elfun in the command window for all details. We introduce a few here. For any variable (or value) x: log(x): Natural logarithm of x. exp(x): Exponential of x or e = to power x. sin(x), cos(x), tan(x), cot(x), sec(x), csc(x): Trigonometric functions of x. abs(x): Modulus or absolute value of x. sqrt(x): Square root of X. conj(x): Complex conjugate of x. imag(x): Complex imaginary part of x. real(x): Complex real part of x.

11 DEFINE A VECTOR Below are common ways to define a horizontal row vector. a:incerment:b Defines a row vector with starting entry a and the last entry (less than or) equal to b in the step of increment. In particular, a:b is the shorthand of a:1:b. linspace(a,b,n) Defines a row vector with starting entry a and the last entry equal to b with n equidistant entries in total.

12 DEFINE A MATRIX When defining a matrix or vector,, separate entries in the same row which can also be replaced by space. ; separate different rows. [ ] is used to cover all its entry values. Besides, [ ] can also combine two matrices to a bigger one. We can also define an empty matrix by C=[]. For example,

13 DEFINE A MATRIX Several functions used to define specific matrices: zeros(m,n) defines an m n zero matrix. ones(m,n) defines an m n matrix with entries equal to one. eye(m,n) defines an m n matrix with main-diagonal entries equal to one and others equal to zero. In particular, eye(m,m) can be shortened to eye(m) and defines a m m identity matrix. rand(m,n) generates an m n random matrix with entries drawn from a uniform distribution over [0, 1]. randn(m,n) generates an m n random matrix with entries drawn from a normal distribution with zero mean and unit variance.

14 DEFINE A MATRIX Provided values x 0, x 1,..., x n, a Vander matrix is a matrix of the form below. x n 0 x 2 0 x 0 1 x n 1 x 2 1 x 1 1 x n 2 x 2 2 x x n n x 2 n x n 1 Such a matrix is useful when doing nth-order interpolation and can be generated by MatLab using vander([x 0 ; x 1 ;... ; x n ]).

15 MATRIX MANIPULATION If two matrices A and B are compatible in dimensions, we can use the following matrix operations. Basics: A + B A B A B Inverse for a square matrix C: Power m: C ˆ m In particular, C ˆ ( 1) = inv(c). Transpose: A inv(c) Entry-by-entry operation:../. ˆ For two compatible matrices A and B: A*inv(B)=A/B inv(b)*a=b A To solve a system of equations Ax=b where A is a square matrix and b is a vector, we can use x=inv(a)*b or x=a b. In general, is faster than using inv.

16 MATRIX MANIPULATION Define A=[1 2; 3 4; 5 6], B=[7 8 9; ] and then compute the examples below:

17 INDEXING ENTRIES Suppose M is a n n matrix, then: M(i,j) is the (i,j) entry of M for any i,j<=n. M(i,:) is the ith. row of M for any i<=n. M(:,j) is the jth. column of M for any j<=n. M([r 1 r 2...r m ], [c 1 c 2...c w ]) is a matrix with the (i,j) entry equal to M(r i, c j ); Try the examples:

18 INDEXING ENTRIES For every subscript (i,j), there is a unique single index s corresponding to it. Note that for a row or column vector, we can simply use such a single index to specify an entry.

19 INDEXING ENTRIES To convert between these two indexing methods, we can use sub2ind(sz,i,j): returns a single index using the subscripts (i,j) for a matrix of size sz. ind2sub(sz,s): returns the subscripts (i,j) using a single index s for a matrix of size sz. A matrix M can be easily vectorized in the single order by reloading M=M(:).

20 SET OPERATIONS OF TWO VECTORS Think of two vectors V1 and V2 as two sets. We can do set operations as below. union(v1,v2) generates their union in terms of a row vector. intersect(v1,v2) generates their intersection in terms of a row vector. setdiff(v1,v2) generates their difference in terms of a row vector.

21 FUNCTIONS ASSOCIATED WITH MATRIX Suppose M is a matrix. size(m) returns the size of M. length(m) amounts to max(size(m)). numel(m) returns the number of entries in M. sign(m) returns the signs of all entries in M. Most elementary functions can be applied to a Matrix in terms of entries.

22 FUNCTIONS ASSOCIATED WITH MATRIX Suppose M is a matrix. We can permute its entries in the following ways. sort(m) sort entries in ascending column by column. This is equivalent using sort(m,1). To sort M row by row using sort(m,2). To sort M in descending order, put descend as the last argument. fliplr(m) flips columns of M in the left-right direction. flipud(m) flips columns of M in the up-down direction.

23 FUNCTIONS ASSOCIATED WITH MATRIX diag is a multipurpose function. If v is a n 1 vector, diag(v) defines a diagonal matrix with the main-diagonal entries equal to those of vector v. Moreover, diag(v,k) defines a (n + k ) (n + k ) matrix with the kth-diagonal entries equal to those of vector v. If M is a n m matrix, diag(m,k) returns the kth diagonal of M in terms of a column vector. In particular, diag(m) defines the column vector equal to the main diagonal of M.

24 FUNCTIONS ASSOCIATED WITH MATRIX spdiags(ds,d,m,n) returns an m-by-n sparse matrix by taking the columns of matrix Ds and placing them along the diagonals specified by d. (Beware the entries assigned to non-main-diagonals!) This is particularly convenient for defining tri-diagonal and banded matrices. For more details, refer to ite help file. Note that MATLAB only shows the nonzeros of a sparse matrix. full(m) converts a sparse matrix M to a full one. In contrast, sparse(m) turns the matrix M into a sparse one when existing numerous zero entries. Example:

25 STRING VARIABLE String is a combination of characters, including space. Each entry of a string is a character. Both can be defined using single quote. (Not double quote!) Similar to a matrix variable, length(), numel() and size() can apply to a string variable. A numerical variable can be transformed to a string using num2str(). str2num() can do the reverse. However, if the input string does not represent a valid number or matrix, str2num() returns the empty matrix in x

26 FUNCTIONS ASSOCIATED WITH STRING Strings can be combined by using square brackets []. disp(a) displays the content of variables where a can be a number or a string. whos shows information of all variables while whos( a ) shows information of the variable a. This command can be shortened to who with less information provided. strcmp(s1,s2) compares two string variables s1 and s2, returns 1 if they are identical and 0 otherwise. strncmp(s1,s2,n) compares the first n characters of two string variables s1 and s2 and returns 1 or 0. findstr(s1,s2) searches any occurrences of the shorter string, which returns the starting index of each occurrence if any and empty array [] otherwise. strrep(s,s1,s2) searches the shorter string s1 in a longer string s and replaces s1 by another string s2.

27 FUNCTIONS ASSOCIATED WITH STRING

28 RELATIONAL OPERATORS The relational operators are <, >, <=, >=, ==, and =, which return logical value 1 (true) if the specified relation holds and 0 (false) otherwise. The operators are performed and return values on the elemental basis. Note that = (assign or define) is different from == (equal to). true and false are preset logical variables with values equal to 1 and 0 respectively.

29 REDUCE MULTIPLE RELATIONS TO ONE Suppose N is a numerical matrix and M is a numerical or logical Matrix. or(expr1, expr2) and expr1 expr2 returns 1 if at least one of expr1 and expr2 is equal to logical 1; otherwise, it returns logical 0. and(expr1, expr2) and expr1 & expr2 returns 1 if both expr1 and expr2 are equal to logical 1; otherwise, it returns logical 0. logical(n) transforms N to a logical matrices with 0 corresponding to zero entry of N and 1 to nonzeros. any(m) returns logical value 1 for the column(s) with at least one nonzero entry and logical value 0 otherwise. all(m) returns logical value 1 for the column(s) with all nonzero entries and logical value 0 otherwise.

30 REDUCE MULTIPLE RELATIONS TO ONE

31 CONDITIONER IF if condition The syntax if condition, operations; end or operations; end states: the operations will be performed only if the condition is true. if condition1 operation1; elseif condition2 A more general form is operation2; else operation3; end which states: if the condition1 is true, the operation1 will be performed; else if condition2 is true, the operation2 will be performed; otherwise, the operation3 will be performed. Note that elseif and else can be omitted if not required. Besides, elseif can be used more than once while else can be used at most once.

32 CONDITIONER IF

33 CONDITIONER IF Tips: In general, a nonzero value can be interpreted as a true logical value when used as a condition. transforms the values of true and false to each other. Example:

34 CONDITIONER SWITCH switch scenario-variable case scenario1 operation1;. case scenarion operationn; otherwise operation-for-others; end The syntax states: operationk will be executed if the scenario-variable is equal to the scenariok and operation-for-others will be executed if no matched scenario is available. Note that only one possibility may happen and only its corresponding operation will be performed in the syntax. Note that, we can use curly brackets {} to combines several scenarios as one.

35 CONDITIONER SWITCH

36 LOOP FOR for IndexVariable = a:increment:b operations; end The syntax states the operations will be performed iteratively. The iteration starts from IndexVariable equal to a, go to the next iteration by adding Increment to IndexVariable and stops until the IndexVariable equal to (or more than) b. If the Increment is equal to 1, then the :Increment can be omitted. Note that in the syntax, the value of IndexVariable is assigned as a single scalar, which is different from IndexVariable=a:Increment:b used out of the syntax.

37 LOOP FOR

38 LOOP WHILE while condition operations; end The syntax states: the operations will be repeatedly carried out until the condition does not hold. Note that if the procedure falls into infinite, use Ctrl+C to break the endless iteration. Example:

39 AUXILIARIES FOR LOOPS Commands below can facilitate the usage of for-loop and while-loop which generally follow some specific conditions. break stops the current iteration of a loop and passes control to the outer scope of the loop. continue stops the current iteration of a loop and passes control to the next iteration of the same loop. return hands over the control to its invoking function or the keyboard. Note both break and continue skip the remaining operations of the current iteration after they are active.

40 AUXILIARIES FOR LOOPS

41 2-D PLOT plot(data1, data2) creates a line graph with data1 on x-axis and data 2 on y-axis. Note that the input data are discrete, but plot( ) draws a line graph by connecting all data points. To show all points without connection, we can use plot(data1, data2,. ). To specify more prpperties such as the line width, point shape, color..., etc, use the help file. Similarly, scatter(data1, data2) can draw a scatter plot. bar(data) generates a bar chart for the vector data.

42 2-D PLOT

43 3-D PLOT The three-dimensional versions of previous plotting functions are plot3(data1, data2, data3), bar3(data) and scatter3(data1, data2, data3). Besides, mesh(x,y,z) draws a wireframe mesh using Z for the color data and height. X and Y are vectors or matrices defining the x and y components of a surface. Another choice is to use surf(x,y,z) to create a shaded surface. When plotting in three dimension, [X,Y] = meshgrid(x,y) is useful, which transforms the domain specified by vectors x and y into arrays X and Y

44 3-D PLOT

45 BASIC CONTROL OF GRAPHS grid on and grid off adds and removes the grid of the graph respectively. title( PlotTitle ) set the plot title as Plottitle. xlabel( xname ), ylabel( yname ) and zlabel( zname ) set the names of x,y and z coordinates as xname, yname and zname individually. legend( string1, string2,...) creates a legend using the specified strings to describe each set of data. hold on retains following pictures in the current graph while hold off stops such combining graphs.

46 BASIC CONTROL OF GRAPHS