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Introduction to MATLAB I

Introduction to MATLAB I. MATLAB. MATLAB is a program for doing numerical computation. It was originally designed for solving linear algebra type problems using matrices. It’s name is derived from MATrix LABoratory.

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Introduction to MATLAB I

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  1. Introduction to MATLAB I

  2. MATLAB • MATLAB is a program for doing numerical computation. It was originally designed for solving linear algebra type problems using matrices. It’s name is derived from MATrix LABoratory. • MATLAB has since been expanded and now has built-in functions for solving problems requiring data analysis, signal processing, optimization, and several other types of scientific computations. It also contains functions for 2-D and 3-D graphics and animation.

  3. Software Development Philosophy • Major software characteristics: • matrix-based numeric computation • high-level programming language • graphics & visualization • toolboxes provide application-specific functionality • BASIC-like syntax, with elements from C, GUI IDE • basic data type: 2- or 3-dimensional floating-point matrix • most operators and functions work on entire matrices • Open & extensible system architecture • Interfaces to other systems • Custom C, Fortran • Extensive data I/O facility

  4. Availability and Documentation • Installed on: • Windows (or Linux) • Full documentation available online in HTML and PDF: • Start Matlab then type helpdesk • http://www.mathworks.com/access/helpdesk/help/helpdesk.html • Read “Getting Started” section of the MATLAB manual • Use the command help function-name

  5. MATLAB Windows • Command Window • enter commands and data • print results • Graphics Window • display plots and graphs • Edit Window • create and modify m-files

  6. MATLAB • The MATLAB environment is command oriented somewhat like UNIX. A prompt appears on the screen and a MATLAB statement can be entered. When the <ENTER> key is pressed, the statement is executed, and another prompt appears. • If a statement is terminated with a semicolon ( ; ), no results will be displayed. Otherwise results will appear before the next prompt. • The following slide is the text from a MATLAB screen.

  7. MATLAB To get started, type one of these commands: helpwin, helpdesk, or demo » a=5; » b=a/2 b = 2.5000 »

  8. MATLAB Variable Names • Variable names ARE case sensitive • Variable names can contain up to 63 characters (as of MATLAB 6.5 and newer) • Variable names must start with a letter followed by letters, digits, and underscores.

  9. MATLAB Special Variables ans Default variable name for results pi Value of  eps Smallest incremental number inf Infinity NaN Not a number e.g. 0/0 i and j i = j = square root of -1 realmin The smallest usable positive real number realmax The largest usable positive real number

  10. MATLAB Math & Assignment Operators Power ^ or .^ a^b or a.^b Multiplication * or .* a*b or a.*b Division / or ./ a/b or a./b or \ or .\ b\a or b.\a NOTE: 56/8 = 8\56 - (unary) + (unary) Addition + a + b Subtraction - a - b Assignment = a = b (assign b to a)

  11. Other MATLAB symbols >> prompt . . . continue statement on next line , separate statements and data % start comment which ends at end of line ; (1) suppress output (2) used as a row separator in a matrix : specify range

  12. MATLAB Matrices • MATLAB treats all variables as matrices. For our purposes a matrix can be thought of as an array, in fact, that is how it is stored. • Vectors are special forms of matrices and contain only one row OR one column. • Scalars are matrices with only one row AND one column

  13. MATLAB Matrices • A matrix with only one row AND one column is a scalar. A scalar can be created in MATLAB as follows: » a_value=23 a_value = 23

  14. MATLAB Matrices • A matrix with only one row is called a row vector. A row vector can be created in MATLAB as follows (note the commas): » rowvec = [12 , 14 , 63] rowvec = 12 14 63

  15. MATLAB Matrices • A matrix with only one column is called a column vector. A column vector can be created in MATLAB as follows (note the semicolons): » colvec = [13 ; 45 ; -2] colvec = 13 45 -2

  16. MATLAB Matrices • A matrix can be created in MATLAB as follows (note the commas AND semicolons): » matrix = [1 , 2 , 3 ; 4 , 5 ,6 ; 7 , 8 , 9] matrix = 1 2 3 4 5 6 7 8 9

  17. Extracting a Sub-Matrix • A portion of a matrix can be extracted and stored in a smaller matrix by specifying the names of both matrices and the rows and columns to extract. The syntax is: sub_matrix = matrix ( r1 : r2 , c1 : c2 ) ; where r1 and r2 specify the beginning and ending rows and c1 and c2 specify the beginning and ending columns to be extracted to make the new matrix.

  18. A column vector can be extracted from a matrix. As an example we create a matrix below: » matrix=[1,2,3;4,5,6;7,8,9] matrix = 1 2 3 4 5 6 7 8 9 Here we extract column 2 of the matrix and make a column vector: » col_two=matrix( : , 2) col_two = 2 5 8 MATLAB Matrices

  19. A row vector can be extracted from a matrix. As an example we create a matrix below: » matrix=[1,2,3;4,5,6;7,8,9] matrix = 1 2 3 4 5 6 7 8 9 Here we extract row 2 of the matrix and make a row vector. Note that the 2:2 specifies the second row and the 1:3 specifies which columns of the row. » rowvec=matrix(2 : 2 , 1 : 3) rowvec = 4 5 6 MATLAB Matrices

  20. Reading Data from files • MATLAB supports reading an entire file and creating a matrix of the data with one statement. >> load mydata.dat; % loads file into matrix. % The matrix may be a scalar, a vector, or a % matrix with multiple rows and columns. The % matrix will be named mydata. >> size (mydata) % size will return the number % of rows and number of % columns in the matrix >> length (myvector) % length will return the total % no. of elements in myvector

  21. Plotting with MATLAB • MATLAB will plot one vector vs. another. The first one will be treated as the abscissa (or x) vector and the second as the ordinate (or y) vector. The vectors have to be the same length. • MATLAB will also plot a vector vs. its own index. The index will be treated as the abscissa vector. Given a vector “time” and a vector “dist” we could say: >> plot (time, dist) % plotting versus time >> plot (dist) % plotting versus index

  22. Plotting with MATLAB • There are commands in MATLAB to "annotate" a plot to put on axis labels, titles, and legends. For example: >> % To put a label on the axes we would use: >> xlabel ('X-axis label') >> ylabel ('Y-axis label') >> % To put a title on the plot, we would use: >> title ('Title of my plot')

  23. Plotting with MATLAB • Vectors may be extracted from matrices. Normally, we wish to plot one column vs. another. If we have a matrix “mydata” with two columns, we can obtain the columns as a vectors with the assignments as follows: >> first_vector = mydata ( : , 1) ; % First column >> second_vector = mydata ( : , 2) ; % Second one >> % and we can plot the data >> plot ( first_vector , second_vector )

  24. Some Useful MATLAB commands • who List known variables • whos List known variables plus their size • help Ex: >> help sqrt Help on using sqrt • lookfor Ex: >> lookfor sqrt Search for keyword sqrt in m-files • what Ex:>> what a: List MATLAB files in a: • clear Clear all variables from work space • clear x y Clear variables x and y from work space • clc Clear the command window

  25. Some Useful MATLAB commands • what List all m-files in current directory • dir List all files in current directory • ls Same as dir • type test Display test.m in command window • delete test Delete test.m • cd a: Change directory to a: • chdir a: Same as cd • pwd Show current directory • which test Display current directory path to test.m • why In case you ever needed a reason

  26. MATLAB Relational Operators • MATLAB supports six relational operators. Less Than < Less Than or Equal <= Greater Than > Greater Than or Equal >= Equal To == Not Equal To ~=

  27. MATLAB Logical Operators • MATLAB supports three logical operators. not ~ % highest precedence and & % equal precedence with or or | % equal precedence with and

  28. MATLAB Logical Functions • MATLAB also supports some logical functions. xor (exclusive or) Ex: xor (a, b) Where a and b are logical expressions. The xor operator evaluates to true if and only if one expression is true and the other is false. True is returned as 1, false as 0. any (x) returns 1 if any element of x is nonzero all (x) returns 1 if all elements of x are nonzero isnan (x) returns 1 at each NaN in x isinf (x) returns 1 at each infinity in x finite (x) returns 1 at each finite value in x

  29. MATLAB Display formats • MATLAB supports 8 formats for outputting numerical results. format long 16 digits format short e 5 digits plus exponent format long e 16 digits plus exponent format hex hexadecimal format bank two decimal digits format + positive, negative or zero format rat rational number (215/6) format short default display

  30. Matlab Selection Structures • An if - elseif - else structure in MATLAB. Note that elseif is one word. if expression1% is true % execute these commands elseif expression2% is true % execute these commands else % the default % execute these commands end

  31. MATLAB Repetition Structures • A for loop in MATLAB for x = array for x = 1: 0.5 : 10 % execute these commands end • A while loop in MATLAB while expression while x <= 10 % execute these commands end

  32. Scalar - Matrix Addition » a=3; » b=[1, 2, 3;4, 5, 6] b = 1 2 3 4 5 6 » c= b+a % Add a to each element of b c = 4 5 6 7 8 9

  33. Scalar - Matrix Subtraction » a=3; » b=[1, 2, 3;4, 5, 6] b = 1 2 3 4 5 6 » c = b - a %Subtract a from each element of b c = -2 -1 0 1 2 3

  34. Scalar - Matrix Multiplication » a=3; » b=[1, 2, 3; 4, 5, 6] b = 1 2 3 4 5 6 » c = a * b % Multiply each element of b by a c = 3 6 9 12 15 18

  35. Scalar - Matrix Division » a=3; » b=[1, 2, 3; 4, 5, 6] b = 1 2 3 4 5 6 » c = b / a % Divide each element of b by a c = 0.3333 0.6667 1.0000 1.3333 1.6667 2.0000

  36. Calculations at the Command Line Assigning Variables • a = 2; • b = 5; • a^b • ans = • 32 • x = 5/2*pi; • y = sin(x) • y = • 1 • z = asin(y) • z = • 1.5708 Semicolon suppresses screen output Results assigned to “ans” if name not given Use parentheses ( ) for function inputs Numbers stored in double-precision floating point format

  37. In MATLAB, a matrix is a rectangular array of numbers Scalars: 1-by-1 matrices Vectors: matrices with only one row or column Matrix elements can consist either of numbers (numeric arrays) or characters (string arrays) Colon generates number sequence: >> 11:14 ans = 11 12 13 14 Specify step size with second colon: >> 1:3:12 ans = 1 4 7 10 MATLAB matrices (1)

  38. Columns (n) 1 2 3 4 5 A (2,4) 1 2 Rows (m) 3 4 5 A (17) MATLAB matrices (2) A = 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25 4 10 1 6 2 8 1.2 9 4 25 7.2 5 7 1 11 0 0.5 4 5 56 23 83 13 0 10 Rectangular Matrix: Scalar: 1-by-1 array Vector:m-by-1 array 1-by-n array Matrix: m-by-n array Matrix elements can be EITHER numbers OR characters

  39. MATLAB matrices (3) Row separator: semicolon (;) Column separator: space / comma (,) Use square brackets [ ] • a=[1 2;3 4] • a = • 1 2 • 3 4 • b=[-2.8, sqrt(-7), (3+5+6)*3/4] • b = • -2.8000 0 + 2.6458i 10.5000 • b(2,5) = 23 • b = • -2.8000 0 + 2.6458i 10.5000 0 0 • 0 0 0 0 23.0000 Matrices must be rectangular. (Undefined elements set to zero) Any MATLAB expression can be entered as a matrix element

  40. MATLAB matrices (4) Array Subscripting / Indexing 1 2 3 4 5 A = 1 6 11 16 21 2 7 12 17 22 3 8 13 18 23 4 9 14 19 24 5 10 15 20 25 4 10 1 6 2 1 2 3 4 5 8 1.2 9 4 25 A(1:5,5) A(:,5) A(21:25) A(1:end,end) A(:,end) A(21:end)’ 7.2 5 7 1 11 0 0.5 4 5 56 A(3,1) A(3) 23 83 13 0 10 A(4:5,2:3) A([9 14;10 15]) • Use () parentheses to specify index • colon operator (:) specifies range / ALL • [ ] to create matrix of index subscripts • 'end' specifies maximum index value

  41. Matrices & Linear Algebra • help ops • help matfun

  42. Matrix Multiplication • Inner dimensions must be equal • Dimension of resulting matrix = outermost dimensions of multiplied matrices • Resulting elements = dot product of the rows of the 1st matrix with the columns of the 2nd matrix • a = [1 2 3 4; 5 6 7 8]; • b = ones(4,3); • c = a*b • c = • 10 10 10 • 26 26 26 [2x4] [4x3] [2x4]*[4x3] [2x3] a(2nd row).b(3rd column)

  43. Matrix Multiplication X = X11 X12 X21 X22 Y = Y11 Y12 Y21 Y22 Then: X*Y = Xrow1.Ycol1 Xrow1.Ycol2 Xrow2.Ycol1 Xrow2.Ycol2 = (X11*Y11)+(X12*Y21) (X11*Y12)+(X12*Y22) (X21*Y11)+(X22*Y21) (X21*Y12)+(X22*Y22)

  44. Matrix Transpose

  45. Example: Solving Equationsusing “Left Division” • Solve this set of simultaneous equations: • A = [-1 1 2; 3 -1 1;-1 3 4]; • b = [2;6;4]; • x = inv(A)*b%If “A” is a square matrix • x = • 1.0000 • -1.0000 • 2.0000 • x = A\b%underdetermined and overdetermined system • x = • 1.0000 %x1 • -1.0000 %x2 • 2.0000 %x3 -x1 + x2 + 2x3 = 2 3x1 - x2 + x3 = 6 -x1 + 3x2 + 4x3 = 4

  46. Matlab Functions • % Matlab Functions • % functions(a,x,y); • % a = 1 for dividing, a = 2 for multiplication, • % a = 3 for adding, a = 4 for subtraction • % calculations x/y , x*y , x-y , x+y • function [s] = functions_all(a,x,y) • if a==1; • s=x/y; • elseif a==2 • s=x*y; • elseif a==3 • s=x+y; • elseif a==4 • s=x-y; • else • disp('a should be 1-2-3-4 '); • disp(' please try again '); • end

  47. Built-in Functions • Sqrt( ), abs( ), sin( ), cos( ), exp( ), tanh( ), acos( ), log( ), log10( ), etc. • These operators are vectorized

  48. Built-in Functions • Certain functions, such as exponential and square root, have matrix definition also • Use “help expm” and “help sqrtm” for details >> A = [1 3 5; 2 4 6; -3 2 -1] A = 1 3 5 2 4 6 -3 2 -1 >> B = sqrt(A) B = 1.0000 1.7321 2.2361 1.4142 2.0000 2.4495 0 + 1.7321i 1.4142 0 + 1.0000i >> C = sqrtm(A) C = 2.1045 + 0.0000i 0.1536 - 0.0000i 1.8023 + 0.0000i 1.7141 - 0.0000i 1.1473 + 0.0000i 1.7446 + 0.0000i -2.0484 + 0.0000i 1.3874 + 0.0000i 0.5210 - 0.0000i

  49. MATLAB Graphics • One of the best things about MATLAB is interactive graphics • “plot” is the one you will be using most often • Many other 3D plotting functions -- plot3, mesh, surfc, etc. • Use “help plot” for plotting options • To get a new figure, use “figure” • logarithmic plots available using semilogx, semilogy and loglog

  50. Plotting Commands • plot(x,y) defaults to a blue line • plot(x,y,’ro’) uses red circles • plot(x,y,’m*’) uses magenta asterisks • If you want to put two plots on the same graph, use “hold on” • plot(a,b,’r:’) (red dotted line) • hold on • plot(a,c,’ko’) (black circles)

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