Chapters covered for the final

Transcription

1 Chapters covered for the final Chapter 1 (About MATLAB) Chapter 2 (MATLAB environment) Chapter 3 (Built in Functions):3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.8, 3.9 Chapter 4 (Manipulating Arrays):4.1,4.2, 4.3 (zeros, ones, diag, magic) Chapter 5 (Plotting): 5.1 (except plots of complex arrays), 5.2, 5.3(Bar graphs and histograms) Chapter 6 (User defined functions): 6.1, 6.2 Chapter 7 (User controlled I/O): 7.1, 7.2, fopen, fgetl, fprintf, fclose Chapter 8 (Logical Functions and control structures): 8.1, 8.2, 8.3, 8.4, 8.5, Chapter 9 (Matrix Algebra): 9.1 (excluding 9.1.5), 9.3 Chapter 10 (Other kind of arrays): 10.1 (excluding ), 10.2, 10.3, 10.4, 10.5

2 MATLAB as a Calculator >> 2+3/4*5 ans= Order of operations First perform calculations inside parenthesis Next, perform exponentiation operation Then perform multiplication and division operations from left to right Finally perform addition and subtraction operations from left to right

3 Numbers and Formats (Chapters 3 and 10.1) MATLAB recognizes several different kinds of numbers. The following two kinds are the most important. Integer format short 4 decimal digits uses 4 bytes (single) format long 14 decimal digits uses 8 bytes (double) Floating point format short e 4 decimal digits uses 4 bytes (single) e+000 format long e 14 decimal digits uses 8 bytes (double) e+000 All computations in MATLAB are done in double precision. The precision can be controlled by the format command. The precision can be changed by using format. >> format short changes the precision of integers to short, and therefore uses 4bytes.

4 variables Variables (Chapters 3 and 10) >> x=1.34; % Legal names consisting of letters and digits starting with a % letter. Note: net cost is not legal. Cannot use built in % function names. Note: You need to decide for every assignment statement (statement with left side = right side) whether the automatic output should be suppressed. If you decide to suppress the automatic output, end the assignment statement with ; Vectors v= [1 3 5]; %row vector: one row and 3 columns v= [1;2;3]; %column vector: 3 rows and one column v % transpose of vector v

5 Variables The Colon notation x=1:4; % will create a row vector x=[ ] x= 1.5:2.5 ; % x=[ 2.5, 1.5,.5, 1.5, 2.5] increment is one unless % specified. x= 1.5:0.5:1.0; % x=[ 1.5, 1.0, 0.5, 0.0, 0.5, 1.0] x=4: 1:1; % x=[4, 3, 2, 1] Matrices (Arrays) (Chapter 4.1) Constructing a simple matrix >> A=[ ; ; 4,11, 17]; % 3 by 3 matrices Specialized matrices: zeros, ones, diag, magic Creating a random matrix >> A=rand(3,5); % Creates a random matrix of size 3 by 5.

6 Variables (Chapters 4.2 and 9.1) Arithmetic Operators + * / \ ^ ' Matrix and array arithmetic Syntax A+B; A B; % Element by element addition A*B; A/B ; % A/B=A*inv(B); Matrix multiplication A.*B; A./B % element by element product or division A^B % A*A*A* *A: A is multiplied B times A.^B % (i,j) element of the product=a(i,j)^b(i,j) A % transpose of the matrix A.

7 For the given two given vectors, the results of various matrix and array operations on them.

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9 Variables Concatenating matrices >> A=ones(2,5); B=rand(3,5); >> C=[A; B] % vertically concatenates A and B; C is a 5 by 5 matrix. % A and B must have the same number of columns >> E=zeros(3,2); F=diag(3); G=[A B] % Horizontally concatenates E and F; % E and F must have the same # of rows.

10 Extracting a sub-matrix

11 Colon Operator

12 One can delete rows and columns from a matrix using just a pair of square brackets. Start with >> A=[ ; ; ; ]; >>X = A; >>X(:,2) = [] % second column of X is deleted. X = Deleting rows and columns

13 MATLAB Special Variables

14 Logical Variables The logical data types represents a logical true or false using a number 1 or 0, respectively. MATLAB functions (such as feof) return logical true of false to indicate whether a certain condition is true or not. Logical functions doesn t have to be only scalar. >> x = magic(4) >= 9 x =

15 Using logicals in array indexing Consider the following example: >> A = [1 2 3; 4 5 6; 7 8 9] A = >> B = logical([0 1 0; 1 0 1; 0 0 1]); % logical is a built in function to convert % a 0 1 array into a logical array. B = >> A(B) ans =

16 Using logicals in array indexing Consider the following example: >> A = [1 2 3; 4 5 6; 7 8 9] A = >> B = logical([0 1 0; 1 0 1; 0 0 1]); A 3x3 72 double % logical is a built in function to convert % a 0 1 array into a logical array. B = >> A(B) ans = >> whos A Name Size Bytes Class >> whos B Name Size Bytes Class B 3x3 9 logical

17 Using logicals in array indexing Consider the following example: >> A = [1 2 3; 4 5 6; 7 8 9] A = >> B = logical([0 1 0; 1 0 1; 0 0 1]); B = >> A(B) ans = >> A(B)=0; >> A A=

18 Character and String variables (section 10.3) Creating a character string >> str = 'Hello ; >> whos str Name Size Bytes Class str 1x5 10 char Creating an array strings >> name = ['Harold A. Jorgensen ';... %The column vector name has 'Assistant Project Manager';... % 3 elements; elements are 'SFTware Corp. ']; % separated by columns; blanks % padded make the length the same. name, defined above, is a two dimensional array. >> name(2,4:7) ans= ista

19 Character and String variables Creating a character string >> str = 'Hello ; >> whos str Name Size Bytes Class str 1x5 10 char Creating an array strings >> whos name Name Size Bytes Class name 3x char >> name = ['Harold A. Jorgensen ';... %The column vector name has 'Assistant Project Manager';... % 3 elements; elements are 'SFTware Corp. ']; % separated by columns; blanks % padded make the length the same. name, defined above, is a two dimensional array. >> name(2,4:7) ans= ista

20 Problem Suppose we have the following vectors each of length n LastName : an n by 10 array of characters FirstName : an n by 10 array of characters Age : an n by 1 column vector (2digits) Grade : an n by 2 array of numbers (7digits number) Prepare for each individual a line of characters containing LastName, FirstName, Age and Grade >> for i= 1:n >> s(1:55)= ; >> s(1:10)=lastname(i); >> s(16:25)=firstname(i); >> s(31:32)=num2str(age(i)); >> s(36:42) = num2str(grade(i,1)); >> s(46:52) = num2str(grade(i,2)); >> line(i,:) = s; >> end

21 Creating Character Arrays by Concatenation You can join two or more character arrays together to create a new character array. This is called concatenation. Example (horizontal concatenation) name = 'Thomas R. Lee'; title = 'Sr. Developer'; company = 'SFTware Corp. ; s = [name, ', ', title, ', ', company]; Vertical concatenation s = char(name; title; company);

22 Cell arrays of strings (section 10.4) Creating strings in a regular MATLAB character array requires that all strings in the array be of the same length. This often means that you have to pad blanks at the end of strings to equalize their length. Cell arrays of string of the following string arrays >> name = ['Harold A. Jorgensen ';... %name has 3 elements 'Assistant Project Manager';... %blanks are padded. 'SFTware Corp. ']; >> cellname=cellstr(name); % trailing blanks are stripped off >> length(cellname{3}); ans= 13

23 Cell arrays Cell arrays of string of the following string arrays >> name = ['Harold A. Jorgensen ';... %name has 3 elements 'Assistant Project Manager';... %blanks are padded. 'SFTware Corp. ']; >> cellname=cellstr(name); % trailing blanks are stripped off >> length(cellname{3}); ans= 13 Each element of cellname is an array. >> cellname{2}(11:17) ans= Project Extend the array by adding an element of different data types: >> cellname{4}=[ ]; % a row vector is the 4 the element >> cellname{4}(2:4) % accessing the 2 nd through 4 th element ans= % of the row vector cellname{4}

24 Structures (Section 10.5) Structures are MATLAB arrays with named "data containers" called fields. The fields of a structure can contain any kind of data. Building Structure Arrays Using Assignment Statements >> patient.name = 'John Doe'; >> patient.billing = ; >> patient.test = [ ; ; ];

25 Structures Building Structure Arrays Using Assignment Statements >> patient.name = 'John Doe'; >> patient.billing = ; >> patient.test = [ ; ; ]; Now entering >> patient at the command line results in Patient= name:'john Doe' billing: 127 test: [3x3 double] To expand the structure array, add subscripts after the structure name: >> patient(2).name = 'Ann Lane'; >> patient(2).billing = 28.50; >> patient(2).test = [ ; ; ];

26 Structures Now entering >> patient at the command line results in 1x2 struct array with fields: name billing test

27 Structures Accessing Data in Structure Arrays Accessing patient records >> patient(1) ans = name: 'John Doe billing: 127 test: [3x3 double] To access a particular field >> str= patient(2).name str = Ann Lane

28 Input and Output (Sections 7.1, 7.2) User defined input >> z=input( Enter a value ) Samples of input values: numeric 12, , 12.09e+01 arrays [1, 2, 3; 4, 5, 6] string hello, it s % takes input from the command window If you are entering a string (a character arrays in MATLAB) without quotes around the string, the input statement should look like >> x=input( Enter your string, s )

29 Syntax >> tline = fgetl(fid) Input from Files (fgetl) tline = fgetl(fid) returns the next line of the file associated with the file identifier fid. If fgetl encounters the end of file indicator, it returns 1. fgetl is intended for use with files that contain newline characters. Examples The example reads every line of the input file fid=fopen( inputfile.dat, r ); %file inputfile.dat is opened for reading while ~feof(fid) tline = fgetl(fid); %reads the contents of the next line as a string disp(tline) %displays the line end fclose(fid);

30 Output Output Options disp(x) % displays the variable x. Note that there is no semicolon(;) disp( Displays the message )

31 Syntax Formatted Output fprintf (Section 7.2) >> count = fprintf( fid, format, A,...) fprintf(fid, format, A,...) formats the data in A under control of the specified format string, and writes it to the file associated with file identifier fid. fprintf returns a count of the number of bytes written if semicolon is missing.

32 fprintf (examples) >> students = 25; >> fprintf('there are %f students in CMPT 102\n', students); There are students in CMPT 102 (if fid was provided, the output would have been to the file. The structure of the line remains the same) >> fprintf('there are %5.2f students in CMPT 102 \n', students); There are students in CMPT 102

33 fprintf (examples) Example Create a text file called exp.txt containing a short table of the exponential function. x = 0:.1:1; y = [x; exp(x)]; %y has two rows fid = fopen('exp.txt', w'); fprintf(fid, '%6.2f %12.8f\n', y); fclose(fid) Now examine the contents of exp.txt: exp.txt=

34 fprintf (examples) Example The commands B = [ ; ] fprintf('x is %6.2f meters or %8.3f mm\n', 9.9, 9900, B) display the lines X is 9.90 meters or mm X is 8.80 meters or mm X is 7.70 meters or mm

35 fprintf (examples) Example To insert a single quotation mark in a string, use two single quotation marks together. For example: fprintf('it''s Friday.\n') displays on the screen: It's Friday.

36 Format String Format String The format argument is a string containing ordinary characters. A conversion specification controls the output format. Conversion specifications begin with the % character and contain these optional and required elements: Width and precision fields (optional) A subtype specifier (optional) Conversion character (required)

37 format string You specify these elements in the following order: Start of conversion %12.5e conversion character specificaton Field width Precision

38 Type field format Type field Result %f fixed point, or decimal point %e exponential notation %g whichever is shorter, %f or %e %d integer %s string of characters

39 Syntax Logical Operators: Elementwise & ~ (Chapter 8) expr1 & expr2 %expr1 and expr2 are any logical expressions expr1 expr2 % an expression is logical if its value is either true or false ~expr The symbols &,, and ~ are the logical array operators AND, OR, and NOT. These operators are commonly used in conditional statements, such as if and while, to determine whether or not to execute a particular block of code.

40 Logical Operators: Elementwise & ~ (Chapter 8) The examples shown in the following table use vector inputs A and B, where >>A = [ ]; B = [ ];

41 Find built in functions Syntax >> ind = find(x) >> [row,col] = find(x) ind = find(x) locates all nonzero elements of array X, and returns the linear indices of those elements in vector ind. [row,col] = find(x) returns the row and column indices of the nonzero entries in the matrix X. >> ind=find(x>20) locates all elements of array X greater than 20, and returns the linear indices of those elements in vector ind. [row,col] = find(x>20) returns the row and column indices of the greater than 20 entries in the matrix X.

42 Example 1 X = [ ]; Find built in functions (Examples) indices = find(x) %returns linear indices for the nonzero entries of X. indices = Example 2 You can use a logical expression to define X. For example, >> find(x > 2) % returns linear indices corresponding to the entries of X that are greater than 2. ans = 3 8 9

43 Example 3 The following find command >> X = [3 2 0; 5 0 7; 0 0 1]; >> [r,c,v] = find(x) Find built in functions (Examples) returns a column vector of row indices of the nonzero entries of X r = a column vector of column indices of the nonzero entries of X c = and a column vector containing the nonzero entries of X. v = Other examples form the text.

44 Basic plotting (Chapter 5.1) Consider two vectors: >>x=[0:2:18]; %x=[ ] >>y=[ ]; >>plot(x,y); %A graph of x versus y is created in %the figure window. >>title( First attempt ); >>xlabel( independent variable ); >>ylabel( dependent variable );

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46 More features of plot >>plot(x,y, m )% connected line is magenta colored. >>xlim([0 25]) % x-axis range >>ylim([0 25]) % y-axis range. the graph now looks like

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48 Multiple plot It is possible to overlay more than one graph in one plot >> hold on %Overlay the following graphs >> plot(x1,y1) >> plot(x2,y2) >>plot(x3,y3) >>hold off %Print now

49 Specifying the Color and Size of Markers You can also specify other line characteristics using graphics properties (refer the text for a description of these properties): *LineWidth Specifies the width (in points) of the line. *EdgeColor Specifies the color of the marker. * FaceColor Specifies the color of the face of filled markers. * Size Specifies the size of the marker in units of points. For example, these statements, x = pi:pi/10:pi; y = tan(sin(x)) sin(tan(x)); plot(x,y,' rs','linewidth',2,... EdgeColor','k',... FaceColor','g',... Size',10)

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51 Scripts and functions (Chapter 6.1) Files that contain code in the MATLAB language are called Mfiles. You create M files using a text editor, then use them as you would any other MATLAB function or command. There are two kinds of M files: Scripts, which do not accept input arguments or return output arguments. They operate on data in the workspace. Functions, which can accept input arguments and return output arguments. Internal variables are local to the function. To view the contents of an M file, for example, myfunction.m, use >> type myfunction

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54 M files: Functions

55 Control of flow Structures The FOR construct The FOR construct is a control construct used to repeat a set of calculations or operations a definite (or pre determined) number of times. The syntax of the for loop is for expression statements end

56 MATLAB Iteration Structures (I)

57 Vectorized Operations (for efficient code)

58 Control of flow Structures The WHILE construct The WHILE construct is related to the FOR construct in that it creates a loop that repeats a set of statments. The primary difference is in the number of times the loop statements are executed. A for loop repeats a predetermined number of times (set by the nature of the loop expression). A while loop repeats an indefinite (and potentially infinite) number of times. The syntax of the while loop is while logical_expression statements end

59 MATLAB Iteration Structures (II)

60 Control of flow Structures The IF/ELSE construct An IF construct is a decision construct as opposed to a repeat construct. The if construct allows you to conditionally (or optionally) execute blocks of code. The simplest form of the if construct has the following syntax if logical_expression block #1 statements else default statements end

61 Control of flow Structures The IF/ELSE construct The general form of the if construct if logical_expression #1 block #1 statements elseif logical_expression #2 block #2 statements : elseif logical_expression #k block #k statements : elseif logical_expression #n block #n statements : else default statements end

62 Control of flow Structures

63 Control of flow Structures The SWITCH construct The syntax is: switch switch_expression case case_expression_list #1 block #1 case case_expression_list #2 block #2 : case case_expression_list #k block #k : case case_expression_list #n block #n otherwise default block end

64 Programming Question1 Compute the value of pi using the following series How many terms are needed to obtain an accuracy where two consecutive terms differ by 1e 12? What is the value of the computed Pi?

65 function [value n]=computepi(delta) Solution to Question 1 %Compute the value of Pi using the equation given in question 1 % till two successive values during the iteration differ by less than delta. format long e previousvalue=realmax; current=0; finish=false; n=1; while ~finish current= current+1/((2*n 1)^2*(2*n+1)^2); if abs(current previousvalue) < delta value=sqrt(current*16+8); finish=true; end previousvalue=current; n=n+1; end >> [a,b]=computepi(1e-12) a = e+000 b = 502

66 Programming Question 2 The Fibonacci numbers are computed according to the following relation: F n = F n 1 + F n 2 with F 0 = F 1 = 1. a. Compute the first 100 Fibonacci numbers. b. For the first 50 Fibonacci numbers, compute the ratio F n / F n 1 It is claimed that this ratio approaches the value of the golden mean ( (1 + sqrt(5))/2 ). What do your results show?

67 Solution to Question 2 function [firstn firstnratio goldenratio]=fibonacci(n) % Solving Question 2 % firstn: vector of length n: firstn(i) contains the ith value % firstnratio: vector of length n:firstnratio(i) contains F(i)/F(i 1). firstn(1)=1; firstn(2)=2; firstnratio(1)=1; firstratio(2)=1; for i=3:n firstn(i)=firstn(i 1)+firstN(i 2); firstnratio(i)=firstn(i)/firstn(i 1); end goldenratio=(1+sqrt(5))/2;

68 Programming Question 3 The present value of an annuity (a yearly sum of money) may be computed from the formula P = (A/i) [(1 + i)n 1]/(1 + i)n where A is the annuity (in $/year), i is the nominal yearly interest rate (in decimal form), n is the number of years over which the annuity is paid and P is the present value ($). (For example computation: If i = 0.15 (15%), A = $100/year and n = 10 years then P = $ ) If you won the $1,000,000 State Lottery and the Lottery offered you the choice of $500,000 today or $50,000/year for 20 years, which would you take? You can assume an inflation (interest) rate of 5%.

69 Programming Question 4 The following algorithm for computing pi is found on a web page*: 1. Set a = 1; b = 1/sqrt(2); t = 1/4; x = 1; 2. Repeat the following commands until the difference between a and b is within some desired accuracy: y = a; a = (a + b)/2; b = sqrt(b*y); t = t x*(y a)^2; x = 2*x; 3. From the resulting values of a, b and t, an estimate of pi is Pi_est = ((a + b)^2)/(4*t); How many repeats are needed to estimate pi to an accuracy of 1e 8? 1e 12? Compare the performance of this algorithm to the result from Exercise 1, above. *

70 Programming Question 5 Create a function that generates random arrays of integers beween a and b, inclusive. Use the definition line function A = randint(a,b,m,n) where a and b define the (inclusive) range and M and N define the size of the output array (rows and columns, respectively). a. Test your function with the following piece of code: x = randint(10,17,100000,1); hist(x,10:17) % The resulting histogram should be nearly flat from 10 to 17. % Pay particular attention to the endpoints of the distribution. b. Test your function with the following pieces of code: x = randint( 30,5,100000,1); hist(x, 30:5) x = randint( 45, 35,100000,1); hist(x, 45: 35) x = randint(7, 2,100000,1); hist(x, 2:7)

71 Example 6

72 Read the file students.dat and store the lines in a cell array. function C=file2cell(fName) % reads the lines from file fname fid=fopen(fname, r ); % First two lines are read and discarded fgetl(fid); % line 1 fgetl(fid); % line 2 linenumber=1; % First data line while ~feof(fid) s=fgetl(fid); C{lineNumber}=s; linenumber=linenumber+1; %next line index end fclose(fid);

73 Extract the information on each line and store them in arrays. (Structure data types are used for elegance) function StudentData=extract(C) % Extracting information in cell array C. for i=1:length(c) StudentData(i).Firstname=C{i}(1:10); % Firstname is extracted StudentData(i).Lastname=C{i}(15:24); % Lastname is extracted StudentData(i).Age=str2num(C{i}(30:31)); %Age is extracted StudentData(i).Sex=str2num(C{i}(37)); % Sex field StudentData(i).Grade(1)=str2num(C{i}(40:43)); %Grades are picked StudentData(i).Grade(2)=str2num(C{i}(50:53)); StudentData(i).Grade(3)=str2num(C{i}(60:63)); StudentData(i).Grade(4)=str2num(C{i}(70:73)); StudentData(i).Grade(5)=str2num(C{i}(80:end)); end

74 Compute statistics, such as average age, average grade in each semester, overall average. function [Average, Overall]= statistics(studentdata); % Average is a structure data containing Age average and grade average Average.Age=0; %initialization for i=1:5 Average.Grade(i)=0; end Overall=0; N=length(StudentData); for i=1:n % all N records are visited Average.Age=Average.Age + StudentData(i).Age; % Age is summed for j=1:5 Average.Grade(j)=Average.Grade(j)+StudentData(i).Grade(j); Overall=Overall+ StudentData(i).Grade(j); end end Average.Age=Average.Age/N; for j=1:5 Average.Grade(j)=Average.Grade(j)/N; end Overall=Overall/N;

75 Remove every 8 th record from the student data. function [NewStudentData NewC]=Delete(StudentData,C) % Removing every 8 th record. Three new records are added. N=length(StudentData); newcount=1; % line count of the reduced data for i=1:n % line count of the current data if rem(i,8) ~= 0 % if it is not the 8 th record, copy. NewStudentData(newCount)=StudentData(i); NewC{newCount}=C{i}; newcount=newcount+1; end end

76 Add at least three new student records (you generate the data by hand). function [EditedStudentData,EditedC]=Insert(NewStudentData,NewC,AddData); % Records in AddData are added to NewStudentData to produce EditedStudentData. % EditedC is the cell array of the data in EditedStudentData. EditedStudentData=NewStudentData; EditedC=NewC; m=length(adddata); N=length(NewStudentData); for i=1:m EditedStudentData(N+1).Firstname=AddData(i).Lastname; EditedStudentData(N+1)Lastname=AddData(i).Firstname; EditedStudentData(N+1).Age=AddData(i).Age; EditedStudentData(N+1).Sex=AddData(i).Sex; for j=1:5 EditedStudentData(N+1).Grade(j)=AddData(i).Grade(j); end N=N+1; end % Create ProcessedC for i=1:m s(1:90)= ; s(1:10)=adddata(i).firstname;s(15:24)=adddata(i).lastname; s(30:31)=num2str(adddata(i).age); s(37)=num2str(adddata(i).sex); s(40:45)=num2str(adddata(i).grade(1)); s(50:55)=num2str(adddata(i).grade(2)); s(60:65)=num2str(adddata(i).grade(3)); s(70:75)=num2str(adddata(i).grade(4)); s(80:85} = num2str(adddata(i).grade(5)); EditedC{N+1}=s; N=N+1; end;

77 Sort the data using the Age key and store the data in a file called Processed-students.dat. function ProcessedC=SortedList(EditedStudentData, EditedC) % ProcessedC contains the records in increasing Age N=length(EditedStudentData); % N = number of records for i=1:n Age(i)=EditedStudentData(i).Age; % Collecting the Age array end; [SortedAge,Index]=sort(Age); % Age array is sorted; Index gives the rank % Copy the data for i=1:n ProcessedC{i}=EditedC{Index(i)}; end % Write the file fid=fopen( Processed-students.dat, w ); for i = 1:N fprintf(fid, %s, ProcessedC{i}); fprintf(fid, \n ); end fclose(fid); end