Post on 12-Jan-2016
WELCOMEWELCOMEEF 105
Fall 2006
Week 10
Topics:
1. Engineering Problem Solving
2. Programming Logic
3. Intro to MATLAB
Engineering Problem Solving
• Define the problem clearly• Work hand examples• Develop the Algorithm (Steps to follow)• Document the Algorithm with a FLOWCHART• Implement the Algorithm (Write computer
program)• Test the Implementation (Run the Program)• Evaluate the Results
PROGRAMMING LOGIC:Top-Down Algorithm Development
• Divide and Conquer
• Break Problem into smaller tasks.
• Define the interaction between tasks.
• Recursively solve the individual tasks.
• Build up the overall solution from the pieces.
Structured ProgrammingStructured ProgrammingStructured Programming
• Combination of– Sequences– Selections– Loops
Sequences
• Series of operations that are performed in order.
Selections
• Choose one path from two or more possible paths.
Loops
• Execute a block of code repeatedly as long as some condition is met.
Basic Flowcharting ElementsBasic Flowcharting Elements
test?test?FF
TT
startstarttasktask
I/O taskI/O taskstopstop
Selection BlockSelection Block
Arrows show the flow - cannot diverge but can converge.Arrows show the flow - cannot diverge but can converge.
Exit PointExit Point
Execution BlockExecution Block
Input/Output BlockInput/Output Block
Entry PointEntry Point
Selection StatementsSelection Statements• Selectively choose one path of execution.
– Based on the evaluation of a test.– Logical Test outcome is either TRUE or FALSE.
test?test?
if_codeif_code
if()if()
TT
FFtest?test?
if_codeif_code else_codeelse_code
if()...elseif()...else
FF
TT
Loop StructuresLoop Structures
Special case of Selection Statement
• One branch eventually leads back to the original selection statement.
• Permits a block of code to be executed repeatedly as long as some test condition is satisfied.
loop_codloop_codee
test?test?FF
TT
inc_codeinc_code
ini_coini_codede
next_codnext_codee
Basic Flowchart Symbols
Entry
ExitTask I/O Q?
T
F
Practice: Algorithm and Flowchart
Compute a sum of all integers from 1 to 100 and displays the result
Step 1: Understand the problem:
Computes a sum of all integers from 1 to 100
Step 2: What is the input?
Step 3: What is the output?
Start from 1, stop if reach 100 or (1 and 100)
Sum
Algorithm
Step 4: How do we compute the output? (first solution)Step 4.1: Start with the current integer: 1
Step 4.3: Add the current integer to SumStep 4.2: Sum starts with 0
Step 4.4: If the current integer is less than 100, keepon adding the current integer to sum and increaseIt by 1(i.e, go back to 4.3).
Step 4.5:Otherwise, print out the sum
Flowchart
Step 4.3: Add the current integer to Sum
Step 4.1: Start with the first integer: 1
Step 4.2: Sum starts with 0
Step 4.4: If the current integer is less than 100, go to next integer and keep on adding the current integer to sum (i.e, go back to 4.3).
Step 4.5:Otherwise, print out the sum
Stop
Current Integer=1Sum=0
Sum = Sum +Current Integer
Current Integer < 100
Increase Current
Integer by 1
Print Sum
Start
Intro to MatlabInside Matlab
This is what you should see once Matlab has loaded. The three most useful areas in the Matlab window are:
1. Right ‘command’ window is used for inputting commands
2. Top left ‘workspace’ window notes size of matrices used
3. Bottom left ‘command history’ window maintains a sequential list of past commands.
Command
History
Workspace
Command
Window
Command
History
Command
Window
Command
Window
Help in MATLABThe following are three ways to access Matlab’s help files:
1. a. From options at top of Matlab window select Help Matlab Helpb. Select ‘Index’ tabc. Type topic to be searched in box
2. In Matlab window type a command preceded by ‘help’ or 'doc' For example,the following commands would produce a help file for the ‘plot’ command.
>> help plotor>>doc plot
3. a. Double click command for which you want helpb. Right-click on commandc. Select ‘Help on Selection’
Creating files in MATLABA. To create a new ‘M-file’ do one of the following:
1. In top left corner of Matlab window select File New M-file2. Select ‘New M-file’ shortcut button located at the top left corner of Matlabscreen
B. Typing the following clears the command window:
>> clcC. Typing a semicolon at the end of a command suppresses output. Note thedifference between typing the following commands:>> x=0:0.5:10>> x=0:0.5:10;
How/Where to write program
Go to MATLAB command windowFile-New-M-FileM-file is an Editor windowWrite your program in M-fileSave in temp/ or your Disk. In command window, Run this file.
• + addition
• - subtraction
• * multiplication
• / right division
• ^ power
Basic Operators
Review Arithmetic Operations and Precedence
Operation Algebraic Matlab Form Scalar
addition a + b a + bsubtraction a – b a – bmultiplication a x b a * bdivision a ÷ b a / bexponentiation ab a ^ b
Precedence Operation
1 Parenthesis, innermost first.
2 Exponentiation, left to right
3 Multiplication & division, left to right
4 Addition & subtraction, left to right
Use Parentheses to Override Operator Precedence
• Normal evaluation of expressions– Left-to-Right if same level and no parentheses
e.g. 3^3-8/4+7-5*2 = 27-2+7-10 = 25+7-10 = 32-10 = 22
• Use parentheses to override
e.g. (3^3-8)/4+(7-5)*2 = (9-8)/4+2*2 =
1/4+4 = 4.25
Overview of MatLab Variables
• Variables are names used to hold values that may change throughout the program.
• MatLab variables are created when they appear on the left of an equal sign.
• >> variable = expression creates the "variable'' and assigns to it the value of the expression on the right hand side. You do not need to define or declare a variable before it is used.
• >> x = 2 % creates a scalar • The variable is x and % indicates a comment
Hands-On DEMO: Expression Evaluation
3 2
2
2 6.3
0.05005 3.14
x x xf
x x
In MatLab, enter the following:
>> x = 1.4;
>> numerator = x^3 - 2*x^2 + x - 6.3;
>> denominator = x^2 + 0.05005*x – 3.14;
>> f = numerator / denominator
Variable Naming• Naming Rules
– must begin with a letter, cannot contain blank spaces– can contain any combination of letters, numbers and
underscore (_)– must be unique in the first 31 characters
• MatLab is case sensitive: “name”, “Name” and “NAME” are considered different variables
• Never use a variable with the same name as a MatLab command (see next slide)
• Naming convention:– Usually use all_lowercase_letters– -or- camelNotation ("hump" in middle)
Reserved Words…• MatLab has some special (reserved) words that
you may not use as variable names:
break
case
catch
catch
continue
else
elseif
end
for
function
global
if
otherwise
persistent
return
switch
try
while
Commands involving variables
– who: lists the names of defined variables
– whos: lists the names and sizes of defined variables
– what: lists all your m-files stored in memory.
– clear: clears all variables, reset the default values of special variables.
– clear name: clears the variable named
– clc: clears the command window
– clf: clears the current figure and the graph window.
Scalars and Vectors and Matrices
• In MatLab, a scalar is a variable with one row and one column.
• A vector is a matrix with only one row OR only one column. The distinction between row and column vectors is crucial.
• When working with MatLab you will need to understand how to properly perform linear algebra using scalars, vectors and matrices. MatLab enforces rules on the use of each of these variables
Scalars• Scalars are the simple variables that we use and
manipulate in simple algebraic equations.
• To create a scalar you simply introduce it on the left hand side of an equal sign. – >> x = 1;– >> y = 2;– >> z = x + y;
Vectors• A row vector in MATLAB can be created by an explicit list, starting with a
left bracket, entering the values separated by spaces (or commas) and closing the vector with a right bracket.
• A column vector can be created the same way, and the rows are separated by semicolons.
• Example:>> x = [ 0 0.25*pi 0.5*pi 0.75*pi pi ]x = 0 0.7854 1.5708 2.3562 3.1416>> y = [ 0; 0.25*pi; 0.5*pi; 0.75*pi; pi ]y = 0 0.7854 1.5708 2.3562 3.1416
x is a row vector.
y is a column vector.
Simple Vector Commands
x = start:end create row vector x starting with start, counting by one, ending at end
x = start:increment:end create row vector x starting with start, counting by increment, ending at or before end
linspace(start,end,number) create row vector x starting with start, ending at end, having number elements
length(x) returns the length of vector x
y = x' transpose of vector x (row to column, or columnn to row)
Hands-On DEMO: Creating Vectors
>> a = 1:10 % leave off semi-colon to see what you get each time
>> b = 0:0.1:1
>> c = [7 8 9]
>> d = [10; 11; 12]
>> length(b)
>> linspace(0,100,21)
Hands-On DEMO: linspace function
% Plotting a function using vector math
x = linspace(0, 20, 100); % define 100 x values (from 0 to 20)
y = 5*exp(-0.3*x).*sin(x); % compute y vector
plot(x,y), xlabel('X'), ylabel('Y'), title('Vector calc')
• linspace( ) function can be very effective for creating the x vector…
0.35 sin( )xy e x0.35 sin( )xy e x
• A = [1 2 3; 4 5 6; 7 8 9] OR• A = [1 2 3 4 5 6 7 8 9]• Must be enclosed in brackets• Elements must be separated by commas or
spaces• Matrix rows must be separated by semicolons
or a return• Matlab is case sensitive
Entering MatricesEntering Matrices
• x = [-1.3 sqrt(3) (1+2+3)*4/5];– Output -> x = -1.3000 1.7321 4.8000
• Matrix Manipulation:– x(5) = abs(x(1));– Let r = [1 2 3 4 5];– xx = [x;r];– z = xx(2,2);– T = xx(2,1:3); %row 2, col. 1-3
• Semicolon at the end of a line means don’t print to the command window.
Matrix Elements
• Format short– 1.3333 0.0000
• Format short e– 1.3333E+000 1.2345E-006
• Format long– 1.333333333333338 0.000001234500000
• Format long e– 1.33333333333333E+000 1.234500000000003E-
006• Format hex
– 3FF555555555555 3EB4B6231ABFD271
• Defaults to format short.
Output Format
• Transpose:
• A = [1 2 3; 4 5 6; 7 8 9];
• C = A’;
• D = [-1 0 2]’;
Matrix Operations
• The term array operations refer to element-by-element operations.
• Preceding an operator (*, /, ^, ‘) by a period indicates an element-by-element operation.
• The addition and subtraction, matrix and array operations are the same and don’t need a period before these operators.
• Example:– X = [1 2 3]; Y=[4 5 6];
– W = X.*Y; % to mult. X and Y arrays
Array Operations(using the period)
• Most often used for a time vector.
• time = 0.0:100.0;
• Time = 10.0:0.5:100.0;
• B_time = 100.0:-0.5:50.0;
• Variable = first:increment:last
Generating Vectors
• Empty Matrix• E = [];• EE([2 4],:) = [];
– Empties rows 2 & 4 and all columns in rows 2 & 4.• Zeros
– Ze = zeros(2,3);– Creates a 2 x 3 matrix consisting all of zeros.
• Ones– O = ones(3,3);– Creates a 3 x 3 matrix consisting all of ones.
• Eye– I = eye(3,3);– Creates a 3 x 3 matrix consisting of an identity matrix. (I’s
on diagonal and 0’s elsewhere)
Useful Matrices
h = [1 2 3];h'
(nothing)ans = 1 2 3
Switches from row to column vector.
h * h'h .* h
ans = 14ans = 1 4 9
* is matrix multiplication, and so the dimensions must line up correctly. (more on this later)
.* is entry-by-entry multiplication.
Hands-On DEMO: Matrix Operations - Transposes
Transpose – (indicated by `)new matrix created by exchanging rows and columns of original matrix
Hands-On DEMO: Functions of Vectors
• Most Matlab functions will work equally well with both scalars and arrays (of any dimension)
>> A=[1 2 3 4 5];
>> sin(A)
ans =
0.8415 0.9093 0.1411 -0.7568 -0.9589
>> sqrt(A)
ans =
1.0000 1.4142 1.7321 2.0000 2.2361
Strings of Characters
• MatLab variables may also contain strings, which are vectors of individual characters. There is no typographical difference in appearance between numerical variables and string variables.
• The type of variable (numerical or string) is determined when the variable is created. – >> x = 5.2 % numeric– >> y = 'Chewbacca' % string
What are Character Strings? Arrays!Example:
» C = 'Hello'; %C is a 1x5 character array.» D = 'Hello there'; %D is a 1x11 character array.» A = 43; %A is a 1x1 double array.» T = 'How about this character string?'» size(T)ans = 1 32» whos % What do you observe? Name Size Bytes Class A 1x1 8 double array C 1x5 10 char array D 1x11 22 char array T 1x32 64 char array ans 1x2 16 double arrayGrand total is 51 elements using 120 bytes
Hands-On DEMO: Strings of Characters
>> h = 'Hello';
>> w = 'World';
>> [h ', ' w] % called concatenation
format Examples(MATLAB performs all computations in
double precision)• The format command described below switches among
different display formats.
• Command Result Example format short 5 digit scaled fixed point 3.1416 format long 15 digit scaled fixed point 3.14159265358979 format short e 5 digit floating-point 3.1416e+00 format long e 15 digit floating-point 3.141592653589793e+00
format short g general purpose 5 or 1.25 or 3.0e-12
format bank Fixed dollars and cents 7.95
format rat Ratio of small integers 355/113
format compact Suppresses excess line feeds. format loose Add line feeds.
Hands-On DEMO: Formatting
p=0:20;
format short e % exponential
[p' pow2(p)' pow2(-p)']
format short g % general purpose
[p' pow2(p)' pow2(-p)']
In-Class Exercise : MatLab Calculations• Do the following in MatLab
– Create a matrix with the form: [2 3]
[5 1] – Create a row (or "horizontal") vector of 2 elements, 3 and 4 (inclusive). – Create a second column (or 'vertical') vector with the elements 2 and 1 in that
order. – Type 'whos' to view your variables. It should read (for example): – >> whos Name Size Elements Bytes Density Complex
a 2 by 2 4 32 Full No b 1 by 2 2 16 Full No
c 2 by 1 2 16 Full No Grand total is 14 elements using 112 bytes Here, a is the matrix, b is the first vector, and c is the second vector.
• Now complete the following exercise: – Multiply your matrix by your first vector, above. – Perform element by element division of your resulting vector, divided by your
second vector transposed. (The result should be a two element horizontal vector with 13 as each entry. )
– Type 'clear' to clear all variables from the workspace. • Add your name as % comment, print Command Window and turn in
Saving/Loading Data Values
• Be sure you have the correct Current Directory set• clear, clc
– clear workspace and command window to start
• assign and calc a few values• save file_name
– saves file_name.mat in current directory– saves all defined variables
• clear; load file_name– brings workspace back
Hands-On DEMO: Saving Workspace
• First make sure your workspace is clear, and that your Current Directory is set to "My Documents" subfolder with your username (e.g. Djackson)
• Create a few scalars and vectors• check workspace window for variables• >> save demo -OR- click corresponding button
• look at your folder in My Documents subfolder• Clear workspace, check Workspace• then Reload variables, check Workspace
Scripts• Scripts allow us to group and save MatLab
commands for future use• If we make an error, we can edit statements and
commands• Can modify later to solve other related problems• Script is the MatLab terminology for a program
• NOTE: Scripts may use variables already defined in the workspace. – This can lead to unexpected results. – It is a good idea to clear the workspace (and the
command window) at the beginning of all scripts.• clear, clc
M-files: Scripts
• A Script is the simplest example of an M-file.• When a script-file is invoked, MatLab simply
executes the commands found in the file.• Any variables created in the script are added to the
MatLab Workspace.• Scripts are particularly useful for automating long
sequences of command.
Writing a MATLAB Script (program)
• File New M-File• usually start with: clear, clc, format compact
• comments after %– for in-class exercises include at least:
• Course, date,section & # (e.g. EF105, Monday 8:00 )• a short title• your name
• semi-colons to suppress variable initialization– omit semi-colons to display results– You can leave off ALL semi-colons to trace a program
Matlab Editor(note "Save and Run" button)
Color keyed text with auto
indents
tabbed sheets for other files being edited
Access to commands
Script Files (filename.m)• Always check Current Directory Window
– Set to MyDocuments\username
• Running scripts– from editor window, click "Save and run" button– -or- just type filename: – -or- type run filename
Hands-On DEMO: Creating our First Script(factorial)
%fact_n – Compute n-factorial, n!=1*2*...*n
% by DFJ
fact = prod(1:n) % no semi-colon so fact displays
%fact_n – Compute n-factorial, n!=1*2*...*n
% by DFJ
fact = prod(1:n) % no semi-colon so fact displays
• File New M-File
• File Save As ... fact_n• Operates on a variable in "global" workspace
– Variable n must exist in workspace– Variable fact is created (or over-written)
Hands-On DEMO: Running your Script
• To Run just type
>> n=5;
>> fact_n
-OR-
>> n=10;
>> run fact_n
Displaying Code and Getting Help
• To list code, use type command
>> type fact_n• The help command displays first
consecutive comment lines
>> help fact_n
MATLAB Exercise 1• See the Word document for this exercise and perform at end of class on your own!!