Lecture 2 f17

Lecture 2—Chapters 2 & 3
Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
3 Control Structures
for loops
Boolean Expressions
if…then…else…end structures
while loops
Che 310 — Chapra 2–3 2 — Introduction August 24, 2017 1 / 49

The plot command
The most basic way to make a plot in Matlab is the command plot
>> x = linspace(1,10,10);
>> y = x .^ 3 - x .^ 2 + x - 1;
>> plot(x,y);
plot plots individual data points
contained in arrays x and y
The default is that all data are
connected by thin blue lines.

Basic plot formatting
You can control how the data appear by changing how you use
the plot command:
>> plot(x,y,’ko’)
1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600
700
800
900
1000

Basic plot formatting
You can control how the data appear by changing how you use
the plot command:
>> plot(x,y,’ko’)
1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600
700
800
900
1000 This example uses a format specifier
string
Color Line style Symbol
k black - o ◦
r red – s
g green : d ♦
b blue -. ^,v,<,> , , ,
See doc linespec for full list.

More plot formatting
Other things you can control:
>> plot(x,y,’kd-’,’linewidth’,2,’markersize’,12,...
’markerfacecolor’,’b’,’markeredgecolor’,’k’);
1 2 3 4 5 6 7 8 9 10
0
100
200
300
400
500
600
700
800
900
1000

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

Limitations of plot
plot is best for plotting arrays of data.
To plot functions with plot, you must create an x array and then use
your function to create a y array.
>> x = [0:6];
>> y = cos(x);
>> plot(x,y);
Command line:

Limitations of plot
plot is best for plotting arrays of data.
To plot functions with plot, you must create an x array and then use
your function to create a y array.
>> x = [0:6];
>> y = cos(x);
>> plot(x,y);
Command line:
0 1 2 3 4 5 6
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Smooth functions can look
rough without enough data
If you change your x-values,
you will have to update the
y-values

fplot
fplot will plot a function directly
Use a function handle:
>> myfunc = @(x) x.^2 * cos(x);
>> fplot( myfunc, [-pi pi], ’k--’);
Command line:
−3 −2 −1 0 1 2 3
−10
−8
−6
−4
−2
0
2

fplot
Shortcut — type the function in as a string
>> fplot( ’cos(x) * x’, [-pi pi], ’k--’);
Command line:
−3 −2 −1 0 1 2 3
−4
−3
−2
−1
0
1
2
3
4

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

Adding labels
xlabel(’text’) Label the x-axis with text
ylabel(’text’) Label the y-axis with text
title (’text’) Give the axis the title text
text(x,y,’text’) Place text at the coordinates (x,y)
legend(’plot 1’,’plot 2’,…) Adds a legend with descriptive text for
each plot on an axis.

More than one plot on an axis
By default, when you make a plot, the last plot you made disappears.
>> plot(linspace(-10,10),linspace(-10,10).*linspace(-10,10));
Command line:
After 1st
plot command:
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
10
20
30
40
50
60
70
80
90
100

By default, when you make a plot, the last plot you made disappears.
>> plot(linspace(-10,10),linspace(-10,10).*-linspace(-10,10));
Command line:
After 2nd
plot command:
−10 −8 −6 −4 −2 0 2 4 6 8 10
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0

hold on causes plots to accumulate on an axis
>> hold on
Command line:
After 1st plot command:
−10 −8 −6 −4 −2 0 2 4 6 8 10
0
10
20
30
40
50
60
70
80
90
100

hold on causes plots to accumulate on an axis
>> hold on
>> plot(linspace(-10,10),linspace(-10,10).*-linspace(-10,10),’r’);
>> legend(’y = x^2’,’y = -x^2’)
Command line:
After 2nd plot command:
−10 −8 −6 −4 −2 0 2 4 6 8 10
−100
−80
−60
−40
−20
0
20
40
60
80
100
y = x2
y = −x2

More than one axis on a figure
subplot is used to make multiple axes in one figure window.
>> subplot(2,3,1)
Command line:
1 2 3
4 5 6
Axis 1 is active (will be plotted on)
The first two numbers indicate how many rows/columns of axes to
make. The last number makes one axis active.
>> subplot(2,3,5)
Command line:
1 2 3
4 5 6
Axis 5 is active (will be plotted on)

More than one axis on a figure
>> subplot(1,3,1)
>> fplot(@(x) sin(x) .* sin(x),[-pi,pi],’r’);
>> title(’sin^2’);
>> subplot(1,3,2)
5 >> fplot(@(x) sin(x) .* cos(x),[-pi,pi],’g’);
>> title(’sin * cos’);
>> subplot(1,3,3)
>> fplot(@(x) cos(x) .* cos(x),[-pi,pi],’b’);
>> title(’cos^2’);
Command line:
−2 0 2
0
0.2
0.4
0.6
0.8
1
sin2
−2 0 2
−0.5
0
0.5
sin * cos
−2 0 2
0
0.2
0.4
0.6
0.8
1
cos2

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

What are m-files?
An m-file is a text file that Matlab recognizes as a command.
There are two varieties of m-file, the script and the function
Begin by opening the Editor window:

The Editor window
This is where you make an m-file

The Editor window
This is where you enter Matlab commands

The Editor window
These buttons can help you debug an m-file

The Editor window
These buttons help you navigate large files

The Editor window
These buttons help you write readable code. Comment, uncomment,
indent, and insert templates.

Scripts
The basic task of an m-file is to operate as a script
A script executes its contents, line-by-line
A script has Base scope. That is, it modifies and can access any
variable showing in the Workspace window.
Remember, the Workspace window shows a
list of variables Matlab currently knows
about. At the start, the Workspace window is
empty.

Scripts
empty.
Now, write and execute basic.m
a = 1;
b = 2;
c = 3;
Script-file: basic.m
>> basic
Command line:
The effect of basic.m is to create a, b, and c
in the Workspace

Scripts
empty.
Now, write and execute basic.m
a = 1;
b = 2;
c = 3;
Script-file: basic.m
>> basic
Command line:
The effect of basic.m is to create a, b, and c
in the Workspace
Script variables affect the Workspace (or
Base) environment

Scripts—scope
Scripts can create and use variables in the Workspace window.

Scripts—scope
Write and execute simplecalc.m
result = a * b + c
Script-file: simplecalc.m
>> simplecalc
result = 5
Command line: Running simplecalc.m

Scripts—scope
result = a * b + c
>> simplecalc
result = 5
Now change a, b, and c
>> a = 10; b = 20; c = 30;
Command line: Change a, b, & c

Scripts—scope
result = a * b + c
>> simplecalc
result = 5
Now change a, b, and c
>> a = 10; b = 20; c = 30;
Command line: Change a, b, & c
Now re-run simplecalc
>> simplecalc
result = 230
Command line: Running simplecalc

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

What is a function?
Functions
A set of commands, executed
1 at a time
Scripts
1 at a time

What is a function?
Functions
1 at a time
Cannot use variables already
defined in the Workspace
Scripts
1 at a time
Can use variables already

What is a function?
Functions
1 at a time
Cannot change Workspace
variables
Scripts
1 at a time
Changes to script variables
effect Workspace variables

What is a function?
Functions
1 at a time
variables
Accepts arguments that
change its behavior
Scripts
1 at a time

What is a function?
Functions
1 at a time
variables
Accepts arguments that
change its behavior
Produces outputs intended for
later use
Scripts
1 at a time

How to make a function
function
First line of m-file
function_name
Name of your function
Functions need their own variables:
function basicfunc
d = a * b + c
Function-file: basicfunc.m
>> basicfunc
??? Undefined function or variable ’a’.
Command line: Using basicfunc.m incorrectly

How to make a function
function
First line of m-file
function_name
Name of your function
Variables within a function have local scope:
function basicfunc2
a = 3; b = 1; c = 2;
d = a * b + c
Function-file: basicfunc2.m
>> basicfunc2
d = 5
>> [a b c ]
ans = 1 2 3
5 >> d
??? Undefined function or variable ’d’.
Command line: Using basicfunc.m correctly. Pay attention to scope!

Naming functions
Good programming practice
Functions are used by their m-file name
NOT the name in the m-file
function badname
disp(’I am mis-named’);
Function-file: misnamed.m

Naming functions
Good programming practice
Functions are used by their m-file name
NOT the name in the m-file
function badname
disp(’I am mis-named’);
Function-file: misnamed.m
>> badname
??? Undefined function or variable ’badname’.
>> misnamed
I am mis-named
Command line:

Functions & Input Arguments
Functions can accept input arguments
function inputfunc (input1, input2)

function inputfunc(input1,input2)
result = input1 + input2
Function-file: inputfunc.m

Unless you do something special, a function with inputs will fail at
the command line without its arguments
>> inputfunc
??? Input argument ”input1” is undefined.
Error in ==> inputfunc at 2
5 >> inputfunc(1,2)
>>
Command line: Using input_func

5 >> inputfunc(1,2)
>>
1
2
inputfunc.m

5 >> inputfunc(1,2)
>>
1
2
inputfunc.m arguments get copied into the function
workspace

5 >> inputfunc(1,2)
>>
1
2
inputfunc.m
input1
1
input2
2
arguments get copied into the function
workspace
This workspace is separate from the
Workspace window

5 >> inputfunc(1,2)
>>
1
2
inputfunc.m
input1
1
input2
2
result
3
workspace
Workspace window
The result is trapped inside of the
function.

5 >> inputfunc(1,2)
>>
1
2
inputfunc.m
input1
1
input2
2
result
3
workspace
Workspace window
The result is trapped inside of the
function.
We can’t use it yet outside of the function!

Adding outputs
For a function to communicate with the outside world, you must
provide an output
function result = outputfunc (input1,input2)
input1
1
input2
2
result1
3
outputfunc.m
1
2

Adding outputs
provide an output
function result = outputfunc (input1,input2)
Inputs go in
input1
1
input2
2
result1
3
outputfunc.m
1
2

Adding outputs
provide an output
function result =
Outputs come out
outputfunc (input1,input2)
input1
1
input2
2
result1
3
outputfunc.m
1
2
ans
3
function result = ...
outputfunc(input1,input2)
result = input1 + input2;
Function-file: outputfunc.m
>> outputfunc(1,2)
ans =
3
Command line:

Adding outputs — Advanced
Multiple outputs
A function can create more than one output
function [ result1 result2 ] = many_output_func(input1,input2)
function [ result1 result2 ] = ...
many_output_func(input1,input2)
result1 = input1 + input2;
result2 = input1 .* ones(input2) + ...
5 input2 .* eye(input2);
end
Function-file: many_output_func.m

Multiple outputs
input1
1
input2
2
result1
3
result2
3 1
1 3
many_output_func.m
1
2
ans
3
end
Only the first output is ‘‘captured’’ by
default
>> many_output_func(1,2)
ans =
3
Command line:

Multiple outputs
input1
1
input2
2
result1
3
result2
3 1
1 3
many_output_func.m
1
2
first
3
second
3 1
1 3
end
Now result1 ⇒ first and result2 ⇒
second
>> [first second] = many_output_func(1,2)
first =
3
second =
5 3 1
1 3
Command line:

Delimiter Review
()
Parentheses are used for…
Grouping portions of expressions
a = 1 ./ (b .* c .* (d + e))

Delimiter Review
()
a = 1 ./ (b .* c .* (d + e))
Assigning values to parts of arrays
A(1,2) = 3

Delimiter Review
()
a = 1 ./ (b .* c .* (d + e))
A(1,2) = 3
Accessing values from parts of arrays
b = A(3,4)

Delimiter Review
()
a = 1 ./ (b .* c .* (d + e))
A(1,2) = 3
b = A(3,4)
Passing arguments to functions
result = sin(pi)

Delimiter Review
()
a = 1 ./ (b .* c .* (d + e))
A(1,2) = 3
b = A(3,4)
Passing arguments to functions
result = sin(pi)
Forming the argument list for functions in an m-file
function output = myfunc(input1,input2)

Delimiter Review
[]
Square brackets are used for…
Creating arrays
A = [ 1 2 3 ];

Delimiter Review
[]
Creating arrays
A = [ 1 2 3 ];
Defining multiple function outputs
function [output1 output2] = multi_output_function(input1,input2)

Delimiter Review
[]
Creating arrays
A = [ 1 2 3 ];
Defining multiple function outputs
function [output1 output2] = multi_output_function(input1,input2)
Using multiple function outputs
>> [ A B ] = calc_a_and_b()

Delimiter Review
’ ’
Single quotes are used to
Start and stop strings
my_string = ’String’;
The double quote ” does not work.

Delimiter Review
{}
The curly braces are used for…
Creating cell arrays
cell_array = {’String 1’, [1;2;3;4], @(x) x.^2 };
Note — As above, cell arrays can store mixtures of any data type

Delimiter Review
{}
The curly braces are used for…
Creating cell arrays
cell_array = {’String 1’, [1;2;3;4], @(x) x.^2 };
Note — As above, cell arrays can store mixtures of any data type
Accessing cell arrays
>> my_string = cell_array{1}
my_string =
String 1
>> cell_array{3}(3)
5 ans =
9
Command line:

Delimiter Review
…
The ellipsis (…) are used to:
Continue a line

Delimiter Review
;
The semicolon…
Supresses output
>> outputfunc(1,2)
ans =
3
>> outputfunc(1,2);
5 >>
Command line:

Delimiter Review
;
The semicolon…
Supresses output
>> outputfunc(1,2)
ans =
3
>> outputfunc(1,2);
5 >>
Command line:
Makes a new row in an array
>> A = [ 1 2 3; 4 5 6 ];

Delimiter Review
;
The semicolon…
Supresses output
>> outputfunc(1,2)
ans =
3
>> outputfunc(1,2);
5 >>
Command line:
Makes a new row in an array
>> A = [ 1 2 3; 4 5 6 ];
Allows more than 1 command on a line
>> A = 2:5; B = 2.*A.^2; plot(A,B);

Delimiter Review
,
Separates arguments in functions (required)
function a = my_function(b,c,d)

Delimiter Review
,
Can be used to separate multiple outputs in functions (optional)
function [ output1, output2 ] = multi_out(input1, input2)

Delimiter Review
,
Can be used to separate multiple outputs in functions (optional)
function [ output1, output2 ] = multi_out(input1, input2)
Can be used to separate columns in arrays (optional)
>> A = [ 1, 2, 3; 4 5 6]

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

for loops
Sometimes we know (or can calcuate) exactly how many times
through a loop we need.

for loops
We want to do something element-by-element to an array

for loops
We want exactly n iterations of some algorithm

for loops
We want to do something over a specific range of numbers.

for loops
These cases are easily coded using a for loop

for loops
Syntax:
for ii
iteration variable
= array_variable
...
Code
...
More Code
...
end

for loops
Syntax:
for ii
iteration variable
= array_variable
...
Code
...
More Code
...
end
Why ii rather than i?
Matlab uses the variables i and j to mean
√
−1
It is therefore good practice in Matlab to avoid
the use of i and j to avoid overlap with these
built-in variables.
Use ii/jj or I/J instead.

for loops — Example
Finite summations are the mathematical equivalent of for loops
3
i=1
i2
= 1 + 22
+ 32
= 14
>> sum_squares(3)
ans =
14
Command line:
function s = sum_squares(n)
% Check that the input argument is positive
if n < 0
error(’n must be a positive number’);
5 end
s = 1; % Initialize the sum
for ii = 2 : n
s = s + ii^2;
end
10 end
Function-file: sum_squares.m

3
i=1
i2
= 1 + 22
+ 32
= 14
n
3
>> sum_squares(3)
ans =
14
Command line:
if n < 0
5 end
for ii = 2 : n
s = s + ii^2;
end
10 end

3
i=1
i2
= 1 + 22
+ 32
= 14
n
3
1:n
1 2 3
i
1
>> sum_squares(3)
ans =
14
Command line:
if n < 0
5 end
for ii = 2 : n
s = s + ii^2;
end
10 end

3
i=1
i2
= 1 + 22
+ 32
= 14
n
3
1:n
1 2 3
i
2
>> sum_squares(3)
ans =
14
Command line:
if n < 0
5 end
for ii = 2 : n
s = s + ii^2;
end
10 end

3
i=1
i2
= 1 + 22
+ 32
= 14
n
3
1:n
1 2 3
i
3
>> sum_squares(3)
ans =
14
Command line:
if n < 0
5 end
for ii = 2 : n
s = s + ii^2;
end
10 end

for loops
Example—Nested loops
Loops can be placed within other loops
ii
jj
M
>> nested_for
ans =
1 2
2 4
Command line:
function M = nested_for
M = zeros(2);
for ii = 1:2
for jj = 1:2
5 M(ii,jj) = ii * jj
end
end
end
Function-file: nested_for.m

for loops
ii
jj
0 0
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
1
ii
jj
0 0
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
1
ii
1
jj
0 0
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
1
ii
1
jj
1 0
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
1
ii
2
jj
1 0
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
1
ii
2
jj
1 2
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
2
ii
2
jj
1 2
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
2
ii
1
jj
1 2
0 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
2
ii
1
jj
1 2
2 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
2
ii
2
jj
1 2
2 0
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

for loops
2
ii
2
jj
1 2
2 4
M
>> nested_for
ans =
1 2
2 4
Command line:
M = zeros(2);
for ii = 1:2
for jj = 1:2
end
end
end

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

What is a boolean expression?
Definition
A is a mathematical expression that has only two possible outcomes
TRUE or FALSE
In Matlab
TRUE ≡ 1
FALSE ≡ 0
Boolean expressions are constructed from boolean operators.

Relational Operators (Comparison operators)
Definition
A relational operator returns a boolean result by comparing two numbers with
each other
Operator Description
<Numerical Expression> < <Numerical Expression> Less than

Definition
each other
<Numerical Expression> <= <Numerical Expression> Less than or equal to

Definition
each other
<Numerical Expression> > <Numerical Expression> Greater than

Definition
each other
<Numerical Expression> >= <Numerical Expression> Greater than or equal to

Definition
each other
<Numerical Expression> == <Numerical Expression> Equal to

Definition
each other
<Numerical Expression> == <Numerical Expression> Equal to
<Numerical Expression> ∼= <Numerical Expression> Not equal to
Both <Numerical Expression>s are evaluated before making the comparison

Relational Operators — Examples
Start with these variables:
>> a = .3;
>> b = .1;
>> c = .2;
Command line:

>> a = .3;
>> b = .1;
>> c = .2;
Command line:
Basic comparisons work as expected
>> c == 2 * b; % Be careful: ’==’ doesn’t always work as expected
5 ans =
1
>> a > b
ans =
1
10 >> a < b
ans =
0
>> a ~= b
ans =
15 1
Command line:

>> a = .3;
>> b = .1;
>> c = .2;
Command line:
Others do not
>> a - 3 * b == 0 % .3 - 3 * .1 equals 0 right?
ans =
0 % FALSE! Why?
Command line:
Why??

>> a = .3;
>> b = .1;
>> c = .2;
Command line:
Others do not
>> a - 3 * b == 0 % .3 - 3 * .1 equals 0 right?
ans =
0 % FALSE! Why?
Command line:
Why??
Numerical errors mean that strict equalities between arithmetically
equivalent #s are not numerically equivalent
>> format long g % Force MATLAB to show full precision
20 >> a-3*b
ans =
-5.55111512312578e-017 % Very close to 0, but NOT EXACTLY 0
Command line:

>> a = .3;
>> b = .1;
>> c = .2;
Command line:
A better way to test for equality of 2 numerical expressions is to use
the abs function:
>> tol = 10^-15;
>> abs( (a-3*b) - 0 ) < tol
25
ans =
1 % The absolute value of (.3 - 3 * .1) is smaller than our tolerance
>> tol = 10^-17;
30 >> abs( (a-3*b) - 0 ) < tol
ans =
0
Command line:

Logical Operators
Definition
A logical operator returns a boolean result based on the relationship between
one or two boolean expressions
Operator Name Result
∼ <Boolean Exp.> Not
∼ 0 = 1
∼ 1 = 0
<Boolean Exp.> && <Boolean Exp.> And
0 && 0 = 0 1 && 0 = 0
0 && 1 = 0 1 && 1 = 1
<Boolean Exp.> || <Boolean Exp.> Or
0 || 0 = 0 1 || 0 = 1
0 || 1 = 1 1 || 1 = 1

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

The if statement
We often would like functions & scripts to behave differently based
on their input.
To look for and report errors
function root = square_root(number)
if(number < 0)
error(’The square root of a negative number is imaginary!’)
end
5 root = sqrt(number);
end
Function-file: Find the square root of non-negative numbers only

The if statement
We often would like functions & scripts to behave differently based
on their input.
To look for and report errors
function root = square_root(number)
if(number < 0)
error(’The square root of a negative number is imaginary!’)
end
5 root = sqrt(number);
end
To make a decision
function signs = signof(number)
if number < 0
signs = -1;
5 elseif number > 0
signs = 1;
else
signs = 0;
end

Outline
1 Making plots
Plotting Arrays
Plotting Functions
Formatting Plots
2 The m-file
Scripts
Functions
for loops
Boolean Expressions
while loops

while loops
The while loop is used to repeat a section of code while a boolean
expression evaluates to TRUE.

while loops
Useful in numerical methods that need to iterate while the result still
needs improvement

while loops
Useful in numerical methods that need to iterate while the result still
needs improvement
Usually a counter variable is introduced to keep track of the
number of times through the loop.

while Loops
A simple example
>> while_demo(3)
ans =
-0.5
Command line:
function result = while_demo(max_iter)
iter = 0;
result = 10;
5 while iter < max_iter && result > 0
result = result / 2 - 1;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3 >> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
0
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
0
result
10
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
0
result
4
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
1
result
4
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
1
result
1
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
2
result
1
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
2
result
-0.5
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
A simple example
max_iter
3
iter
3
result
-0.5
>> while_demo(3)
ans =
-0.5
Command line:
iter = 0;
result = 10;
iter = iter + 1;
end
Function-file:

while Loops
Another Example—estimating π
π can be evaluated numerically using the Gregory-Leibniz series:
π = 4
∞
i=0
(−1)i 1
2i + 1 >> [ pi_est n_iter ] = pi_estimate(12,.1)
pi_est =
3.2837
n_iter =
5 7
Command line:
estimate ii err
0.1
tol
lastestimate
12
maxiter
function [ estimate ii ]= pi_estimate(maxiter, tol)
% Initialization section
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
while err > tol && ii < maxiter
lastestimate = estimate;
estimate = estimate + 4*(-1)^ii / (2*ii + 1);
10 err = abs((estimate-lastestimate)/estimate);
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
0
estimate
ii err
0.1
tol
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
0
estimate
0
ii
err
0.1
tol
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
0
estimate
0
ii
1.1
err
0.1
tol
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
0
estimate
0
ii
1.1
err
0.1
tol
0
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
4
estimate
0
ii
1.1
err
0.1
tol
0
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
4
estimate
0
ii
1
err
0.1
tol
0
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
4
estimate
1
ii
1
err
0.1
tol
0
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
4
estimate
1
ii
1
err
0.1
tol
4
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
2.667
estimate
1
ii
1
err
0.1
tol
4
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
2.667
estimate
1
ii
0.5
err
0.1
tol
4
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
2.667
estimate
2
ii
0.5
err
0.1
tol
4
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
2.667
estimate
2
ii
0.5
err
0.1
tol
4
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:
Keep repeating until while condition is false

while Loops
π = 4
∞
i=0
(−1)i 1
pi_est =
3.2837
n_iter =
5 7
Command line:
3.2837
estimate
7
ii
0.0937
err
0.1
tol
2.976
lastestimate
12
maxiter
estimate = 0;
ii = 0;
5 err = tol + 1;
% Loop section
ii = ii + 1;
end
Function-file:

Lecture 2 f17

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Lecture 2 f17