Chapter 4 MATLAB Programming Combining Loops and Logic Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

Chapter 4MATLAB Programming

Combining Loops and Logic

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Combining Loops and Logic

• Complex programs usually contain a both loops and logical statements to control the sequence of calculations

• As an example, let’s consider this example: We want to create an identity matrix (a square array with all ones as diagonal elements, zeros as off-diagonal elements) for a given size. For size = 3, our matrix would be:

Engineering Computation: An Introduction Using MATLAB and Excel

1 0 0 0 1 0 0 0 1

Example

• There is a built-in MATLAB command “eye” that creates an identity matrix, but we will ignore that for this example

• We will call the matrix A, so we need to define each element A(m,n) where m and n both range from 1 to the specified size

• This will require two nested for loops• For each element, a conditional statement is

required: if m = n, then A(m,n) = 1, else A(m,n) = 0


Example

• Can you de-scramble these lines of code to create an identity matrix and print it to the screen?


A(m,n) = 0;for m = 1:SizeendSize = 10;AA(m,n) = 1;elseendif m == nendfor n = 1:Size

Flow Chart


No

Yes

Define Size

for m = 1: Size

m = n ?

for n = 1: Size

A(m,n) = 1 A(m,n) = 0

Output A

Solution


Size = 10;for m = 1:Size for n = 1: Size if m == n A(m,n) = 1; else A(m,n) = 0; end endendA

Example


Example

• Let’s set the Size to 3 and repeat:

• No change! What happened?


Example

• A was stored as a a 10 X 10 matrix in memory before the m-file was modified

• When the new file was executed, the elements of A up to (3,3) were overwritten; the rest were unchanged

• Good practice to clear your variables (especially arrays) at the beginning of an m-file:


Example

• Now it works:


Example (Example 4.2)

• In many textbook problems, a 3-4-5 triangle is encountered:

• Note that the hypotenuse (5) is a perfect square: an integer that is the square root of another integer (25)

• Many calculations are made simple for this configuration: cosθ = 4/5; sinθ = 3/5, etc.

θ 3

4

5

Problem Description

• Can we find the other perfect squares for integer side lengths x and y from 1 to 25?

• h = hypotenuse


θy

x

h

Program Planning

• We will need to use two nested loops, since we have two independent variables: x and y

• Things to consider:– How will we identify perfect squares mathematically?– Do we want to store the perfect squares that we find,

and/or do we want to print them to the screen or a file?– Do we want to count the number of perfect square

combinations found?


Begin Flow Chart with Loop Statements

• How do we determine if h is an integer? Consider the MATLAB function floor:

>> help floor

FLOOR Round towards minus infinity.

FLOOR(X) rounds the elements of X to the nearest integers towards minus infinity.


for x = 1:25

for y = 1:25

h = sqrt(x^2 + y^2)

Check for Perfect Square

• We can compare h and floor(h) directly in an if statement:

if h == floor(h)

• If this statement is true, then we will print the values of x, y, and h to the screen

• If it is false, then we will do nothing



for x = 1:25

Is h an integer?

Print m

No

Yes m = m + 1Print x, y, h

for y = 1:25

h = sqrt(x^2 + y^2)

m = 0

MATLAB Code

for x = 1:25

for y = 1:25

h = sqrt(x^2 + y^2);

• What type of conditional statement do we need?• if• if-else• if-elseif


MATLAB Code

• A simple if statement works here – we either do the next steps or we skip them

for x = 1:25

for y = 1:25

h = sqrt(x^2 + y^2);

if h == floor(h)

x

y

h

end


MATLAB Code

• Close both loops with end statementsfor x = 1:25 for y = 1:25 h = sqrt(x^2 + y^2); if h == floor(h) x y h end endend


Results

• Save file as “PerSquares” and run:>> PerSquares

x =

3

y =

4

h =

5

x =

4

y =

3

h =

5

(Screen output continues for many lines)


Add Formatting to Output

for x = 1:25

for y = 1:25

h = sqrt(x^2 + y^2);

if h == floor(h)

fprintf('%10i %10i %10i\n',x,y,h)

end

end

end

More about the fprintf command later


Results

>> PerSquares 3 4 5 4 3 5 5 12 13 6 8 10 7 24 25 8 6 10 8 15 17 9 12 15 10 24 26 12 5 13 12 9 15 12 16 20 15 8 17 15 20 25 16 12 20 18 24 30 20 15 25 20 21 29 21 20 29 24 7 25 24 10 26 24 18 30>>

• How would we add a counter to report the number of combinations found?

• How do we eliminate duplicates? (such as 3, 4, 5; 4, 3, 5)


Counter Added

m = 0;

for x = 1:25

for y = 1:25

h = sqrt(x^2 + y^2);

if h == floor(h)


m = m + 1;

end

end

end

fprintf('\n Combinations found = %i\n',m)


Eliminating Duplicates

• The first time though the x-loop, x = 1 and y = 1:25:

x = 1

y = 1,2,3,4….25• The second time though the x-loop, x = 2 and y = 1:25:

x = 2

y = 1,2,3,4….25• But since we have already looked at the combination

x = 1, y = 2, we do not want to look at x = 2, y = 1


Eliminating Duplicates

• If we start the y-loop at 2 when x = 2:

x = 2

y = 2,3,4,5….25• And start the y-loop at 3 when x = 3:

x = 3

y = 3,4,5,6….25• And similar for the other values of x, then we have

eliminated duplicate combinations


y-Loop Adjusted

m = 0;

for x = 1:25

for y = x:25

h = sqrt(x^2 + y^2);

if h == floor(h)


m = m + 1;

end

end

end

fprintf('\n Combinations found = %i\n',m)


Results

>> PerSquares

3 4 5

5 12 13

6 8 10

7 24 25

8 15 17

9 12 15

10 24 26

12 16 20

15 20 25

18 24 30

20 21 29

Combinations found = 11

>>

• For you to think about:

• How would we eliminate combinations that are multiples of other combinations?

(Example: 6, 8, 10 is a multiple of 3, 4, 5 – forms a similar triangle)


The fprintf Command

• This command writes formatted output to the screen• The format of the command is:

fprintf(fid, ’Text to be written, including conversion

specifications for any variables to be printed’, variables)

• The file ID (fid) is omitted for output to the screen• Conversion specifications are instructions for how the

variables are to be formatted, inserted at the points where the variables are to be written


Conversion Specifications

• Conversion specifications begin with a % symbol• Next comes the number of digits and decimal

places• Last is designator of format type. Most common

are:• f = fixed number of decimal places• E or e = exponential notation• i = integer


Examples

• >> fprintf('Pi = %8.3f',pi)

Pi = 3.142


Text to be written to the screen, including the conversion specification for the variable (pi)

pi is output over 8 spaces, including 3 decimal places

Examples

• >> fprintf('Pi = %e',pi)

Pi = 3.141593e+000


Note that the number of digits and the number of decimal places are optional

pi is output in exponential notation, with the number of digits corresponding to the default “short” format

Examples

• >> fprintf('\nPi = %.4f\n',pi)

Pi = 3.1416

>>


The characters \n within the output string start a new line

pi is output to 4 decimal places

Examples

• >> m = 12;

>> fprintf('\n\nThe value of m is %i\n\n',m)

The value of m is 12

>>


m is output as an integer

Writing to a File

• Before writing to a file, you must first open a file and assign it to “fid” (file ID):

• Example: fid = fopen('amtable','wt');• 'wt' indicates write access; text file format. If the

file does not exist, it will be created. If it already exists, its contents will be overwritten

• 'at' instead of 'wt' indicates that an existing file will be appended – the new results will be added to the end of an existing file


Modifications to “PerSquares” File:

m = 0;fid = fopen('output.txt','wt');fprintf(fid,' x y h\n');fprintf(fid,' === === ===\n');for x = 1:25 for y = x:25 h = sqrt(x^2 + y^2); if h == floor(h) fprintf(fid,'%5i %5i %5i\n',x,y,h); m = m + 1; end endendfprintf('\n Combinations found = %i\n',m)


New file “output.txt” opened for write access

Column headers written to file

Integer triangles written to file

Number of triangles found written to screen

Output

• The table is now printed to the file “output,” which can be opened in Word or Notepad, or imported into Excel


Documents

Chapter 4 MATLAB Programming Combining Loops and Logic Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display