Lecture 3 Creating M - files docx

An example of technical computing Let us consider using the hyperbolic tangent to model a down-hill section of a snowboard or snow ski facility..  Let us first examine the hyperbolic t

Lecture 3 Creating M-files

Programming tools:

Input/output (assign/graph-&-display)

Repetition (for) Decision (if)

© 2007 Daniel Valentine All rights reserved Published by

Elsevier.

 Arrays

– List of numbers in brackets

 A comma or space separates numbers (columns)

 A semicolon separates row

– Zeros and ones Matrices:

 zeros()

 ones()

– Indexing

 (row,column)

– Colon Operator:

 Range of Data first:last or first:increment:last

– Manipulating Arrays & Matrices

 Transpose

 Examples of input to arrays:

– Single number array & scalar: 1 × 1

>> a = 2

– Row array & row vector: 1 × n

>> b = [1 3 2 5]

– Column array & column vector: n x 1

>> b = b’ % This an application of the transpose.

– Array of n rows x m columns & Matrix: n × m

>> c = [1 2 3; 4 5 6; 7 6 9] % This example is 3 x 3.

Basic elements of a program

 Input

– Initialize, define or assign numerical values to

variables

 Set of command expressions

– Operations applied to input variables that lead

to the desired result

 Output

– Display (graphically or numerically) result.

An example of technical computing

 Let us consider using the hyperbolic tangent

to model a down-hill section of a snowboard

or snow ski facility

 Let us first examine the hyperbolic tangent function by executing the command:

>> ezplot( ‘tanh(x)’ )

We get the following graph:

Ezplot(‘tanh(x)’)

Hyperbolic tangent: tanh(X)

 Properties (as illustrated in the figure):

– As X approaches –Inf, tanh(X) approaches -1.

– As X approaches Inf, tanh(X) approaches +1. – X = 0, tanh(X) = 0.

– Transition from -1 to 1 is smooth and most

rapid (roughly speaking) in the range

-2<X<2.

– REMARK: With this familiarity with tanh(X), let

us apply it to solve the following problem.

Problem background

 Let us consider the design of a slope that

is to be modeled by tanh(X) Consider the range -3<X<3, and assume the slope

changes negligibly for |X|>3

 Thus, for –3 < X < 3, –1 < tanh(X) < 1

 We want to design a downhill slope such that in 10 meters the hill drops 2 meters

 Thus, as shown next, the following

formula is needed: y = 1 – tanh(3*X/5)

Formula for snow hill shape

Problem statement

 Find the altitude of the hill at 0.5 meter

intervals from -5 meters to 5 meters using the shape described and illustrated in the previous two slides

 Tabulate the results

Structure plan

 Structure plan

– Initialize the range of X in 0.5 meter intervals.

– Compute y, the altitude, with the formula:

y = 1 – tanh(3*X/5).

– Display the results in a table.

 Implementation of the structure plan

– Open the editor by typing edit in the

command window.

– Translate plan into the M-file language.

This is from editor: It is lec3.m

%

% Ski slope offsets for the

% HYPERBOLIC TANGENT DESIGN

% by D.T Valentine January 2007

%

% Points at which the function

% is to be evaluated:

% Step 1: Input x

x = -5:0.5:5;

% Step 2: Compute the y offset,

% that is, the altitude:

y = 1 - tanh(3.*x./5);

%

% Step 3: Display results in a table

%

disp(' ') % Skips a line

disp(' X Y')

disp([x' y'])

Command window OUTPUT

X Y -5.0000 1.9951 -4.5000 1.9910 -4.0000 1.9837 -3.5000 1.9705 -3.0000 1.9468 -2.5000 1.9051 -2.0000 1.8337 -1.5000 1.7163 -1.0000 1.5370 -0.5000 1.2913

0 1.0000 0.5000 0.7087 1.0000 0.4630 1.5000 0.2837 2.0000 0.1663 2.5000 0.0949 3.0000 0.0532 3.5000 0.0295 4.0000 0.0163 4.5000 0.0090 5.0000 0.0049

SOLUTION TO IN-CLASS EXERCISE

Use of repetition (‘for’)

 Repetition: In the previous example, the fastest way to compute y was used An alternative way is as follows:

Replace:

With:

for n=1:21

y(n) = 1 - tanh(3*x(n)/5);

end

Remark: Of course, the output is the same.

This is from editor: It is lec3_2.m

%

% Ski slope offsets for the

% HYPERBOLIC TANGENT DESIGN

% by D.T Valentine January 2007

%

% Points at which the function

% is to be evaluated:

% Step 1: Input x

x = -5:0.5:5;

% Step 2: Compute the y offset

for n=1:21

y(n) = 1 - tanh(3*x(n)/5);

end

%

% Step 3: Display results in a table

disp(' ') % Skips a line

disp(' X Y')

disp([x' y'])

Command window OUTPUT

X Y -5.0000 1.9951 -4.5000 1.9910 -4.0000 1.9837 -3.5000 1.9705 -3.0000 1.9468 -2.5000 1.9051 -2.0000 1.8337 -1.5000 1.7163 -1.0000 1.5370 -0.5000 1.2913

0 1.0000 0.5000 0.7087 1.0000 0.4630 1.5000 0.2837 2.0000 0.1663 2.5000 0.0949 3.0000 0.0532 3.5000 0.0295 4.0000 0.0163 4.5000 0.0090 5.0000 0.0049

SOLUTION TO IN-CLASS EXERCISE

Decision: Application of ‘if’

– Convert C to F or F to C.

% Temperature conversion from C to F

% or F to C as requested by the user

%

Dec = input(' Which way?: 1 => C to F? 0 => F to C: ');

Temp = input(' What is the temperature you want to convert? ');

%

% Note the logical equals sgn (==)

TF = (9/5)*Temp + 32;

disp(' Temperature in F: ')

disp(TF)

else

TC = (5/9)*(Temp-32);

disp(' Temperature in C: ')

disp(TC)

end

Decision: Application of ‘if’

 Temperature conversion problem:

– Convert C to F or F to C.

– SOLUTION:

 Introduced, by an example, the structure plan approach to design approaches to

solve technical problems

 Input: Assignment with and without ‘input’ utility

 Output: Graphical & tabular were shown

 Illustrated array dot-operation, repetition (for) and decision (if) programming tools