Learning Objectives:
By completing this Lab exercise you should be competent in the following learning objectives:
Instructions:
Study the material in lectures 1 and 2.
Read the instructions below carefully.
Each student should perform the following on an individual basis.
Part 1 Matlab constants
A string is a list (vector) of any (keyboard) characters. In Matlab,
the keyboard characters are literal constants.
At the Matlab prompt type:
>> last = 'Square Pants'
>> first = 'Sponge Bob'
We can combine string vectors just like we did (in lab 1) numeric
valued vectors,
At the Matlab prompt type:
>> name = [ first , last]
Or, better:
>> name = [first , ' ' , last]
You can include numeric digits in your string
> > movie = [ name, ' Movie 1.23456']>> 1 + '1'Answer question #1 on Lab 2 answer sheet for Part 1.
For this question first type the following expression at the Matlab prompt:
>> clear
>> number = 1.2341234
We want to create the string 'test1.2341234' in
Matlab.
Fill in the blank below that creates the string described above. You
must include the variable , number
in your answer.
(Hint: use the Matlab num2str
function. Use Matlab Help or type help num2str to find
out how this function works.)
Part 2. Matlab Operators
In lab 1 we studied the operators +
- * / \
^ and in this lab we will introduce
.* ./ .\ .^ to complete the set of
ten arithmetic operators.
What you need to know (for the first midterm)
concerning the ten Matlab arithmetic operators:
Assuming that you have already typed in values for a and b
(vectors or matrices) and (op is one of the ten operators
above) then consider the expression,
>> a
op b
We will call .* .\ ./ .^
the "dot" operators.
The operators * or .* give the same result
when a
and b are scalars. In fact the following pairs of
operators give the
same values when a and b are scalars:
/ or ./
\ or .\
^ or .^
If op is a dot operator then a op b is defined
only when the size of a is the same as the size of b.
From lab 1,
the size of a is m x n
where m is the number of rows and n is the number of columns. Use the
size function in Matlab to find the size of a variable, for example
consider the following row vector a with one row and 3 columns,
>> a = [1 2 3];
>> size(a)
Except for the case when a and b are scalars we will
not cover
/ in CS101.
The * and \ operators will be
covered in a
future lab where a and b are not scalars.
Answer questions #2-#13 on Lab 2 answer sheet for Part 2.
To answer the following questions type the following commands at the Matlab prompt:
>> a = 1 : 5 : 15;
>> b = 4:6;
2. Fill in the blanks with the response Matlab gives when you type
the following at the
Matlab prompt. If the result is an error then write ERROR.
>> a
a =
_______________________
3. >> length(a)
ans =
_______________________
4. >> size(a) , size(b)
ans =
_______________________
ans =
________________________
5. >> 2 a
ans =
__________________________
6. >> 2 * a
ans =
__________________________
7. >> 2 .* a
ans =
__________________________
8. >> a * b
ans =
___________________________
9. >> a .* b
ans =
___________________________
10. >> a .^ 2
ans =
_____________________________
11. >> 2 .^ b
ans =
______________________________
12. >> a ./ b
ans =
_____________________________
13. >> a .\ b
ans =
______________________________
Part 3. Vector carving (subscripting)
Given a Matlab vector (for example),
>> x = .5 + (1:4)
x =
1.5 2.5 3.5 4.5
we can carve up the vector x using parentheses and an index value from 1 to 4:
>> x(1)
ans =
1.5
>> x(2)
ans =
2.5
>> x(4)
ans =
4.5
We can also carve a vector by specifying a vector of index values:
>> x([ 1 2])
ans =
1.5 2.5
>> x(1:2)
ans =
1.5 2.5
>> x([ 2 1])
ans =
2.5 1.5
>> x([ 4 2])
ans =
4.5 2.5
>> x(4:-2:2)
ans =
4.5 2.5
Recall that a:b:c means start at a, increase by b, repeatedly while less than or equal to c. Thus 4:-2:2 is [4 2]
Matlab subscripting works on column vectors in a similar manner.
Subscripting on matrices will be covered in a future lab.
Matlab also allows you to use subscripting to change value(s) of a
vector.
>> y = [ 1.5 ; 2.5 ; 3.5];
>> y(1) = -1.5
y =
-1.5
2.5
3.5
>> y( 2:3) = 0
y =
-1.5
0.0
0.0
Answer questions #14-#20 on Lab 2 answer sheet for Part 3.
Type the following expression at the Matlab prompt then either answer the question or fill in the blanks with the response Matlab returns. If there is an error write ERROR.
>> x = linspace(-2*pi,2*pi,200);
>> y = [ 1 3 5 7 11] ;
14. How many elements does the vector x have?
________________________
15. >> y(3:end)
ans =
_______________________
16. >> y(1:end-1)
ans =
________________________
17. >> x( [1 2])
ans =
_________________________(approx.)
18. >> x(1:2)
ans =
__________________________(approx.)
19. Fill in the blank below to carve out the first three
values of y and prepend zero to these three values. The correct
answer would display
z =
0
1 3 5
>> z = [ 0 ,
______________________________ ]
(The correct answer must use the variable y and
subscripting on y)
20. Fill in the blanks below to flip the values of y. The correct
answer would display
flipy =
11 7 5 3 1
>> flipy = y( end : ______ : _______
)
Part 4 Matlab functions
The Matlab function rand generates uniformly (equally likely) random numbers between zero and one. The Matlab function floor(x) rounds the elements of x to the nearest integer less than or equal to x. The Matlab function ceil(x) rounds the elements of x to the nearest integers greater than or equal to x. For example,
>> floor([1.9 3.4
-4.9 -5.3])
ans =
1 3
-5 -6
>>ceil([1.9 3.4 -4.9 -5.3]
)
ans =
2 4
-4 -5
In lab 1 we used the Matlab function diff to compute the derivative using symbolic differentiation. However diff also work on vectors (and matrices) by computing consecutive differences, for example,
>> diff( [ 1 2 4 8] )
ans =
1 2 4
since 2 -1 = 1 , 4 - 2 = 2
and
8 - 4 = 4.Answer questions #21-#22 on Lab 2 answer sheet for Part 4.
21. Complete the following Matlab expression to simulate the roll of
a single six sided die. Note the possible die values are 1,2,3,4,5, or
6 and are
integers.
(Hint: use the function rand with proper scaling)
>>floor( ____________ * _____________ ) + 122. Write a Matlab expression to display random integers(not reals) in the range from 3 to 7 inclusive. (Hint: use the rand and floor function)
=
______________________________________________________________Part 5 Mathematical functions
A function in mathematics is defined as a set of ordered pairs {
(x,y) | x in Domain, y in Range} with the restriction that if (a,b) and
(a,c) are in the set then b=c which means that if you plot the set of
points in the plane then the plot passes the "vertical line test"- no
vertical line cuts the plot in more than one point.
In algebra and calculus classes most functions are represented
symbolically, for example the parabola , y = x2 where the
Domain is the set of all Real numbers and the Range is the set of all
non-negative numbers.
Programs like Mathematica and Matlab can consider functions defined
either as a set of ordered pairs or symbolically. Matlab excels at
manipulating sets of ordered pairs whereas Mathematica excels at the
symbolic representation of functions.
No questions for this Part 5 on the answer sheet.
Part 6 Symbolics in Matlab
The function y = x2 can be
represented
in various equivalent forms. Matlab has a built-in function simple to help simplify
complicated symbolic expressions.
For example since sin(x)^2 + cos(x)^2 = 1,
>> syms x
>> simple( x^2 / ( sin(x)^2+cos(x)^2) )
(after some screen output)
x^2
The Matlab function factor can be used to factor polynomials.
Answer questions #23-#25 on Lab 2 answer sheet for Part 6.
To answer the following questions, first type,
>> syms x
23. Write the response Matlab returns after typing the following,
>> factor(x^3 - 1)
ans = __________________________________________________________
24. Write the response Matlab returns after typing the following,
( i is the "imaginary" number )
>> simple( exp(i * x) / (cos(x) +
i* sin(x) ) )
(Just write the answer, don't write the intermediate results)
ans =
____________________________________
(Euler's formula)
25. Fill in the blank below to simplify the mathematical
function cos(2x) / (cos2(x)-sin2(x)).
>>
simple(________________________________________________________________)
Part 7 Discretization in Matlab
A mathematical function y = f(x) can be discretized by
sampling (selecting) a finite set of x values for the domain and
computing the y values only at these finite x. In Matlab we often
choose the values of x to be equally spaced although this is not
required. Once a function is
discretized we need to know how we can perform the following operations
on the discretized function:
26. For the function y = x2 choose the
sample of x values as [ -1 0 1] type the following command at the
Matlab prompt
>> x =
[-1 0 1];
Fill in the blank below to
compute the values for y, (you should type this at the Matlab
prompt for use in Part 8)
>> y =
___________________________________
That's it, we have now discretized y = x2
. How good is our discretization? We find out in Part 8.
27. y = sin(x) Fill in the blanks below to discretize this
function. Choose the sample of x values as 50 evenly spaced
values from 0 to 2*pi. We will use x2 and y2 so that
we can keep these separate from the x and y in problem #26.
Part 8 Plotting
One Matlab command to do plotting is called plot . (see lecture notes 2-7 , 2-8
2-9) If x and y are two vectors of the same length, say
for example
>> x = [ 0 1 2]
>> y = [ 3 4 5]
then
>> plot(x,y)
plots the points (0, 3) (1, 4) and (2, 5) in the plane. The
function plot by default
connects these points with straight lines. We can choose to just plot
the data points and not connect the points with lines by typing
>> plot(x,y , '.' ) (that's a period in
single
quotes)
You can pass just one argument to the Matlab plot function. Type,
>> plot(y)
then use the hold on command so that we can hold (keep) the existing
plot while we plot again in the same figure window,
>> hold on
>> plot([1,2,3], y)
Notice that the two plots are identical!
Answer questions #28-#29 on Lab 2 answer sheet for Part 8.
28. Type the following command to plot the discretized parabola we
created in Part 7.
>> plot(x,y,'g') ('g' for green)
Was the discretization in #26 good? Yes, if we wanted to know whether
the parabola was concave up or down but No in the sense that the plot
doesn't look like a parabola. Fix the problem by plotting y = x2
using 100 points from -1 to 1. The plot should be in red, no
lines connecting the points and each point should be marked with a
*(star) not a . (period).
>> x = ____________________________
>> y = ____________________________
>> plot( _____________ , _____________________, ________________)
29. We will now plot the discretized function from problem #27.
>> plot(y2,x2)
Now put two plots on one figure window, type,
>> hold on
>> plot( x2,y2)
>> grid on
Does plot(x2,y2) give the same results as plot(y2,x2)
________________(YES/NO)?
Part 9 Differentiation
From Calculus, when a function y = f(x) has a tangent line at x = c then its slope is called the derivative. (see graph below)As an approximation of the derivative at x = c we could choose two points a = x - h/2 and b = x + h/2 where h is a small positive number and compute the slope of the "secant line" (which we can use as an approximation for the slope of the tangent line) as follows:
Slope of secant line (at x = c)
= rise / run
= (f(b) - f(a) ) / h
= (f(x + h/2) - f(x - h/2)) / h (Centered
Difference Formula)
Answer question #30 on Lab 2 answer sheet for Part 9.
Note how we will work with the discretized version of f(x) and not do a symbolic manipulation (differentiation) of the function.
The discretized function y = f(x) is available in vector form:
x = [x1 , x2 , x3 , ... , xn]where
y = [y1 , y2 , y3 , ... , yn]
y1 = f(x1)How can we compute and plot the derivative function y = f'(x)?
y2 = f(x2)
.
.
.
yn = f(xn)
If we know only two values of y = f(x), that is x = [x1 , x2] and y = [y1 , y2] then to plot the derivative function f'(x)we can use the Centered Difference Formula
yprime = (y2 - y1)/(x2 - x1)midpoint = (x1 + x2)/2
so that we have approximated the value f'(midpoint) = yprime. Alternatively, we say that the point (midpoint, yprime) lies on the curve of the function of y = f'(x).
For three points x = [x1 , x2 , x3] and y = [y1 , y2 , y3] then as above,
midpoints = [(x1+ x2)/2 , (x2 + x3)/2] = ( [ x1 , x2] + [ x2 , x3])/2
yprimes = [(y2 - y1)/(x2 - x1) , (y3 - y2)/(x3 - x2)] = [ (y2 - y1) , (y3 - y2) ] ./ [ (x2 - x1) , (x3 - x2)]
and we can plot these points that lie on the curve of the function y = f'(x) by typing,
plot(midpoints,yprimes)Therefore, given x and y we can compute an approximation of the discretization of the derivative y = f'(x)in Matlab by typing,
>> midpoints = (x(1:end-1) + x(2:end))/2 ;
>> yprimes = diff(y) ./ diff(x) ;
Answer question #31-#32 on Lab 2 answer sheet for Part 9.
>> midpoints = (x(1:end-1) + x(2:end))/2 ;
>> yprimes = diff(y) ./ diff(x) ;
fill in the blanks below in order to compute and plot the derivative
of y = sin(x) on the interval [0, 2*pi] using 10 equally spaced points.
>> x2 = linspace(0,2*pi,10);
>> y2 = sin(x2);
31.
>> midpoints = ___________________________________ ;
Part 10 Logical values and relational operators
Logical values are created when you use relational operators such as
( < , <= , > , >= , == , ~=). Matlab treats the value '1'
as True and the value '0' as False.
For example, at the Matlab prompt type,
>> clear
>> x = 5 > 1
>> y = 5 < 1
>> z = 5 == 1 % == means test for equality
>> w = 5 ~= 1 % ~= means test for inequality
>> whos
Note that the "Class" of logical variables is "logical array".
Logical operators can be used with arrays,
>> x = [5 -1.5 0.0 6.2] > 1
>> y = [5 -1.5 0.0 6.2] < 1
>> z = [5 -1 5 2 5] == 1
>> w = [5 -1 5 2 5] ~= 1
An important use of logical values is doing subscripting. For
example, suppose you are given a vector v = [ 1 2 3 4 5 6] and you want
to create a vector with only values in v that are >= 3. To do this
in Matlab type,
>> v = 1:6
>> large = v( v >= 3)
Note that if you manually type
>> large = v([ 0 0 1 1 1 1])
you will get an error. This seems strange since the values you get
from the expression v>=3 are the identical to [0 0 1 1 1 1]
however the Class of each of these expressions is different, the first
one being a logical array the latter being double array and Matlab
processes these differently when doing subscripting. For example, if
you type,
>> v = 1:6 ;
>> test1 = v >=3 ;
>> test2 = [ 0 0 1 1 1 1]
>> whos test1 test2
Name Size Bytes Class
test1 1x6 6 logical array
test2 1x6 48 double array
Answer question #33-#36 on Lab 2 answer sheet for Part 10.
33. Type the following assignment statement at the Matlab prompt,
>> v = [ -1 0 2 0 3 0
4 0 5 0 6]
Fill in the blank in the Matlab expression below that uses
subscripting to extract all non-zero values from v and assign these to
the variable w.
>> w = v ( ___________________________________________)
w =
-1 2 3
4 5 6
34. Fill in the blank in the Matlab expression below that uses subscripting to extract all zero values from v and assign these to the variable w.
>> w = v ( ___________________________________________)
w =
0 0 0 0
0
35. Suppose that we are given the discretized values of
a function y = f(x) as below. We want to find all the roots of the
function f . That is, we want to find all the values r such that 0 =
f(r).
Type the following assignment statements at the Matlab prompt,
>> y = [ -1 0 5 0
3 0
4 0 5 0 ] ;
>> x = [ .1 .2 .3 .4 .5
.6
.7 .8 .9 1.0 ] ;
result = ______________________________________________ ;
Congratulations, you are done!