Write the program ball_function.m as given in the text and
confirm that the program runs correctly. Then save a copy of the
program and use that program during the following error testing.
You are supposed to introduce errors in the code, one by one. For each error introduced, save and run the program, and comment how well Matlab's response corresponds to the actual error. When you are finished with one error, re-set the program to correct behavior (and check that it works!) before moving on to the next error.
a)
Change the first line from function ball_function() to
ball_function(), i.e. remove the word function.
b)
Change the first line from function ball_function()
to function ball_func(), i.e., change the name of the function.
c)
Change the line function result = y(t) to function y(t).
d)
Change the line function result = y(t) to
function result = y(), i.e., remove the parameter t.
e)
Change the first statement that calls y from
vertical_position = y(time); to vertical_position = y();.
Filename: introducing_errors.m.
Explain briefly, in your own words, what the following program does.
a = input('Give an integer a: ');
b = input('Give an integer b: ');
if a < b
fprintf('a is the smallest of the two numbers\n');
elseif a == b
fprintf('a and b are equal\n');
else
fprintf('a is the largest of the two numbers\n');
end
Proceed by writing the program, and then run it a few times with
different values for a and b to confirm that it works as
intended. In particular, choose combinations for a and b so that
all three branches of the if construction get tested.
Filename: compare_a_and_b.m.
Write a program that takes a circle radius r as input from the user
and then computes the circumference C and area A of the
circle. Implement the computations of C and A as two separate
functions that each takes r as input parameter. Print C and A to
the screen along with an appropriate text. Run the program with \( r =
1 \) and confirm that you get the right answer.
Filename: functions_circumference_area.m.
Write a program that computes the area \( A = b c \) of a rectangle. The values of \( b \) and \( c \) should be user input to the program. Also, write the area computation as a function that takes \( b \) and \( c \) as input parameters and returns the computed area. Let the program print the result to screen along with an appropriate text. Run the program with \( b = 2 \) and \( c = 3 \) to confirm correct program behavior.
Filename: function_area_rectangle.m.
One of the most important mathematical problems through all times has been to find the area of a polygon, especially because real estate areas often had the shape of polygons, and it was necessary to pay tax for the area. We have a polygon as depicted below.
The vertices ("corners")
of the polygon have coordinates \( (x_1,y_1) \), \( (x_2,y_2) \),
\( \ldots \), \( (x_n, y_n) \), numbered either in a clockwise or
counter clockwise fashion.
The area \( A \) of the polygon can amazingly be computed by just knowing the
boundary coordinates:
$$ A = \frac{1}{2}\left| (x_1y_2+x_2y_3 + \cdots + x_{n-1}y_n + x_ny_1)
- (y_1x_2 + y_2x_3 + \cdots + y_{n-1}x_n + y_nx_1)\right|\thinspace .$$
Write a function polyarea(x, y) that takes two coordinate arrays
with the vertices as arguments and returns the area.
Assume that x and y are either lists or arrays.
Test the function on a triangle, a quadrilateral, and a pentagon where you can calculate the area by alternative methods for comparison.
Filename: polyarea.m.
Write a program that gets an integer \( N > 1 \) from the user and computes the average of all integers \( i = 1,\ldots,N \). The computation should be done in a function that takes \( N \) as input parameter. Print the result to the screen with an appropriate text. Run the program with \( N = 5 \) and confirm that you get the correct answer.
Filename: average_1_to_N.m.
Assume some program has been written for the task of adding all integers \( i = 1,2,\ldots,10 \):
some_number = 0;
i = 1;
while j < 11;
some_number += 1
print some_number
a) Identify the errors in the program by just reading the code and simulating the program by hand.
b) Write a new version of the program with errors corrected. Run this program and confirm that it gives the correct output.
Filename: while_loop_errors.m.
Consider one circle and one rectangle. The circle has a radius \( r =
10.6 \). The rectangle has sides \( a \) and \( b \), but only \( a \) is known from
the outset. Let \( a = 1.3 \) and write a program that uses a while loop
to find the largest possible integer \( b \) that gives a rectangle area
smaller than, but as close as possible to, the area of the circle. Run
the program and confirm that it gives the right answer (which is \( b =
271 \)).
Filename: area_rectangle_vs_circle.m.
Consider two functions \( f(x) = x \) and \( g(x) = x^2 \) on the interval \( [-4,4] \).
Write a program that, by trial and error, finds approximately for which values of \( x \) the two graphs cross, i.e., \( f(x) = g(x) \). Do this by considering \( N \) equally distributed points on the interval, at each point checking whether \( |f(x) - g(x)| < \epsilon \), where \( \epsilon \) is some small number. Let \( N \) and \( \epsilon \) be user input to the program and let the result be printed to screen. Run your program with \( N = 400 \) and \( \epsilon = 0.01 \). Explain the output from the program. Finally, try also other values of \( N \), keeping the value of \( \epsilon \) fixed. Explain your observations.
Filename: crossing_2_graphs.m.
The built-in function rand may be used to draw pseudo-random numbers
for the standard uniform distribution between \( 0 \) and \( 1 \) (exclusive
at both ends). See help rand.
Write a script that generates an array of \( 6 \) random numbers between \( 0 \) and \( 10 \). The program should then sort the array so that numbers appear in increasing order. Let the program make a formatted print of the array to the screen both before and after sorting. The printouts should appear on the screen so that comparison is made easy. Confirm that the array has been sorted correctly.
Filename: sort_numbers.m.
Up through history, great minds have developed different computational schemes for the number \( \pi \). We will here consider two such schemes, one by Leibniz (\( 1646-1716 \)), and one by Euler (\( 1707-1783 \)).
The scheme by Leibniz may be written $$ \begin{equation*} \pi = 8\sum_{k=0}^{\infty}\frac{1}{(4k + 1)(4k + 3)} , \nonumber \end{equation*} $$ while one form of the Euler scheme may appear as $$ \begin{equation*} \pi = \sqrt[]{6\sum_{k=1}^{\infty}\frac{1}{k^2}} . \nonumber \end{equation*} $$ If only the first \( N \) terms of each sum are used as an approximation to \( \pi \), each modified scheme will have computed \( \pi \) with some error.
Write a program that takes \( N \) as input from the user, and plots the
error development with both schemes as the number of iterations
approaches \( N \). Your program should also print out the final error
achieved with both schemes, i.e. when the number of terms is N.
Run the program with \( N = 100 \) and explain briefly what the graphs show.
Filename: compute_pi.m.
Consider an ID number consisting of two letters and three digits, e.g., RE198. How many different numbers can we have, and how can a program generate all these combinations?
If a collection of \( n \) things can have \( m_1 \) variations of the first thing, \( m_2 \) of the second and so on, the total number of variations of the collection equals \( m_1m_2\cdots m_n \). In particular, the ID number exemplified above can have \( 26\cdot 26\cdot 10\cdot 10\cdot 10 =676,000 \) variations. To generate all the combinations, we must have five nested for loops. The first two run over all letters A, B, and so on to Z, while the next three run over all digits \( 0,1,\ldots,9 \).
To convince yourself about this result, start out with an ID number on the form A3 where the first part can vary among A, B, and C, and the digit can be among 1, 2, or 3. We must start with A and combine it with 1, 2, and 3, then continue with B, combined with 1, 2, and 3, and finally combine C with 1, 2, and 3. A double for loop does the work.
a)
In a deck of cards, each card is a combination of a rank and a suit.
There are 13 ranks: ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, jack (J),
queen (Q), king (K), and
four suits: clubs (C), diamonds (D), hearts (H), and spades (S).
A typical card may be D3. Write statements that generate a
deck of cards, i.e., all the combinations CA, C2, C3, and so on
to SK.
b)
A vehicle registration number is on the form DE562, where the letters
vary from A to Z and the digits from 0 to 9. Write statements that compute
all the possible registration numbers and stores them in a list.
c) Generate all the combinations of throwing two dice (the number of eyes can vary from 1 to 6). Count how many combinations where the sum of the eyes equals 7.
Filename: combine_sets.m.
Write a program that takes a positive integer \( N \) as input and then draws
\( N \) random integers in the interval \( [1,6] \) (both ends inclusive). In the program,
count how many of the numbers, \( M \), that equal 6 and write out
the fraction \( M/N \). Also, print all the random numbers to the screen so that
you can check for yourself that the counting is correct. Run the program with
a small value for N (e.g., N = 10) to confirm that it works as intended.
Hint.
Use 1+floor(6*rand()) to draw
a random integer between 1 and 6.
Filename: count_random_numbers.m.
For large \( N \), this program computes the probability \( M/N \) of getting six eyes when throwing a die.
Consider some game where each participant draws a series of random integers evenly distributed from \( 0 \) and \( 10 \), with the aim of getting the sum as close as possible to \( 21 \), but not larger than \( 21 \). You are out of the game if the sum passes \( 21 \). After each draw, you are told the number and your total sum, and is asked whether you want another draw or not. The one coming closest to \( 21 \) is the winner.
Implement this game in a program.
Hint.
Use floor(11*rand()) to draw
random integers in \( [0,10] \).
Filename: game_21.m.
Some measurements \( y_i \), \( i = 1,2,\ldots,N \) (given below), of a quantity \( y \) have been collected regularly, once every minute, at times \( t_i=i \), \( i=0,1,\ldots,N \). We want to find the value \( y \) in between the measurements, e.g., at \( t=3.2 \) min. Computing such \( y \) values is called interpolation.
Let your program use linear interpolation to compute \( y \) between two consecutive measurements:
b) Write another function with in a loop where the user is asked for a time on the interval \( [0,N] \) and the corresponding (interpolated) \( y \) value is written to the screen. The loop is terminated when the user gives a negative time.
c) Use the following measurements: \( 4.4, 2.0, 11.0, 21.5, 7.5 \), corresponding to times \( 0,1,\ldots,4 \) (min), and compute interpolated values at \( t=2.5 \) and \( t=3.1 \) min. Perform separate hand calculations to check that the output from the program is correct.
Filename: linear_interpolation.m.
Assume the straight line function \( f(x) = 4x + 1 \). Write a script that tests the "point-slope" form for this line as follows. Within a chosen interval on the \( x \)-axis (for example, for \( x \) between 0 and 10), randomly pick \( 100 \) points on the line and check if the following requirement is fulfilled for each point: $$ \frac{f(x_i) - f(c)}{x_i - c} = a,\hspace{.3in} i = 1,2,\ldots,100\thinspace , $$ where \( a \) is the slope of the line and \( c \) defines a fixed point \( (c,f(c)) \) on the line. Let \( c = 2 \) here.
Filename: test_straight_line.m.
Assume some measurements \( y_i, i = 1,2,\ldots,5 \) have been collected, once every second. Your task is to write a program that fits a straight line to those data.
a) Make a function that computes the error between the straight line \( f(x)=ax+b \) and the measurements: $$ e = \sum_{i=1}^{5} \left(ax_i+b - y_i\right)^2\thinspace . $$
b) Make a function with a loop where you give \( a \) and \( b \), the corresponding value of \( e \) is written to the screen, and a plot of the straight line \( f(x)=ax+b \) together with the discrete measurements is shown.
c) Given the measurements \( 0.5, 2.0, 1.0, 1.5, 7.5 \), at times \( 0, 1, 2, 3, 4 \), use the function in b) to interactively search for \( a \) and \( b \) such that \( e \) is minimized.
Filename: fit_straight_line.m.
Fitting a straight line to measured data points is a very common task. The manual search procedure in c) can be automated by using a mathematical method called the method of least squares.
A lot of technology, especially most types of digital audio devices for processing sound, is based on representing a signal of time as a sum of sine functions. Say the signal is some function \( f(t) \) on the interval \( [-\pi,\pi] \) (a more general interval \( [a,b] \) can easily be treated, but leads to slightly more complicated formulas). Instead of working with \( f(t) \) directly, we approximate \( f \) by the sum $$ \begin{equation} S_N(t) = \sum_{n=1}^{N} b_n \sin(nt), \tag{1} \end{equation} $$ where the coefficients \( b_n \) must be adjusted such that \( S_N(t) \) is a good approximation to \( f(t) \). We shall in this exercise adjust \( b_n \) by a trial-and-error process.
a)
Make a function sinesum(t, b) that returns \( S_N(t) \), given the
coefficients \( b_n \) in an array b and time coordinates in an
array t. Note that if t is an array, the return value is also
an array.
b)
Write a function test_sinesum() that calls sinesum(t, b) in a)
and determines if the function computes a test case correctly.
As test case, let t be an array with values \( -\pi/2 \) and \( \pi/4 \),
choose \( N=2 \), and \( b_1=4 \) and \( b_2=-3 \). Compute \( S_N(t) \) by hand
to get reference values.
c)
Make a function plot_compare(f, N, M) that plots the original
function \( f(t) \) together with the sum of sines \( S_N(t) \), so that
the quality of the approximation \( S_N(t) \) can be examined visually.
The argument f is a Matlab function implementing \( f(t) \), N
is the number of terms in the sum \( S_N(t) \), and M is the number
of uniformly distributed \( t \) coordinates used to plot \( f \) and \( S_N \).
d)
Write a function error(b, f, M) that returns a mathematical measure
of the error in \( S_N(t) \) as an approximation to \( f(t) \):
$$ E = \sqrt{\sum_{i} \left(f(t_i) - S_N(t_i)\right)^2},$$
where the \( t_i \) values are \( M \) uniformly distributed coordinates on
\( [-\pi, \pi] \).
The array b holds the coefficients in \( S_N \) and f is a Matlab
function implementing the mathematical function \( f(t) \).
e)
Make a function trial(f, N) for interactively giving \( b_n \)
values and getting a plot on the screen where the resulting
\( S_N(t) \) is plotted together with \( f(t) \). The error in the
approximation should also be computed as indicated in d).
The argument f
is a Matlab function for \( f(t) \) and N is the number of terms \( N \) in
the sum \( S_N(t) \). The trial function can run a loop where
the user is asked for the \( b_n \) values in each pass of the
loop and the corresponding plot is shown.
You must find a way to terminate the loop when the
experiments are over. Use M=500 in the calls to plot_compare
and error.
f)
Choose \( f(t) \) to be a straight line
\( f(t) = \frac{1}{\pi}t \) on \( [-\pi,\pi] \). Call trial(f, 3)
and try to find through experimentation
some values \( b_1 \), \( b_2 \), and \( b_3 \) such that
the sum of sines \( S_N(t) \) is a good approximation to the straight line.
g) Now we shall try to automate the procedure in f). Write a function that has three nested loops over values of \( b_1 \), \( b_2 \), and \( b_3 \). Let each loop cover the interval \( [-1,1] \) in steps of \( 0.1 \). For each combination of \( b_1 \), \( b_2 \), and \( b_3 \), the error in the approximation \( S_N \) should be computed. Use this to find, and print, the smallest error and the corresponding values of \( b_1 \), \( b_2 \), and \( b_3 \). Let the program also plot \( f \) and the approximation \( S_N \) corresponding to the smallest error.
Filename: fit_sines.m.
In the analysis of genes one encounters many problem settings involving searching for certain combinations of letters in a long string. For example, we may have a string like
gene = 'AGTCAATGGAATAGGCCAAGCGAATATTTGGGCTACCA'
length(gene), so with a loop index i, for i = 1:length(gene)
will produce the required index values. Letter number i is then reached through
gene(i), and a substring from index i up to and including j,
is created by gene(i:j).
a)
Write a function freq(letter, text) that returns the frequency of the
letter letter in the string text, i.e., the number of occurrences of
letter divided by the length of text.
Call the function to determine the frequency of C and G in the
gene string above. Compute the frequency by hand too.
b)
Write a function pairs(letter, text)
that counts how many times a pair of the letter letter (e.g., GG)
occurs within the string text. Use the function to determine
how many times the pair AA appears in the string gene above.
Perform a manual counting too to check the answer.
c)
Write a function mystruct(text) that counts the number of a certain
structure in the string text. The structure is defined as G followed by
A or T until a double GG. Perform a manual search for the structure
too to control the computations by mystruct.
Filename: count_substrings.m.