Computer Science 106: MATLAB
HW 5, due 3/26
Please include your name in your homework
1. The US population figures from 1900 to 2000 (from Moler Chapter 5)
t
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
Pop (million)
75.995
91.972
105.711
123.203
131.669
150.697
179.323
203.212
226.505
249.633
281.422
a. Use the Matlab spine function to estimate the population in 2010.
b. Use Matlab’s polyfit function to estimate the population in 2010 using a cubic
polynomial.
The actual census count for 2010 is 308,745,538
2. Solve the triangles problem from Project Euler
Go to projecteuler.net and look at problem #102. A thousand triangles are scattered
randomly near the origin of the plane. How many of them have the origin embedded
inside? I have saved the triangles file in Ella under Resources as triangles.txt with
spaces rather than commas as separators.
It is very easy to load a text file of numbers (delimited by spaces) into Matlab. The
file triangles.txt is stored, in the resources area of Ella, as a table 6 numbers per line
(coordinates of a,b,c of triangle) separated by spaces. The way to load this file is to
type:
load triangles.txt
and the numbers will be stored in an N x 6 table called triangles. More complex data
is better stored in .mat files, so see help load for information on that.
To solve this problem, look at each side of the triangle and treat it as a vector.
Given two points A (Ax, Ay) and B (Bx,By) on a plane, how do you find the
vector from A to B and what is the z component value?
Use the cross-product to determine if the origin is to the right or left of each side of
the triangle. You will need the vector from each vertex to 0 for this. If all of the
cross-products are of the same sign, the origin is in the triangle. (You will need to go
consistently in one direction or the other around the triangle.)
The diagram below is incorrect: a,b,c are the vertices, not the sides
You are to solve the problem by writing functions that
1. compute the cross-product of two vectors. (there is a Matlab function)
2. returns the Z-component of the cross product of two vectors
3. uses the Z- components of the three cross products of the three sides and the
origin to determine if the origin is inside the triangle.
Write these functions and test on a triangle with the origin inside and one with the origin
outside, returning true or false. Show your test results (incorporate them as a few lines of
comments in your last function.
Then write a script to read in the triangles file and use the three functions to determine
how many triangles contain the origin.
Submit the code for the functions and the script and also the answer.
For extra credit, try vectorizing some of the whole process. To do this you might use
these statements:
a = triangles(:, 1:2);
b = triangles(:, 3:4);
c = triangles(:, 5:6);
which will read an entire column of number pairs into the variable a (and b and c)
where a will be 1000x2 in size.
3. Learn about strings to decide on palindromes
Write a script that asks for a string and then checks to see if it is a palindrome, a sentence
or phrase in which the letters read the same forward or backwards. Some examples of
palindromes:
Madam I’m Adam
(Adams introduction to Eve)
Able was I ere I saw Elba
(Napoleon’s lament on being exiled)
To ask for input: pal = input(‘Type a string: ’); and type the response in single quotes
like ‘Help me!’ If you have a single quote IN a string, type it twice:
‘Madam I’’m Adam’
To check for a palindrome, strip out all spaces and punctuation and test the resulting
string. Takes all of six lines, perhaps fewer if you nest operations. Look at help for these
functions:
strcmp
fliplr
isletter
upper
logical vectors
This can all be done with vector commands, no loops needed
The basic idea is this: Use the isletter operation to create a logical vector with a 1
wherever the string contains a letter and a zero elsewhere. Use the ability of Matlab
to use a logical vector to create a set of subscripts to extract only the letters into a
new string. Use upper to convert all to uppercase. Use fliplr and strcmp to compare
the forward and reverse strings to see if they are equal.
See
https://www.mtholyoke.edu/courses/pdobosh/cs106/features/logicalvectors.html
for how to use the logical vectors