Homework 6, Math 307, Fall 2011

PROBLEM 1(a). Consider a model y=a+bx+cx^3 that you are fitting to the
data (x,y)=(0,y_0),(1,y_1),(2,y_2).  In general, given y_0,y_1,y_2,
will there be a unique solution a,b,c?  You may calculate this by
hand or use a computer.

PROBLEM 1(b). Same a PROBLEM 1(a), except for the data
(x,y)=(-1,y_(-1)),(0,y_0),(1,y_1).


PROBLEM 2(a). The following program finds the least squares fit 
y(x) = alpha + beta x + gamma x^2 (i.e., fitting with a quadratic function in x)
to y(x)=sin(x) for the values x=-.2,-.1,0,.1,.2:
(The program below works in Matlab; in Octave you have to replace
"clearvars A b" with "clear")

clearvars A b
for i=1:5
x = (i-3)*.1;
A(i,1)=1;
A(i,2)=x;
A(i,3)=x^2;
b(i,1)=sin(x);
end
A
b
inv(A' * A) * A' * b

Explain how this program creates A,b and finds the vector whose components
are alpha, beta, and gamma.  How do you explain the fact that alpha=gamma=0
and beta is slightly less than 1, given what you know about the function
sin(x)?


PROBLEM 2(b) Guess the rough values of alpha, beta, and gamma that you
will see when you replace "sin" with "cos".  [As usual, there is no
"correct" guess; hopefully you are not entirely off, but hopefully you
will still learn something.] Now perform the calculation and explain
the results.


PROBLEM 2(c) Do the same as in 2(b), except replace cos(x) with 10 * x^4.

PROBLEM 2(d) Do the same as in 2(b), except replace cos(x) with abs(x)
(the absolute value of x).