Description

1. (10 points) Cubic Splines
Here, we will assume cubic splines for a set of n ordered data points (x1,y1),…,(xn,yn) are of the
form:
 s1(x)
 …
S(x) =
 sn−1(x)
where the i = 1,…,n − 1 splines are defined as: x1 ≤ x ≤ x2
…
xn−1 ≤ x ≤ xn
si(x) = ai(x − xi)3 + bi(x − xi)2 + ci(x − xi) + di
It turns out we can use properties of an interpolating function and assumptions of continuity of splines to determine equations for the coefficients in terms of bi = Mi/2 where:

with h = xi − xi−1 for i = 2,…,n. With Boundary Conditions specified for conditions on M1 and
, we can write this system of equations as a linear system AM~ = R~ where we need to solve for
(a) (5 points) It turns out that the matrix A is diagonally dominant. Explain what this property is, why you can see it is true for rows 2,…,n − 2, and how it leads to knowing we can solve for coefficients uniquely.
(b) (5 points) We would call A a sparse matrix since it has many entries that are 0’s. As we know from earlier discussions about error, each 0 entry will be represented by its floating point representation in the computer where 0 = . From an error and computational storage standpoint, why does it make sense to use a sparse matrix representation where you only store non-zero entries in the matrix? (Note that Matlab and most programming languages have sparse matrix utilities)
2. (15 points) Discrete Least Squares
(a) (7 points) Write a code that takes an input of D for the degree of the least squares polynomial that you want to fit to the data. The data points are x =[1, 1.1, 1.3, 1.5, 1.9, 2.1], y =[1.84, 1.96, 2.21, 2.45, 2.94, 3.18].
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(b) (5 points) Determine the discrete least squares polynomials of degrees 1, 2, and 3 for the data.Create a plot for each polynomial fit. Plot both the data points (using a symbol for data in Matlab such as ‘og’ or ‘xg’) and also plot the polynomial at additional points xeval=linspace(0,3,100) with the data.
(c) (3 points) Compute the error in each polynomial approximation of degree pn(xi))2 for m data points.
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