MA3457 Homework 3 Solved

Description

1. Cubic Splines

Here, we will assume cubic splines for a set of n ordered data points (x₁,y₁),…,(x_n,y_n) 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

s_i(x) = a_i(x − x_i)³ + b_i(x − x_i)² + c_i(x − x_i) + d_i

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 b_i = M_i/2 where:

with h = x_i − x_i−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

1. M. Once we have M^~ , we then have b_i = M_i/2 and d_i = y_i for i = 1,…,n − 1. Then c_i and a_i can be determined based on b_i for i = 1,…,n−1. Refer to slides 16-17 of power point from Tuesday Nov 1st class.
   • (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.
   • (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
   • (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].

1

• (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.
• (3 points) Compute the error in each polynomial approximation of degreep_n(x_i))² for m data points.