Homework MATH 4446
Introduction to Numerical Analysis: Spring 2009
1. Plot the following data points using MATLAB.
x−3 1 2 5
f(x)−23 −11 −23 1
2. Plot f(x) = sin(2x) on the interval [0, π] using MATLAB.
3. Convert P(x) = 4x5+5x4−3x3+ 7x2−8x+ 10 to nested form (by hand) and evaluate
P(3). How many fewer multiplications are required for the function evaluation if Pis
in nested form? Horner’s method is often used to obtain the nested representation of a
polynomial. Use Horner’s method in MATLAB to obtain the nested form of P. (Use
the MATLAB help-function to obtain more information about horner.) To evaluate
the function at x= 3, use the MATLAB command subs.
Chapter 3
Section 3.1
1. Section 3.1 #1 (a)
2. Consider the following interpolating points for the function f(x) = ln x.
xi2 2.5 3
f(xi) 0.69315 0.91625 1.09861
2.1 Construct a linear, P1, and a quadratic, P2, interpolating polynomial to the func-
tion f(x) = ln xto approximate f(2.2). Compute Pn(2.2), n = 1,2.
2.2 Graph the Lagrange interpolating polynomials, Ln,j(x), n = 1,2, on the interval
[2,3].
2.3 Compare, by graphing, the two interpolants in 2.1 to fon the interval [2,3].
Which interpolant would you say is the most accurate and why?
2.4 Compute the absolute error, |f(2.2) −Pn(2.2)|, for n= 1,2.
2.5 Find an error bound for the two approximations. (Theorem 3.3.)
3. Let xi=i. Compute and plot L2, L3, and L5on the intervals [0,4],[0,6],and [0,10]
respectively.
Based on your results, comment on the following statement:
At each of the points xithere is only one non-zero Lagrange polynomial, Li, and one
expect that Liwill have the most influence on the shape of the curve near xi. This will
be the case if Litakes its maximum value at xiand decays in magnitude further away
from xi.
