Description
This assignment is a more extensive investigation of the rounding error example we studied in class using
the Matlab code roundex.m (posted on Canvas). The polynomial function (x−a)
n with a any real number
and n a positive integer can be written in expanded form as
(x − a)
n =
Xn
k=0
n
k
x
n−k
(−a)
k = x
n −
n
1
x
n−1
a +
n
2
x
n−2
a
2 −
n
3
x
n−3
a
3 + … +
n
n
(−a)
n
(∗)
where
n
k
=
n!
(n−k)! k!
are binomial coefficients.
1. Plot f(x) = (x − 2)n
for n = 1, 2, 3, 4, 5, 6 on the domain x ∈ [0, 4], using the factored (unexpanded)
form. Display your 6 curves together on a single plot, and use different colors, line styles and/or
a legend to clearly identify the various curves. Choose appropriate limits for the y-axis that ensure
important features of the function are visible. Consider this your “exact result”.
2. Write a Matlab function in an m-file named fexpand.m that computes the polynomial in the expanded form (∗). The first line of your code should be this function definition statement:
function fx = fexpand(a, n, x)
The function has 3 input parameters (a, n and x) and returns a single output argument (the computed
value of f(x)). You can make use of the built-in Matlab function nchoosek to compute the binomial
coefficients. Test your code completely, making sure that it is able to exit gracefully with a suitable
error/warning message for any values of the input arguments that might generate an invalid result.
3. Produce 6 plots depicting your “expanded” f(x) curves that “zoom in” near the point x = 2 using a
series of successively smaller x-intervals, x ∈ [2 − δ, 2 + δ] for δ = 0.5, 0.1, 0.05, 0.025, 0.01, 0.005†
.
In
your report, you should only include the plots for δ = 0.5, 0.05, 0.005, with two select values of the
polynomial degree n, for a total of 3 × 2 = 6 plots – and choose the two n values that you think best
illustrate your conclusions! Describe any interesting behaviour or differences that arise between the
various n, δ values.
In addition, be sure to comment on the following:
(a) Identify the smallest value of the exponent n for which your plots of the expanded polynomial
(∗) differ significantly from the “exact” plots (in part 1).
(b) Describe what happens as n increases (it might help to try a few n even larger than 6).
(c) Explain why performing these computations in double-precision floating point arithmetic gives
rise to the errors you observe.
Be specific!
†Note that you should choose your plotting points to be consistent with the size of each x-interval! In other words, as δ gets
smaller, you should also decrease the spacing between plotting points so that you can clearly resolve any interesting features
