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Problem 1
Consider the polynomial interpolation for the following data points
x 0 2 3 4
y 7 11 28 63
(a). Write down the linear system in matrix form for solving the coecients
ai (i = 0, ··· , n)
of the polynomial pn(x).
(b). Use the Lagrange interpolation process to obtain a polynomial to approximate these data
points.
Problem 2
The polynomial p(x) = x4
x3 + x2
x + 1 has the values shown.
x -2 -1 0 1 2 3
p(x) 31 5 1 1 11 61
Find a polynomial q(x) that takes these values (you don’t need expand it):
x -2 -1 0 1 2 3
q(x) 31 5 1 1 11 30
(Hint: This can be done with little work. Try the Lagrange form.)
Problem 3
Let P3(x) be the interpolating polynomial for the data (0, 0), (0.5, y), (1, 3) and (2, 2). Find y if
the coecient
of x3 in P3(x) is 6.
Matlab Problem 1
Ccompute the numerical derivative of f(x) = xex on [0, 1] by using the formula below.
Write a matlab code to test the convergence order numerically (Please hand in your code).
Matlab Problem 2
Consider the polynomial interpolation on the interval [1,
1] with two types of f(x):
f1(x) = cos(x), f2(x) = 1
1 + x2 .
Write a matlab script for computing the error of polynomial interpolations of fi(x), and fill
Errn for di↵erent polynomial interpolations in the following table. The error of polynomial
interpolation is defined as
En = kpn(x)
f(x)k
where x is a vector representing the uniform grid points on [1,
1].
Hint: Using the element-wise division ./ and the element-wise power .^.
What to hand in? Your script file to get the results
1
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