MATH 128A, SUMMER 2009: PROGRAMMING ASSIGNMENT 1
In this assignment, we will use Newton’s divided difference algorithm to approximate the function cos(x)
by a Lagrange polynomial in divided-difference form.
There are three steps involved. The first step is to write a general function that computes the divided-
difference coefficients ak=f[x0, . . . , xk] of the Lagrange interpolating polynomial, given as input a vector
containing x0, . . . , xn. The second step will be to evaluate the Lagrange interpolating polynomial (in its
divided-difference form Pn(x) = Pn
k=0 f[x0,...xk]Qk
j=0(x−xj)) at a few sample points. The third step
will be to use the MATLAB plot function to graph the interpolating polynomial for the Cosine function,
together with the Cosine function on the same set of axes over an appropriately-chosen interval.
We begin by giving some example code. The following recursive MATLAB function will evaluate a0+
a1(x−x0) + · · · +an(x−x0). . . (x−xn−1), given as input vectors containing a0, . . . , anand x0, . . . , xn.
When n > 0, it works by evaluating
a0+ (x−x0)[a1+a2(x−x1) + · · · +an(x−x1). . . (x−xn−1)].
function y=newton_eval(x,ai,xi)
if length(xi)==1
y = ai(1);
else
y = ai(1) + (x-xi(1)) .* newton_eval(x,ai(2:end),xi(2:end));
end
1. Write a recursive function y=divided_difference(f, xi), which computes f[xi(1),...,xi(end)]
directly from its definition:
f[x0] = f(x0)
f[x0, . . . , xn] = f[x1,. . . , xn]−f[x0, . . . , xn−1]
xn−x0
,if n > 0
Note: The algorithm given in section 3.2 is more efficient.
2. Create a set of 6 equally-spaced nodes in the interval [0,10]:
xi = linspace(0, 10, 6);
Calculate a vector ai, where ai(k) =f[xi(1),...,xi(k)].
Use newton_eval to evaluate the interpolating polynomial at x=π/3, x=π/2, and x=π. Give
these estimates, and the absolute error, accurate to 6 significant digits.
3. Plot y= cos(x) and the interpolating polynomial y=P(x) on the same set of axes:
xdata = linspace(0, 10);
ydata1 = newton_eval(xdata, ai, xi);
ydata2 = cos(xdata);
plot(xdata, ydata1, xdata, ydata2)
Your report should contain your code for divided_difference, the MATLAB commands you used, the
numerical results from problem 2, and the graph from problem 3.
Date: Due Monday 7/20.
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