349A ASSIGNMENT #6

1. (a) (2 points) Determine the Lagrange form of the interpolating polynomial P(x)
that interpolates a function f(x) at x = 0, 2h and 3h, where h 0. (Multiply the
linear factors together, but leave P(x) as a sum of 3 quadratics in the variable x.)
DELIVERABLES: All your work in constructing the polynomial. This is to be
done by hand not MATLAB.
(b) (2 points) Derive the quadrature formula of the form
a0f(0) + a1f(2h) + a2f(3h)
for approximating I =
R 3h
0
f(x)dx that results from approximating the integral I
by I ≈
R 3h
0
P(x)dx.
Note: if you know only 3 function values of f(x) and they are at 3 unequallyspaced points 0, 2h and 3h, then this kind of quadrature formula can be used to
approximate I.
DELIVERABLES: All your work in deriving the quadrature formula.
(c) (2 points) Suppose that you know only the following function values of f(x):
x f(x)
0.0 0.5
0.24 0.50727
0.36 0.51656
Use the quadrature formula from (b) to approximate R 0.36
0
f(x)dx.
Note: the above data corresponds to the function f(x) = 1/(1 + cos(x)), and
the exact value is about R 0.36
0
f(x)dx = 0.1819695.... Use this information only
to assess the accuracy of your computed approximation. If you do not obtain a
fairly good approximation in (c), then your answer in (b) is incorrect. That is,
the relative error between your answer and the true answer should be way less
than 1%.
DELIVERABLES: All your work to calculate the quadrature.
2. (a) (4 points) Determine the degree of precision of the quadrature formula
5
9
f

−
r
3
5
!
+
8
9
f(0) + 5
9
f
r
3
5
!
for approximating
Z 1
−1
f(x)dx.
Use the definition of ”degree of precision” to do this, and show all of your work.
See Handout #27. Note that the interval of integration here is fixed at [−1, 1].
Use exact arithmetic here – do not use a numeric approximation for p
3/5.
(b) (2 points) Use the quadrature formula in (a) to approximate
Z 1
−1
e
−x
√
x + 2dx.
2
Either use your calculator or MATLAB to do the computations for evaluating the
quadrature formula (use at least 7 significant digits).
Note: the exact value of this integral is 3.01747... Your computed approximation
based on a sampling of f(x) at only three points is remarkably accurate (as this
quadrature formula has a fairly high degree of precision).
DELIVERABLES: Show all your work.
3. (a) (4 points) Fill in the blanks in the following MATLAB function M-file trap
so that it implements the composite trapezoidal rule, using a sequence of m =
1, 2, 4, 8, 16, . . . trapezoids, to approximate R b
a
f(x)dx.
This M-file uses the MATLAB built-in function trapz. If
x = (x1, x2, . . . , xm+1) and y = (f(x1), f(x2), . . . , f(xm+1)),
then the execution of
z = trapz(x, y)
approximates
Z xm+1
x1
f(x)dx
using the composite trapezoidal rule (with m trapezoids). (Note that the entries
of x are labeled starting at x1 because MATLAB does not allow an index of 0.)
The function M-file trap has 4 input parameters:
• a and b (the lower and upper limits of the integral);
• maxiter (the maximum number of iterations that are allowed; the above
program will implement the composite trapezoidal rule with a sequence of
1, 2, 4, 8, 16, . . . trapezoids, up to a maximum of 2maxiter trapezoids);
• tol (a tolerance for testing the relative error of the computed approximation).
If the difference between 2 successive approximations (with respect to relative
error) is < tol, then the algorithm terminates.
You also must define and store a MATLAB function M-file
y = f(x)
which must be written to accept a vector x as the argument. Note that this means
that you must use arithmetic operators like .* , .^ , ./ and so on, rather than
just *, ^, /.
function trap(a, b, maxiter, tol)
m = 1;
x = linspace(a, b, m+1);
y = f(x);
approx = trapz(x, y);
disp(’ m integral approximation’);
fprintf(’ %5.0f %16.10f \n ’, m, approx);
3
for i = 1 : maxiter
m = ________________________________ ;
oldapprox = __________________________________ ;
x = linspace ( _______ , ________ , __________ ) ;
y = f(x);
approx = trapz(x, y);
fprintf(’ %5.0f %16.10f \n ’, m, approx);
if abs( _______________________________ ) < tol
return
end
end
fprintf(’Did not converge in %g iterations’, maxiter)
DELIVERABLES: A copy of your function trap. Feel free to add an extra
parameter f at the end of parameter list to make your function calls simpler.
(b) (4 points)
Use the above MATLAB script to approximate the following two integrals:
Z 3
0.1
sin(1/x)dx using maxiter = 20 and tol = 10−6
,
Z 1
0
e
3x
√
x
3 + 1
dx using maxiter = 20 and tol = 10−10
.
DELIVERABLES: The MATLAB calls and output for each. If you use function
files to evaluate each of the functions I also want the .m files for those.
4