Math 151A Homework #5 – due Wednesday 11/29, in class
Show all your work!
1. Forward and backward differences
Suppose we use Newton’s divided differences on the points x0= 0, x1= 1 and x2= 2 to construct
the following matrix
0 0.
0
6 2 1
2
a. Fill in the missing elements.
b. Use the Newton foward difference formula to estimate f(0.1).
c. Use the Newton backward difference formula to estimate f(1.9).
2. Numerical differentiation
Let f(x) = ex. Approximate the value of f0(1) using:
a. The 3-point forward difference formula with h= 0.1
b. The 3-point backward difference formula with h= 0.1
c. The 3-point centered difference formula with h= 0.1
Which do you expect to be a better approximation, and why?
3. Richardson’s extrapolation (problem 9 of section 4.2)
Suppose that N(h) is an approximation to Mat every h > 0 and that
M=N(h) + K1h+K2h2+K3h3+. . . ,
for some constants K1,K2,K3, . . . . Use the values N(h), N(h/3), and N(h/9) to produce an
O(h3) approximation to M.
4. Numerical integration (problems 1a, 3a, 5a, 7a, 9a, and 11a of section 4.3)
Approximate the integral
Z1
0.5
x4dx
using
a. The trapezoidal rule
b. Simpson’s rule
c. The midpoint rule
For each of the above, find a bound for the error using the error formula, and compare this to the
actual error.