Math 4340 – Numerical Methods
Homework 6.1
Due Thursday, Sept 17, 2015
Instructions: Check your finite difference expressions for #4, 5 and 6, using the standard
Taylor’s series chart.
1. Let f ( x)  ln( x) . Create a chart with the following information at x=2.3, using the
central difference approximation. Compute an error bound for the truncation error using
the observation that the absolute value of all of the derivatives of ln( x ) are monotonically
decreasing for x>0. Thus, we will make the assumption that f ' ' ' (c)  f ' ' ' (2.2) and
show that the actual error is less than the error bounds. Use Richardson extrapolation to
determine an improved estimate based on these results.
N (h)
h
N RE (h)
E(h)  N (h)  Exact
0.1
0.01
0.001
2. Let f ( x)  e x . Create a chart with the following information at x=2.3, using the
fourth-order central difference approximation. Compare the results from this calculation
with the results from problem 1. Compute an error bound for the truncation error using
the assumption that f (5) (c)  f (5) (2.4) and show that the actual error is less than the
error bounds.
h
N (h)
E(h)  N (h)  Exact
0.1
0.01
0.001
3. The distance D  D (t ) traveled by an object is given in the following table
t
D(t)
12.0
61.41426
12.5
58.57686
13.0
55.73545
13.5
52.89012
14.0
50.04098
a. Find the velocity V(t) of the particle at t=13.0 , using the double forward difference
with h=0.5, using the central difference approximation with h=0.5, using the central
difference approximation using h=1 and using the fourth order central difference
approximation using h=0.5.
b. Compare your answers with the velocity obtained from the generating function
D(t )  140  6t  12 e t / 20 by calculating the error between the exact velocity at t=13.0
with the numerical approximations.
4. By considering the Taylor’s series expansion of the finite difference expression for the
second derivative, combine the two expressions shown below to generate a finite
difference expression for the fourth derivative
f x  h   2 f x   f x  h 
f  x  2h   2 f  x   f  x  2h 
2
h
( 2h) 2
Check your answer by calculating the leading order error term for the new formula.
5. Combine the following finite difference expressions to generate a third-derivative
approximation:
 f  x  2h   4 f  x  h   3 f  x 
 f  x  4h   4 f  x  2h   3 f  x 
2h
4h
6. Determine what the following finite difference expressions approximate. What are the
orders of accuracy and what are the leading order error terms?
f  x  h   3 f  x   3 f  x  h   f  x  2h 
h3
f  x  2h   2 f  x  h   2 f  x  h   f  x  2h 
b.
2h 3
f  x  2h   4 f  x  h   6 f  x   4 f  x  h   f  x  2h 
c.
h4
a.
7. Improve the order of accuracy of the finite difference expression in 6.a. by using
Richardson extrapolation and calculate the order of accuracy and leading order error term
of the new formula.