Lab for Simpson's Quadrature Rule
Composite Simpson Quadrature Rule for Numerical Integration. To approximate the integral
by sampling f(x) at the 2m+1 equally spaced points .
Report to be handed in.
Computer project.
Consider the following function used by chemists .
Using the Fundamental Theorem of Calculus, we see that the value of the function
is the integral of over the integral 0 <= t <= x.
Use the composite Simpson rule to construct numerical approximations to
Exercise 1. First define f[t] and be sure to include the definition when t = 0.0
Exercise 2. Plot f[t] for 0 <= t <= 5 . Estimate the area under the curve y = f[t] for 0 <= t <= 5
Exercise 3. In order to apply Simpson's Rule it is desirable that f(t) be continuous. We defined f(0) = 0.
Did this make f(t) continuous at t = 0 ? Why ? Find .
Exercise 4. Use the composite Simpson rule with h = 0.5 and numerically approximate values for g(1), g(2), g(3), g(4) and g(5). Show the details for finding g(1) and g(2).
Use the composite Simpson rule to construct a numerical approximations to
.
Use the composite Simpson rule to construct a numerical approximations to
.
Now use the subroutine for the computations and numerically approximate values for g(1), g(2), g(3), g(4) and g(5).
Use the six points (0,g(0)), (1,g(1)), (2,g(2)), (3,g(3)), (4,g(4)), (5, g(5)) and plot a crude graph
of y=g(x).
Exercise 5. Use the composite Simpson rule with h = 0.25 and numerically approximate values for g(1), g(2), g(3), g(4) and g(5). Show the details for finding g(1) and g(2).
Use the composite Simpson rule to construct a numerical approximations to
.
Use the composite Simpson rule to construct a numerical approximations to
.
Now use the subroutine for the computations and numerically approximate values for g(1), g(2), g(3), g(4) and g(5).
Exercise 6. An accurate answer requires that many subintervals be used in the composite Simpson Rule.
Use the composite Simpson Rule and find numerically approximations for using the step sizes
h = 0.5, 0.25, 0.125, 0.0625, 0.03125
The approximations should improve when more subintervals are used.
Exercise 7. Assume that . Find the absolute errors for the above approximations.
Exercise 8. The remainder term for the composite Simpson rule is .
Does the absolute error found in 7. exhibit the pattern expected ? Why ?
Yes. Because the error decreases by approximately 1/16 when the step size is cut in half.
