A Computer Project
Part 1
Use (a) the left endpoint rectangle rule, (b) the midpoint rectangle rule,
(c) the trapezoid rule, (d) Simpson's rule,
and (e) the two-point Gaussian quadrature to compute the integral
with the number of subintervals of equal length
n = 1, ..., 10, respectively. Compute also all the corresponding absolute errors.
- Make a table of six columns with one for n and
the other five for the the computed values by the five rules, respectively.
Keep eight digits after decimal points.
- In a single plot, display five curves showing the absolute errors in the log-log scale
(i.e. log(error) vs. log(n))
for the five corresponding rules.
- Discuss convergence rates for these quadrature rules
based on the computational result.
Part 2
Use the trapezoid rule and Simpson's rule to compute the above integral
with the number of subintervals of equal length
n = 2k, k = 0, ..., 8.
Then, apply the Richardson extrapolation procedure
to the computed values corresponding to the trapezoid rule for the
pairs (2k-1,2k), k = 1, ..., 8.
Compute all the corresponding absolute errors.
- Make a table of four columns with one for n, one for the
computed values by the trapezoid rule, one for that by
the Richardson extrapolation, and one for that by Simpson's rule.
Keep eight digits after decimal points.
- In a single plot, display three curves showing in the log-log scale
the absolute errors (i.e. log (error) vs. log (n))
for the two corresponding rules and the Richardson extrapolation.
- Discuss the computational result in terms of convergence rates.