Matlab代写|数学代写 - Math481
Problem 1: Analysis of performance of the Richardson extrapolation.
• Prepare a function implementing the Composite Trapezoidal Rule for calculating the
√111x2 dx. Keep your programming so that you can provide the parameter,
h, the size of the interval in the Composite Trapezoidal Rule.
• Calculate approximations of the integral by the Composite Trapezoidal rule using step
sizes h = 0.01, h = 0.005, h = 0.0025, h = 0.00125, h = 0.000625. Compare to the
exact value of the integral. Arrange results in a table:
step size, h Comp. Trap. Rule, S(h) absolute error, red. factor, m order, α
0.01 . . . . . .
0.005 . . . . . . . . . . . .
0.0025 . . . . . . . . . . . .
0.00125 . . . . . . . . . . . .
0.000625 . . . . . . . . . . . .
Table 1. Convergence of the composite trapezoidal method in evaluation of R00.99
Here the error reduction factor is computed as m = 1/2, where 1 and 2 are taken
from two consecutive rows in the third column. Also, the order α is computed as
α = ln(1/2)/ ln 2. (HINT: You may want to use an Excel or similar spreadsheet to
prepare the tables)
Do results of your simulations agree or disagree with the error formula in Theorem 4.5?
In particular, if h is reduced to h/2, how much reduction in error is expected? How
much reduction is observed?
• Develop a formula for the third order Richardson approximation of the integral based on
the Composite Trapezoidal Rule. For that, start from an assumption that the Composite
Trapezoidal Rule S(h) satisfies the following error expansion,
Z0 0.99 1 √1 鯦 x2
dx x S(h) = c2h2 + c3h3 + . . .
Use ideas of Section 4.2 to derive the third order Richardson approximation
(1) M3(h) = S(h/2) + S(h/2) ) S(h) 3 . 1
2 • Use results obtained by the Composite Trapezoidal Rule and the third order Richardson
interpolation (1) to populate the following table
step size, h Rich. Extrp. M3(h) error, Rich. Extrp., e red. factor, m order, α
0.01 . . .
0.005 . . . . . . . . .
0.0025 . . . . . . . . .
0.00125 . . . . . . . . .
Table 2. Convergence of the composite trapezoidal method in evaluation of R00.99
Please note that in this table the absolute error is computed by comparing results of
Richardson extrapolation and the exact integral.
• Compare errors for the Composite Trapezoidal Rule for h = 0.00125 and for Richardson
extrapolation for h = 0.01. Are they similar or different? Which of the two formulas
is more efficient to use: Composite Trapezoidal Rule for h = 0.00125 or Richardson
extrapolation for h = 0.01. Explain your answer.
• Write a report. Include the above tables and any other results you want to share with
comments. Include the derivation of formula (1). Attach the code.