Numerical Analysis - Richardson Extrapolation on Riemann Sum

In summary, numerical analysis is a field of mathematics that uses algorithms and computer methods to solve mathematical problems. The Richardson extrapolation method is a numerical technique that improves the accuracy of approximations by using a sequence of approximations with different step sizes. In the context of Riemann sums, Richardson extrapolation involves combining two different step sizes to get a more precise result. It has the advantage of reducing computational time and resources, but may not work well for certain types of functions or with large step sizes.
  • #1
Graham87
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Homework Statement
Apply Richardson Extrapolation on a definite integral using Riemann sum.
Then prove that it has the same order of convergence as Trapezoidal rule.
Relevant Equations
Richardson Extrapolation
Riemann Sum
I got something like this, but I'm not sure it is correct, because if it has the same order of convergence as trapezoidal rule which is 2, it should yield the same result as trapezoidal rule but mine doesn't (?).

For example sin(x) for [0,1], n with trapezoidal rule = 0.420735...
With my own formula I get 0 or much above 0.4207.Cheers!
 

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  • #2
Having the same order of convergence does not mean that they will necessarily give the same result. Those are two different things. They should converge to the same value and at the same rate, but may not agree along the way.
 
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1. What is numerical analysis?

Numerical analysis is a branch of mathematics that deals with the development and implementation of methods for solving mathematical problems using numerical approximations.

2. What is Richardson extrapolation?

Richardson extrapolation is a numerical technique used to improve the accuracy of a numerical method by combining multiple approximations with different step sizes.

3. What is Riemann sum?

Riemann sum is a method for approximating the area under a curve by dividing the region into smaller rectangles and summing their areas.

4. How does Richardson extrapolation work on Riemann sum?

Richardson extrapolation on Riemann sum involves using a series of Riemann sums with different step sizes and then combining them to get a more accurate approximation of the area under the curve.

5. What are the applications of Richardson extrapolation on Riemann sum?

Richardson extrapolation on Riemann sum is commonly used in numerical integration, which has applications in various fields such as physics, engineering, and economics.

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