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MathGroup Archive 1996

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Re: Re: NIntegrate

  • Subject: [mg3008] Re: [mg2987] Re: NIntegrate
  • From: richard at seuss.math.wright.edu (Richard Mercer)
  • Date: 22 Jan 1996 05:55:47 -0600
  • Approved: usenet@wri.com
  • Distribution: local
  • Newsgroups: wri.mathgroup
  • Organization: Wolfram Research, Inc.
  • Sender: mj at wri.com

>   as I recall, NIntegrate approximates an integral in
>  exactly the same way that a person would, through
>  trapezoidal approximation.
>  

>   so for the function f[x] it selects  points on f[x] and
>  finds the area of all the trapizoids formed by connecting
>  these points to eachother and the x axis over the
>  specified range of x {x,a,b}.
>  

>  Mathmatica continues to refine these trapezoids by making
>  them thinner, and there by getting a more accurate
>  approximation. It continues to refine the approximations
>  untill it reaches the prescribed accuracy.

I hope not! The trapezoidal approximation is not very accurate, in fact  
less accurate than Riemann sums using midpoints. Even as a "person", I  
would use Simpson's Rule, which is much more accurate for the same  
amount of work.

At the very least NIntegrate would use an adaptive Simpson's Rule,  
using smaller intervals where the function changes rapidly. I suspect  
(but do not know) it uses more sophisticated methods as well.

Richard Mercer


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