numerical integration algorithm details

05312019, 04:12 AM
Post: #1




numerical integration algorithm details
A few years ago Bernard Parisse shared this link to an article that explains the numerical integration algorithm used by the Prime. The algorithm described works quite well. I'm curious about the mathematical reasons behind a couple of decisions made in the algorithm.
The method uses a standard 15point Gaussian Quadrature and then a 14point and a 6point calculation to estimate the error. The 14 and 6 point calculations reuse nodes from the 15point method but with different weights. I've tried to show this visually below using X to represent the nodes used. (df=degrees of freedom) Code: XXXXXXXXXXXXXXX  15 nodes, df=30, exact for degree 29 polynomial My question relates to why 6 nodes were chosen. As the least accurate part of the calculation it seems like you'd want to use more nodes. Using 7 nodes seems more natural. Why skip the middle node when you've already skipped the two adjacent nodes? The original 15 nodes are already distributed nonuniformly towards the endpoints. Code: X X X X X X X  7 nodes, df=7, exact for degree 6 polynomial Or why not use even more, say 13 nodes Code: XXXXXX X XXXXXX  13 nodes, df=13, exact for degree 12 polynomial So my question boils down to, "Why 6?" My second question pertains to the error calculation. Using ERR1 as the difference between the 15 and 14 point calculation, and ERR2 as the difference between the 15 and 6 point calculation, the error is estimated as error = ERR1*(ERR1/ERR2)^2. This seems to work well, but I don't see how it was derived. Why squared? Or perhaps it was simply found by experimenting and they found that this worked reasonable well. Thanks. 

05312019, 06:10 AM
Post: #2




RE: numerical integration algorithm details
I think the reason it works well is because the error ERR1*(ERR1/ERR2)^2 is in h^30, where h is the step, like the most accurate method theoretical error.


05312019, 11:22 AM
(This post was last modified: 05312019 11:22 AM by yangyongkang.)
Post: #3




RE: numerical integration algorithm details
The numerical integration algorithm is introduced in the Wolfram reference documentation.
* NIntegrate uses symbolic preprocessing to resolve function symmetries, expand piecewise functions into cases, and decompose regions specified by inequalities into cells. *With Method>Automatic, NIntegrate uses "GaussKronrod" in one dimension, and "MultiDimensional" otherwise. *If an explicit setting for MaxPoints is given, NIntegrate by default uses Method>"QuasiMonteCarlo". *"GaussKronrod": adaptive Gaussian quadrature with error estimation based on evaluation at Kronrod points. *"DoubleExponential": nonadaptive doubleexponential quadrature. *"Trapezoidal": elementary trapezoidal method. *"Oscillatory": transformation to handle certain integrals containing trigonometric and Bessel functions. *"MultiDimensional": adaptive Genz\[Dash]Malik algorithm. *"MonteCarlo": nonadaptive Monte Carlo. *"QuasiMonteCarlo": nonadaptive Halton\[Dash]Hammersley\[Dash]Wozniakowski algorithm. study hard, improve every day 

05312019, 01:26 PM
Post: #4




RE: numerical integration algorithm details
In certain parts we can not compare the symbolic calculation kernel and others of Xcas with the Mathematica Kernel, Wolfram has several brains working for it. XCas only BP and some other volunteer. Enthusiasts are sought to increase the power of Xcas, for example, that Xcas can read external data from the console like most languages.
contributions to Giac computer algebra system https://github.com/marohnicluka/giac Quote:marohnicluka contributions to Giac/Xcas computer algebra system, including: SBASIC > https://smallbasicpublicwebsite.azurewebsites.net/ 

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