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gauss-kronrod rule data

  • To: mathgroup at smc.vnet.net
  • Subject: [mg131422] gauss-kronrod rule data
  • From: Alex Krasnov <akrasnov at cory.eecs.berkeley.edu>
  • Date: Sun, 21 Jul 2013 21:39:56 -0400 (EDT)
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This question has already been asked on this list but received no answer. 
NIntegrate`GaussKronrodRuleData takes number of points and precision, and 
returns three lists as follows:

In:	{abscissas, weights, errorweights} = GaussKronrodRuleData[3, 3]
Out:	{{0.020, 0.113, 0.283, 0.500, 0.717, 0.887, 0.980},
 	 {0.0523, 0.134, 0.201, 0.23, 0.201, 0.134, 0.0523},
 	 {0.0523, -0.144, 0.201, -0.219, 0.201, -0.144, 0.0523}}

The first and second lists contain the abscissas and corresponding 
weights, respectively. The third list supposedly contains the weights for 
error estimation. We have the following relationship:

In:	errorweights/weights
Out:	{1.00, -1.07, 1.00, -0.97, 1.00, -1.07, 1.00}

For a Gauss-Kronrod rule, the error is typically estimated as the 
difference between the integral result from the extended Kronrod rule and 
that from the base Gauss rule. Therefore, we expect to see a list of 1 
interleaved with a list of -1 as follows:

In:	errorweights/weights
Exp:	{1.00, -1.00, 1.00, -1.00, 1.00, -1.00, 1.00}

Is there an explanation for the small deviation from -1 in the actual 
result?

Alex



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