       Error estimates in the Gauss-Kronrod rule

• To: mathgroup at smc.vnet.net
• Subject: [mg127010] Error estimates in the Gauss-Kronrod rule
• From: João Paulo Pereira <joaopereira9 at gmail.com>
• Date: Sat, 23 Jun 2012 04:17:29 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hi there,

I need some information regarding the Gauss-Kronrod rule data provided by
Mathematica. When we write  NIntegrate`GaussKronrodRuleData[n, precision],
we get the gauss-kronrod nodes, weights and an error estimate. In this link
http://reference.wolfram.com/mathematica/tutorial/NIntegrateIntegrationRules.html
it
is explained how this rule can be applied through the example:

Total@MapThread[{f[#1] #2, f[#1] #3} &, {absc, weights, errweights}]
understand
with  {absc, weights, errweights} = NIntegrate`GaussKronrodRuleData[n,
precision]

It also explains that "The command (...) implements the integration rule
weighted sums for the integral estimate, , and the error estimate, , where are
the abscissas,  are the weights, and  are the error weights."

I know how to compute the nodes and I do get the same values as
Mathematica's but I haven't been able, so far, to figure out how the  are
computed. I need to understand this for the program I'm writing and it
seems Mathematica's Help does not provide any explanation on the subject.

Can someone point me to the reference  (article, etc ) from which I can
understand how this Methematica command computes the error weights
estimates?