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Interpretation of PrecisionGoal and AccuracyGoal

• To: mathgroup at smc.vnet.net
• Subject: [mg80684] Interpretation of PrecisionGoal and AccuracyGoal
• From: "Andrew Moylan" <andrew.j.moylan at gmail.com>
• Date: Wed, 29 Aug 2007 04:10:08 -0400 (EDT)

The article "ref/PrecisionGoal" in the Mathematica documentation says:

----
With PrecisionGoal->p and AccuracyGoal->a, Mathematica attempts to make the
numerical error in a result of size x be less than (10^-a)+|x| 10^-p.
----

On the other hand, the article at
"tutorial/NIntegrateIntegrationStrategies#280704764" says:

----
"GlobalAdaptive" stops if the following expression is true:
globalError<=globalIntegral 10^-pg \[Or] globalError<=10^-ag
----

So GlobalAdaptive has a different interpretation of the meaning of
PrecisionGoal and AccuracyGoal than that given in the article
"ref/PrecisionGoal".

Which of these two interpretations is dominant in Mathematica's actual
numerical methods? Or does it vary from method to method?