Re: Interpretation of PrecisionGoal and AccuracyGoal

• To: mathgroup at smc.vnet.net
• Subject: [mg80850] Re: Interpretation of PrecisionGoal and AccuracyGoal
• From: Chris Chiasson <chris.chiasson at gmail.com>
• Date: Tue, 4 Sep 2007 03:44:17 -0400 (EDT)
• References: <fb39l4\$q54\$1@smc.vnet.net>

```On Aug 29, 3:10 am, "Andrew Moylan" <andrew.j.moy... at gmail.com> wrote:
> 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?

I am interested in knowing this also.

The documentation seems to be self-contradictory because it says:

Under ref/AccuracyGoal (and ref/PrecisionGoal):

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

...

"Find a minimum with convergence criteria |x_k-xstar| <= max(10^-10,|
x_k| 10^-8) and del f(x_k) <= 10^-10"

"f[x_]=Sin[x^2];"

"FindMinimum[f[x],{x,2},AccuracyGoal->10,PrecisionGoal->8]"

Under ref/FindMinimum:

"FindMinimum continues until either of the goals specified by
AccuracyGoal or PrecisionGoal is achieved."