Re: How can I use FindMaximum to get a result better than MachinePrecision?

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• Subject: [mg127958] Re: How can I use FindMaximum to get a result better than MachinePrecision?
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Wed, 5 Sep 2012 03:09:10 -0400 (EDT)
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• References: <20120904095037.527DA6332@smc.vnet.net>

```expr = 8 E^(-x) Sin[x] - 1;

FindMaximum[expr, {x, 0, 8}] //
InputForm

{1.5791755355586754, {x -> 0.7853981516735594}}

FindMaximum[expr, {x, 0, 8},
AccuracyGoal -> 20, PrecisionGoal -> 20] //
InputForm

{1.5791755355586754, {x -> 0.7853981553245005}}

Precision[%]

MachinePrecision

>From documentation for AccuracyGoal:
"In most cases, you must set WorkingPrecision to be at least as large
as AccuracyGoal. "

>From documentation for PrecisionGoal:
"In most cases, you must set WorkingPrecision to be at least as large
as PrecisionGoal. "

FindMaximum[expr, {x, 0, 8},
AccuracyGoal -> 20, PrecisionGoal -> 20,
WorkingPrecision -> 20] // InputForm

{1.579175535558675594017576308550564195414708583669983385\
2472`20., {x -> 0.785398163420404303959134242845118197357\
3041737258458441009`20.}}

Precision[%]

20.

>From documentation for WorkingPrecision:
"Even if internal computations are done to n-digit precision, the
final results you get may have much lower precision. "

FindMaximum[expr, {x, 0, 8},
WorkingPrecision -> 20] // InputForm

{1.579175535558675594017576301475807582647461879911073128\
3986`20., {x -> 0.7853981634204044}}

Precision[%]

MachinePrecision

FindMaximum[expr, {x, 0, 8},
WorkingPrecision -> 25] // InputForm

{1.579175535558675594018935476273188075390077304522331264\
0472`25., {x -> 0.785398163397457286632920167624848350070\
0473014719814661246`25.}}

Precision[%]

25.

Bob Hanlon

On Tue, Sep 4, 2012 at 5:45 AM, David Kirkby <drkirkby at gmail.com> wrote:
> I've tried this:
>
> In[2]:= FindMaximum[8 E^(-x) Sin[x] -1,{x,0,8}]
>
> Out[2]= {1.57918, {x -> 0.785398}}
>
>
> Then played around to try to get a more accurate result.
>
> In[9]:= FindMaximum[8 E^(-x) Sin[x] -1,{x,0,8},AccuracyGoal->20, PrecisionGoal->20]
>
> Out[9]= {1.57918, {x -> 0.785398}}
>
> In[10]:= Precision[%]
>
>
> Also:
>
>
> In[7]:= N[FindMaximum[8 E^(-x) Sin[x] -1,{x,0,8},AccuracyGoal->200, PrecisionGoal->200],100]
>
> Out[7]= {1.57918, {x -> 0.785398}}
>
> In[8]:= Precision[%]
>
> Out[8]= MachinePrecision
>
>
> Any suggestions?
>

```

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