       Re: Find all minima, maxima and saddles

• To: mathgroup at smc.vnet.net
• Subject: [mg117683] Re: Find all minima, maxima and saddles
• From: "J.Frazer" <j.frazer at sussex.ac.uk>
• Date: Tue, 29 Mar 2011 06:59:57 -0500 (EST)

```Thanks for the tip but I can't get that approach to work. This is a D=2 example of the type of things I'm interested in:

d = 2;
trunc = 5;
mv = 1;
mh = Sqrt[8 =CF=80]*Sqrt[d*trunc];
vec=CF=86 = Table[Symbol["=CF=86" <> ToString[i]], {i, d}];
veck = Table[Symbol["k" <> ToString[i]], {i, d}];
vecj = Table[Symbol["j" <> ToString[i]], {i, d}];
=CF=83 = Exp[-(vecj.vecj)/(2 trunc)];
airrat = Table[RandomReal[NormalDistribution[0, =CF=83]], {j1, 5}, {j2, 5}=
];
birrat = Table[RandomReal[NormalDistribution[0, =CF=83]], {j1, 5}, {j2, 5}=
];
a = Table[Rationalize[airrat[[j1, j2]], 0], {j1, 5}, {j2, 5}];
b = Table[Rationalize[birrat[[j1, j2]], 0], {j1, 5}, {j2, 5}];
V = mv*Sum[
a[[k1, k2]] Cos[(veck.vec=CF=86)/mh] + b[[k1, k2]] Sin[(veck.vec=CF=86)/mh], {k1,
5}, {k2, 5}];

Plot3D[V, {=CF=861, -50, 50}, {=CF=862, -50, 50},
AxesLabel -> {"   SubscriptBox[\"=CF=86\", \"1\"] ",
"   SubscriptBox[\"=CF=86\", \"2\"] "}]

The plot approximately shows the size of box I would like to count over, though its not critical.

Cheers,

Jonny

On 24 Mar 2011, at 18:51, Daniel Lichtblau wrote:

> Could set all partial derivatives to zero and use Solve or Reduce to find=
all roots in your specified region. How well this works will depend on the=
>
> Daniel Lichtblau
> Wolfram Research
>
> ----- Original Message -----
>> From: "J.Frazer" <j.frazer at sussex.ac.uk>
>> To: mathgroup at smc.vnet.net
>> Sent: Thursday, March 24, 2011 6:29:51 AM
>> Subject: [mg117608] Find all minima, maxima and saddles
>> Hello,
>>
>> I imagine this is a well known problem but so far I can't spot an
>>
>> The problem is simply to find all minima, maxima and saddle points in
>> some finite region of a function in D variables where D< 10 or so.
>>
>> There is nothing particularly awkward about the functions I have in
>> mind, nor do I expect vast numbers of maxima and minima, 10s or maybe
>> 100s, not thousands.
>>
>> Is there a simple way of doing this?
>>
>> Cheers,
>>
>> Jonny

```

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