       Re: Question about Solve and Reduce (mathematica 8)

• To: mathgroup at smc.vnet.net
• Subject: [mg114530] Re: Question about Solve and Reduce (mathematica 8)
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 7 Dec 2010 06:44:14 -0500 (EST)

```On 6 Dec 2010, at 03:53, olfa wrote:

> Hi Mathematica Community,
>
> 1)now that Solve can handle inequalities in Mathematica 8, what is the
> main difference between Solve and Reduce in finding solutions for a
> given variable(s)?
> 2)is it possible to let Solve give solutions not in terms of rules of
> the form x->sol but as it is displayed by Reduce?
>
> Thank you very much.
>

There are a number of differences. For example, Solve will return {{}} when the set of solutions is full dimensional:

Solve[Sqrt[x+1]<1-Sqrt[3x^3+1],x,Reals]
During evaluation of In:== Solve::fulldim: The solution set contains a full-dimensional component; use Reduce for complete solution information.
>>
{{}}

while Reduce will give you the complete solution

Reduce[Sqrt[x+1]<1-Sqrt[3x^3+1],x,Reals]
Root[3 #1^3+1&,1]<==x<Root[9 #1^6-6 #1^4-6 #1^3+#1^2-2 #1-3&,1]

Also, Solve only finds generic solutions, so:

Solve[a^2 x^2>==a^2 (x^2+1),x,Reals]
{}

while

Reduce[a^2 x^2>==a^2 (x^2+1),x,Reals]
a====0

There are other differences too. On the whole Solve has the advantage in speed, Reduce in the completeness of the solutions it returns.

Andrzej Kozlowski

```

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