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http://tinyurl.com/33pb6gq
I think this is a fairly stunning illustration of the way in which the
difficulty of solving a polynomial equation skyrockets as the degree of
the equation increases.
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Invisible wrote:
> http://tinyurl.com/33pb6gq
Actually, you know what?
http://tinyurl.com/3686x2y
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Le 10/08/2010 13:56, Invisible a écrit :
> Invisible wrote:
>> http://tinyurl.com/33pb6gq
>
> Actually, you know what?
>
> http://tinyurl.com/3686x2y
Solving Third & Fourth degres polynomials is known since a long time (I
have a book with that which was printed in the 60')
Now, the sad thing is it soon stop working for fifth and higher.
--
A: Because it messes up the order in which people normally read text.<br/>
Q: Why is it such a bad thing?<br/>
A: Top-posting.<br/>
Q: What is the most annoying thing on usenet and in e-mail?
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Le_Forgeron wrote:
> Solving Third & Fourth degres polynomials is known since a long time (I
> have a book with that which was printed in the 60')
I'll bet to hell nobody actually *uses* the formulas above though. (!)
> Now, the sad thing is it soon stop working for fifth and higher.
I thought it only stops working in terms of "elementary" functions? As
in, there are more specialised functions you can use to go a little bit
higher still.
Also, I thought it was a case of being impossible to write a single
closed-form formula for the solution to an arbitrary high-order
polynomial. Like, if the polynomial has a special form, it might still
be solvable (possibly easily).
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Invisible a écrit :
> Le_Forgeron wrote:
>
>> Solving Third & Fourth degres polynomials is known since a long time (I
>> have a book with that which was printed in the 60')
>
> I'll bet to hell nobody actually *uses* the formulas above though. (!)
>
Probably not. If you consider that most - if not all - computers
calculate square roots by using an iterative method, it would probably
be faster to use it on the original polynomial than to use it 8 times on
smaller polynomials, as in the solution you linked to.
--
/*Francois Labreque*/#local a=x+y;#local b=x+a;#local c=a+b;#macro P(F//
/* flabreque */L)polygon{5,F,F+z,L+z,L,F pigment{rgb 9}}#end union
/* @ */{P(0,a)P(a,b)P(b,c)P(2*a,2*b)P(2*b,b+c)P(b+c,<2,3>)
/* gmail.com */}camera{orthographic location<6,1.25,-6>look_at a }
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Le 10/08/2010 14:17, Invisible a écrit :
> Le_Forgeron wrote:
>
>> Solving Third & Fourth degres polynomials is known since a long time (I
>> have a book with that which was printed in the 60')
>
> I'll bet to hell nobody actually *uses* the formulas above though. (!)
>
Humans have a tendancy to make substitution of variables instead.
>> Now, the sad thing is it soon stop working for fifth and higher.
>
> I thought it only stops working in terms of "elementary" functions? As
> in, there are more specialised functions you can use to go a little bit
> higher still.
>
> Also, I thought it was a case of being impossible to write a single
> closed-form formula for the solution to an arbitrary high-order
> polynomial. Like, if the polynomial has a special form, it might still
> be solvable (possibly easily).
Yes, of course x^9 + a x^6 + b x^3 + c = 0 is just a rewrite of y^3 + a
y^2 + b y + c = 0, with y = x^3.
Solving fifth and higher requiers usually to get a first root, then
dividing the polynomial by (x-root) to reduce the degre (and start
over). Sometimes it is possible to extract a second degre polynomial
instead... that is where all the "(a+b)²=a²+2ab+b²" and the like come
handy when trying to move to the formula to a product instead of a sum)
--
A: Because it messes up the order in which people normally read text.<br/>
Q: Why is it such a bad thing?<br/>
A: Top-posting.<br/>
Q: What is the most annoying thing on usenet and in e-mail?
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Le_Forgeron <lef### [at] free fr> wrote:
> Now, the sad thing is it soon stop working for fifth and higher.
More info on that subject:
http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
--
- Warp
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Warp wrote:
> More info on that subject:
>
> http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem
Hmm, interesting. I thought the theorum states that there is no single
formula that covers all possible polynomials of a given degree. However,
Wikipedia asserts something far stronger: It seems to claim that you can
construct a single, fixed polynomial who's solutions (which are also
fixed) cannot be expressed by radicals.
This indicates that there are some algebraic numbers that are not
expressible by radicals. (And yet, the *definition* of an algebraic
number is one expressible by radicals...)
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Am 10.08.2010 13:56, schrieb Invisible:
> Invisible wrote:
>> http://tinyurl.com/33pb6gq
>
> Actually, you know what?
>
> http://tinyurl.com/3686x2y
Yeah, sure. I recite that daily from memory before I go to sleep...
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clipka wrote:
> Yeah, sure. I recite that daily from memory before I go to sleep...
Now there's a question: How do you unambiguously recite complex
mathematical expressions out loud?
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