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General Solution for Polynomial Equations?
Posted by
michael
on Fri Sep 10, 2004 07:56 AM
from the higher-order-geekdom dept.
from the higher-order-geekdom dept.
An anonymous reader writes "On september 9, several media reported that a young Dutch student found a formula to determine the roots of any polynomial equation. Does this conflict with Abel's proof that such a formula cannot exist? Here is the news item (in Dutch) on his school's homepage." Another reader writes "A Dutch student at the Fontys school of physics has solved a math problem of several centuries old: finding the roots of any polynomial equation. Arxciv copy here. Although an exact solution has been proven impossible for higher orders, this is not the case for numeric solutions."
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The fish (Score:5, Informative)
Re:The fish (Score:5, Funny)
Parent
Re:The fish (Score:5, Funny)
I'm guessing your degree was for English Mathematics. This paper is clearly in Dutch Mathematics (so, just head over to Babelfish).
Parent
Re:The fish (Score:5, Funny)
At least it isn't in Polish Mathematics. Not only would it be difficult to decipher, you'd also have to read it backwards.
Parent
Re:The fish (Score:5, Funny)
Parent
Re:The fish (Score:5, Funny)
Parent
Re:The fish (Score:4, Funny)
Parent
slashdot walkabout (Score:5, Funny)
1) put poly in standard form and take the first n-1 derivatives.
2) put the derivatives in terms of x(s) (for 1..n-1), or remember why you dropped calculus and goto step 9.
3) Use the derivatives to write a differential equation with coefficients m1..mn, or remember why you dropped differential equations and goto step 9.
4) Use the original equation to reduce the differential equation to order n, and note the use of "then" instead of "than" in the mit write-up. (sorry, mit).
5) Substitute a formula for x(s), multiply resulting eq by it's denominator, getting another diffEq. Whee! ask a Grad student.
6) Now substitute a power series representation. All 's' should be zero. (mutter: Aha! I knew it) Solve b_sub_i for 1..n-2 (Grad student).
7) Substitute another power series to get an equation. (The grad students are gone, ask your hallmates, one of 'em has to be a math major.)
8) Let b_sub_n-1 equal the determinant of a funky, unexplained matrix (here, have an aspirin).
9) Everyone else in the class is out drinking by now, so don't worry about the next matrix, it's even funkier. Write a note on your hand to memorize it this weekend. Go drinking with peers.
10) Wake up at 3pm tomorrow, and try to remember what the hell all those squiggles meant.
11) Change your minor from math to polisci. Don't worry about taking Calc 1-3, DiffEq, or linear algebra. Note: many girls do not care about the roots of arbitrary polynomials, so no worries there. 8^)
Parent
Right in the middle of my Calc class too... (Score:5, Funny)
Dang it, that means I'll have to buy a new math book for this quarter's Calc class, won't I?
Ah, the world, she is a changin'...
Re:Right in the middle of my Calc class too... (Score:5, Insightful)
Abel's proof showed that polynomials with a degree higher than 4 could not be solved algebraically (i.e through a finite number of additions, subtractions, multiplications, etc.). Abel's proof did no say it was impossible to solve the equations (indeed, numerical solutions to these equations are solved regularly).
This is similar to how some integral equation solutions cannot be expressed in simple terms. However numerical answers are rather easy to obtain (even easier with a computer)
The method presented is a simpler way to find the roots of polynomial equations numerically by treating it like a power series (x, x^1, x^2,...,x^n) and applying standard differential techniques.
Pretty cool if you ask me.
~X~
Parent
Re:Right in the middle of my Calc class too... (Score:5, Interesting)
Parent
I wonder if anyone will get this... (Score:5, Funny)
Less Drang.
Parent
Re:Right in the middle of my Calc class too... (Score:5, Funny)
Parent
Re:Right in the middle of my Calc class too... (Score:5, Informative)
Parent
There's no conflict... (Score:5, Interesting)
Although an exact solution has been proven impossible for higher orders, this is not the case for numeric solutions.
No conflict here. Saying that an exact solution does not exist is consistent with saying that numeric solutions do exist.
A numberic solution is a solution that is "close enough", but not exact. Sort of like saying 2.0000000000000001 = 2. They aren't equal, but for many purposes, they are equivalent.
Better formula (Score:4, Funny)
Re:Better formula (Score:5, Funny)
Parent
Maths == Dutch (Score:5, Funny)
Re:Maths == Dutch (Score:5, Funny)
Parent
Re:Maths == Dutch (Score:5, Funny)
Parent
Man does the impossible (Score:5, Insightful)
i) Abel's proof contains a flaw that generations of extremely talented mathematicians have failed to spot in their years and years of teaching it.
ii) Student mistaken; popular media talking out of arse.
(Can't read PDF; slashdotted)
Re:Man does the impossible (Score:5, Insightful)
iii) Student comes up with interesting (and possibly new, I don't know) result of generating infinate series which converges to root of polynomial. Someone (popular media?) believes that this violates Abel's proof (it doesn't, his proof is for finite representations of the roots).
This has NOTHING to do with Abel, or Godel, or anyone other related theories, as they do not consider the case of infiniate series.
Parent
Re:Man does the impossible (Score:5, Interesting)
Now I'll RTFA to determine which it really is....
Parent
Re:Man does the impossible (Score:5, Insightful)
Yeah, but Newton's method isn't guaranteed to converge. This method claims to converge; although you don't get the exact answer in a finite number of steps.
Whether this method is useful or not, probably depends on how fast it converges and how long it takes to do each step.
Parent
Re:Man does the impossible (Score:5, Funny)
Parent
Mirror (Score:5, Informative)
The Roots of any Polynomial Equation [mit.edu]
Its no big deal (Score:5, Informative)
The student just used the method of formal power series to solve the equation. This approach dates back at least to Cauchy ~1850 and probably can be found in the works of Euler.
Re:Its no big deal (Score:5, Funny)
Parent
Re:Its no big deal (Score:5, Funny)
In the U.S., that is a big deal.
Parent
No sample code or algorithm? (Score:5, Insightful)
I guess this is obsolete now... (Score:4, Funny)
LISTER: Yeah, the Skutters managed to smuggles something out of the medi-lab for us, y'know that stuff that helps impotent guys put the zest back in their love lives?
KRYTEN: 'Boing!', the virility enhancement drug!?
LISTER: That's the stuff, and we've Mickey Finn'd their drinks.
RIMMER: Within seconds, you're harder than a quadratic equation, and, it doesn't wear off for seven hours.
KRYTEN: For seven hours those guys are going to be like catapults!
Red Dwarf, Series 8, Episode 6 [blueyonder.co.uk]
Rule of equations in school (Score:5, Funny)
The more complicated the equations for the math problem looks, the more likely the answer is 1.
Re:Rule of equations in school (Score:5, Funny)
Parent
Re:Rule of equations in school (Score:5, Insightful)
Time for some mega nerdiness: I was captain of the math team when I was in high school.
Feh, screw that "nerdiness" crap. Good for you. Math is a powerful tool, worthy of dedication. I wish I were better at it, and respect those who are. I think being captain of the math team is far and away a better thing than being the captain of the freakin football team.
Parent
Isn't it an approximation method? (Score:5, Interesting)
Re:Isn't it an approximation method? (Score:5, Insightful)
Parent
My brain seems to have shut down. (Score:4, Funny)
europe (Score:5, Funny)
european academic finds solution to very hard problem.
2 years later:
a) americans find way of turning said solution into entertainment technology and make billions of dollars.b) European academic still unemployed and eating pasta all week.
We need more GREED in europe..
This doesn't seem likely (Score:5, Interesting)
However, from what I remember, Abel's theorem was proven using Galois Groups and Field extensions. This implies that what it actually proves is that analytical solutions using a particular set of functions -- in particular, the field operations (addition, subtraction, multiplication, division by non-zero) extended to include radicals (square, cubic, etc roots), composed in any way possible (as in a ruler and compass construction proof) cannot possibly generate an analytical formula depicting the solution for polynomials of order greater than 4.
Does this mean that an analytical formula using other functions is impossible? Not at all. Trivially, I will define a function called, say, omega, which, given a n-dimensional complex vector, gives a solution to one of the roots of the function a_n * x^n + a_(n-1) * x^(n-1) +
Clearly, this solution is analytical in the sense that it a) provides an exact solution and b) is algebraic in nature. However, it isn't useful, because it depends on a function (omega) which cannot itself be defined analytically in terms of other functions (or at least, not ones we know how to compute).
The reason Abel's proof is so important is because it deals with the 4 fundamental operations that polynomials themselves use (the field ops) and adds radicals, which are inverse ops to the building blocks of polynomials themselves. So it essentially says, we cannot use the functions that we constructed the polynomial with to solve it.
Now, my omega function may seem a little bit contrived to non-math types, but actually a large number of functions are arbitrarily defined this way. Logarithms are a good simple example. An analytical formula for the likes of log n wouldn't be possible either, and yet we study logarithms without having an express analytical means of calculating them.
What you should ask yourself is, what does analytical mean, anyway? It really isn't useful (or correct) to say that no analytical solution exists unless you explicitly restrict what particular set of 'basic' functions/operators the analytical solution can contain. In Abel's case (and it's a beautiful proof, by the way) he uses the field operators plus radicals. But what if you added logarithms into the mix? Exponential functions?
It's impossible to say. If you don't restrict your base, you open yourself up to the attack that I just used with the omega function (which certainly exists, after all, I just defined it.)
No closed formula (Score:5, Insightful)
I did something similar (Score:5, Informative)
a x^b + c x^d + e x^f = 0
where the exponents are integers and the coefficients can be complex. I tried to generalize it for complex exponents but I quit after a while. Google should provide some preliminary information on using hypergeometric functions to solve the quintic
a x^5 + b x^c + e = 0
where c is less than 5 and greater than zero.
This is an analytic solution to the general trinomial that I found empirically (without proof). If one wants to solve to solve the quartnomial then two dimensional structures, quintnomials need 3 dimensional structures. This was computationally taxing on me and my computer so I didn't even consider the quartnomial equations.
By the way, I have implemented a Jenkins-Traub algorithm not so long ago that gives numerical approximations to general polynomial roots. It is fast and well known.
Here's a math prof's take on the paper (Score:5, Insightful)
I don't know what his point is. He says its a "method of solving the roots"
of a polynomial. Well, we already have very fine methods for doing that,
interval Newton methods for instance. Using circular disk arithmetic in the
complex plane we can find all the complex roots as well.
There is no need whatever to make things more complicated such as going to
differential equations. That is unneccessary. Root finding is an algebraic
problem."
Looks flawed (Score:5, Informative)
Finally, he puts back a_0 into s, but conveniently forgets the case that a_0 is bigger than 1.
Also, it is not clear whether this is in the complex plane or not. For example, for finding real roots of real polynomials, you could use Sturm Sequences. There's even sample code in graphics Gems IV (IIRC).
In any case, the student was studying at the "hogeschool" which roughly translates to "higher professional education", a school which doesn't teach mathematics, and whose level which significantly lower than Dutch the MSc., BSc. or engineering degree.
Han-Wen
(yes, I am a mathematician)
Mathematical Elucidation (Score:5, Informative)
Abel's theorem merely says you can not solve the general quintic (5th degree) or higher in terms of radicals. That is entierly in terms of multiplication, addition, and taking nth roots. If we don't put that restriction about radicals the solution is trivial. Let x be such that P(x)=0 is one obvious solution.
Going through this again the write up is *entierly wrong*. It is completly possible to give an exact solution for the general polynomial (I just did in the paragraph above). Furthermore this distinction between exact and numerical solutions which is made so much of by our high school and college teachers is really illusionary. Writing a solution in terms of sin(3) isn't an exact value, we just have a good algorithm to approximate sin. Really what we mean when we talk about exact solutions is solvable in elementary functions, which is nothing but a certain commonly used set of functions for which we have good approximations. Unfortunatly, we still insist on students 'solving' differntial equations rather than just finding some quickly converging numerical solution even though at a deep level these are not differnt.
Now since abel's theorem there has been considerable research on other ways to solve polynomial equations. For instance one big result was that a certain degree of polynomials could be solved in a terms of continous two place function. Possibly this result in question is another result like this one but I imagine it is much less significant. For one I'm not entierly convinced he is correct, nor novel. (Don't make the mistake of assuming if he is right he has given a continous solution of any polynomial..it isn't clear his solution is continous in the coefficents).
Re:Damn, 11 Years too late (Score:5, Informative)
To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries' coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent.
The short paper has more details.
Parent
Re:/.ed after 4 comments (Score:5, Funny)
Parent
Re:/.ed after 4 comments (Score:5, Funny)
Dang, I just started reading this, and you allready beat me to it! ;-)
However, I am still typing up my GUT (I prove that there are only 17 dimensions, string theory is wrong, the Multiverse doesn't REALLY exist, and that the cat is alive or dead BEFORE you open the box), and should have it available for subscribers shortly.
Parent
Re:run away! (Score:5, Informative)
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Re:run away! (Score:5, Funny)
(Mods, parent was mistaken, but not a troll).
My favorite word in the 503 message was geblokkeerd. That's what I'm going to use instead of "slashdotted" from now on -- "Oh no! The site is geblokkeerd!"
Parent
Re:Easy! (Score:5, Funny)
5) Profit!
(awaits an ass-whooping by the mods)
Parent