FAQFAQ   SearchSearch   MemberlistMemberlist   UsergroupsUsergroups 
 ProfileProfile   PreferencesPreferences   Log in to check your private messagesLog in to check your private messages   Log inLog in 
Forum index » Science and Technology » Math
inverting a cubic polynomial as a series
Post new topic   Reply to topic Page 1 of 1 [7 Posts] View previous topic :: View next topic
Author Message
Paul Abbott
science forum addict


Joined: 19 May 2005
Posts: 99

PostPosted: Thu Jul 20, 2006 4:12 am    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

In article <1153198491.059962.290590@b28g2000cwb.googlegroups.com>,
"pluton" <plutonesque@gmail.com> wrote:

Quote:
I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ? (invert means find x(t) such
as P(x(t))=t)

The exact solution to an arbitrary cubic,

p[x_] = a x^3 + b x^2 + c x + d;

is, of course, well known.

Quote:
If not, what about an infinite series?

Such series can be obtained using series reversion. For example, with

p[x_] = a x^3 + b x^2 + c x + d;

then (using Mathematica),

x[t_] = InverseSeries[ p[x] + O[x]^4, t]

(where the expansion point is x = 0), yields

2 2 3
(-d + t) b (-d + t) (2 b - a c) (-d + t) 4
-------- - ----------- + ---------------------- + O[-d + t]
c 3 5
c c

Check that this is the inverse series:

p[x[t]] // Normal // Simplify

t

Quote:
It looks like it does not converge over a wide range of t?

You only require the inverse series at t = 0:

Normal[x[t]] /. t -> 0

2 2 3
d b d (2 b - a c) d
-(-) - ---- - ---------------
c 3 5
c c

This agrees with the expansion of the exact solution.

For more on this topic, see

http://library.wolfram.com/examples/quintic/

and, in particular,

http://library.wolfram.com/examples/quintic/main.html#series

where series methods are illustrated.

Cheers,
Paul

_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
Back to top
pluton
science forum beginner


Joined: 16 Feb 2006
Posts: 8

PostPosted: Thu Jul 20, 2006 2:12 am    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

Quote:
Yes. It's been known for centuries. The cubic formula can be found, for
example, in section 3.8.2 of Abramowitz and Stegun:
http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=17&Submit=Go
It really puzzles me why no previous respondents mentioned the formula!

ok thank you. I do not know why I did not think about it either. I was
more focusing on a function than finding the roots but it is a solution
as well.

Pluton
Back to top
David W. Cantrell
science forum Guru


Joined: 02 May 2005
Posts: 352

PostPosted: Wed Jul 19, 2006 2:54 am    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

"pluton" <plutonesque@gmail.com> wrote:
Quote:
Hi all,

I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ?

Yes. It's been known for centuries. The cubic formula can be found, for
example, in section 3.8.2 of Abramowitz and Stegun:
<http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=17&Submit=Go>
It really puzzles me why no previous respondents mentioned the formula!

Quote:
(invert means find x(t) such as P(x(t))=t)

Of course, that inverse relation is not necessarily single-valued. But
that's not a "problem", IMO.

Quote:
If not, what about an infinite series ?

Considering that there is an exact inverse in terms of radicals, I suppose
you will not be interested in using series. Nonetheless, if you were still
interested, you could search for "reversion of series".

Quote:
It looks like it does not converge over a wide range of t ?

An important observation, indeed. This would make it crucial to choose the
correct point about which to expand, before doing the reversion. But as I
said, I doubt that you will want to go down this path...

David
Back to top
Achava Nakhash, the Lovin
science forum beginner


Joined: 14 Sep 2005
Posts: 35

PostPosted: Tue Jul 18, 2006 5:25 pm    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

pluton wrote:
Quote:
Hi all,

I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ? (invert means find x(t) such
as P(x(t))=t) If not, what about an infinite series ? It looks like it
does not converge over a wide range of t ? Any suggestion ? Thank you,

Pluton

One potential problem with inverting it exactly is that it might not be
1-1. You need to check that condition first. I rather doubt that you
can find an explicit inverse in terms, say, of radicals, except in
special cases, and without an explicit polynomial in front of me, I
can't really comment in that. In general, there is a formula for the
inverse of a power series that gives you another power series.
Obviously there are some restirictions on the applicability of this
result, but they are minimal. A cubic, or any other degree, polynomial
is simply a short power series as far as this theorem is concerned. It
is probably a big mess and might or might not be useful to you, but it
is not hard (if you know anything about power series and how their
coefficients are derived) to figure out the first several.terms of the
power series of your inverse.

Hope this helps,
Achava
Back to top
alex.lupas@gmail.com
science forum beginner


Joined: 23 Feb 2006
Posts: 47

PostPosted: Tue Jul 18, 2006 7:04 am    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

pluton wrote:
Quote:
Hi all, I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ? (invert means find x(t) such
as P(x(t))=t) If not, what about an infinite series ? It looks like it
does not converge over a wide range of t ? Any suggestion ? Thank you,
Pluton
===========

Perhaps try Lagrange series as someone inform you.
Also of interest are the works by Gauss , and
Hans von Mangoldt ,
"Ueber die Darstellung Wurzeln einer dreigliedrigen algebraischen
Gleichung durch unendlichen Reihen",Dissertation,Berline,1878.
See also Umbral Calculus(Gian-Carlo Rota,Steven
Roman,Odlyzko,Kahane,...)
Back to top
mariano.suarezalvarez@gma
science forum addict


Joined: 28 Apr 2006
Posts: 58

PostPosted: Tue Jul 18, 2006 5:25 am    Post subject: Re: inverting a cubic polynomial as a series Reply with quote

pluton wrote:
Quote:
Hi all,

I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ? (invert means find x(t) such
as P(x(t))=t) If not, what about an infinite series ? It looks like it
does not converge over a wide range of t ? Any suggestion ? Thank you,

(The polynomial you mention can be factored into a product of
of x and a quadratic polynomial. I guess you meant some other
polynomial Wink )

The equation

a x^3 + b x^2 + c x + d = 0

defines x implicitely as a function of a, b, c, d near points
where the implicit function theorem applies (basically, away
from multiple roots). Moreover, since the left hand side of
the equation is analytic, the function implicitely defined by
it is analytic, so you can in fact develop x(a,b,c,d) as a
multiple power series of a, b, c and d.

This is the tip of the iceberg known as the theory of
algebraic functions, btw.

You will find information about the implicit function theorem in any
book on calculus of functions of several variables. To see how it is
that one can conclude that the function x(a,b,c,d) is in fact analytic,

an introductory text on complex analysis should be of help.

-- m
Back to top
pluton
science forum beginner


Joined: 16 Feb 2006
Posts: 8

PostPosted: Tue Jul 18, 2006 4:54 am    Post subject: inverting a cubic polynomial as a series Reply with quote

Hi all,

I have a polynomial of the form : P(x)=a x^3+b x^2+c x. Is there a way
to invert it exactly knowing a, b and c ? (invert means find x(t) such
as P(x(t))=t) If not, what about an infinite series ? It looks like it
does not converge over a wide range of t ? Any suggestion ? Thank you,

Pluton
Back to top
Google

Back to top
Display posts from previous:   
Post new topic   Reply to topic Page 1 of 1 [7 Posts] View previous topic :: View next topic
The time now is Sat Jun 21, 2014 12:14 am | All times are GMT
Forum index » Science and Technology » Math
Jump to:  

Similar Topics
Topic Author Forum Replies Last Post
No new posts A series related to the Hilbert transform larryhammick@telus.net Math 0 Fri Jul 21, 2006 3:28 am
No new posts Entire functions, polynomial bounds david petry Math 2 Thu Jul 20, 2006 11:09 pm
No new posts Function from Taylor series? Nathan Urban Research 1 Thu Jul 20, 2006 12:48 am
No new posts Complex Analysis question, Taylor series James1118 Math 6 Tue Jul 18, 2006 2:17 pm
No new posts why are the polynomials in this series all solvable by ra... Lee Davidson Symbolic 5 Mon Jul 17, 2006 4:34 pm

Copyright © 2004-2005 DeniX Solutions SRL
Other DeniX Solutions sites: Electronics forum |  Medicine forum |  Unix/Linux blog |  Unix/Linux documentation |  Unix/Linux forums  |  send newsletters
 


Powered by phpBB © 2001, 2005 phpBB Group
[ Time: 0.1753s ][ Queries: 20 (0.1424s) ][ GZIP on - Debug on ]