MATLAB Central Newsreader - polynomial factorization

polynomial factorization

Dejun Wang — Fri, 22 Feb 2008 23:05:03 -0500

Hi, I have a polynomial in the following form:
H(z)=[z^4+a3*z^3+a2*z^2+a1*z+a0]/
[z^5+b4*z^4+b3*z^3+b2*z^2+b1*z+b0], where a3-a0,b4-b0 are
known.
I want to obtain a factorized polynomial in the form of
H(z)=A/(z-p0)+B/(z-p1)+C/(z-p2)+D/(z-p3)+E/(z-p4), I know
I can use roots to get p0-p5, but is there a function that
I can use to get the value of A,B,C,D,E? Or do I need to
create my own symbolic equations to solve this?

Thanks.

Re: polynomial factorization

John D'Errico — Fri, 22 Feb 2008 23:26:03 -0500

"Dejun Wang" <dejunwang@yahoo.com> wrote in message
<fpnkeu$h3g$1@fred.mathworks.com>...
> Hi, I have a polynomial in the following form:
> H(z)=[z^4+a3*z^3+a2*z^2+a1*z+a0]/
> [z^5+b4*z^4+b3*z^3+b2*z^2+b1*z+b0], where a3-a0,b4-b0 are
> known.
> I want to obtain a factorized polynomial in the form of
> H(z)=A/(z-p0)+B/(z-p1)+C/(z-p2)+D/(z-p3)+E/(z-p4), I know
> I can use roots to get p0-p5, but is there a function that
> I can use to get the value of A,B,C,D,E? Or do I need to
> create my own symbolic equations to solve this?

It is impossible in general to factorize a
symbolic polynomial of degree 5 or higher.

http://en.wikipedia.org/wiki/Abel-Ruffini_theorem

John

Re: polynomial factorization

Arthur G — Sat, 23 Feb 2008 00:31:10 -0500

On Feb 22, 6:05=A0pm, "Dejun Wang" <dejunw...@yahoo.com> wrote:
> Hi, I have a polynomial in the following form:
> H(z)=3D[z^4+a3*z^3+a2*z^2+a1*z+a0]/
> [z^5+b4*z^4+b3*z^3+b2*z^2+b1*z+b0], where a3-a0,b4-b0 are
> known.
> I want to obtain a factorized polynomial in the form of
> H(z)=3DA/(z-p0)+B/(z-p1)+C/(z-p2)+D/(z-p3)+E/(z-p4), I know
> I can use roots to get p0-p5, but is there a function that
> I can use to get the value of A,B,C,D,E? Or do I need to
> create my own symbolic equations to solve this?
>
> Thanks.

residue is the function you want.

Re: polynomial factorization

John D'Errico — Sat, 23 Feb 2008 01:10:20 -0500

Arthur G <gorramfreak@gmail.com> wrote in message <7d356194-01f0-
472f-93d1-04bdad5a1e64@60g2000hsy.googlegroups.com>...
> On Feb 22, 6:05=A0pm, "Dejun Wang" <dejunw...@yahoo.com> wrote:
> > Hi, I have a polynomial in the following form:
> > H(z)=3D[z^4+a3*z^3+a2*z^2+a1*z+a0]/
> > [z^5+b4*z^4+b3*z^3+b2*z^2+b1*z+b0], where a3-a0,b4-b0 are
> > known.
> > I want to obtain a factorized polynomial in the form of
> > H(z)=3DA/(z-p0)+B/(z-p1)+C/(z-p2)+D/(z-p3)+E/(z-p4), I know
> > I can use roots to get p0-p5, but is there a function that
> > I can use to get the value of A,B,C,D,E? Or do I need to
> > create my own symbolic equations to solve this?
> >
> > Thanks.
>
> residue is the function you want.

But don't expect success. No matter what,
it will need to factor a symbolic polynomial
of the 5th degree with symbolic coefficients.

Its not gonna work.

John

Re: polynomial factorization

Arthur G — Sat, 23 Feb 2008 12:55:43 -0500

On 2008-02-22 20:10:20 -0500, "John D'Errico"
<woodchips@rochester.rr.com> said:

> Arthur G <gorramfreak@gmail.com> wrote in message <7d356194-01f0-
> 472f-93d1-04bdad5a1e64@60g2000hsy.googlegroups.com>...
>> On Feb 22, 6:05=A0pm, "Dejun Wang" <dejunw...@yahoo.com> wrote:
>>> Hi, I have a polynomial in the following form:
>>> H(z)=3D[z^4+a3*z^3+a2*z^2+a1*z+a0]/
>>> [z^5+b4*z^4+b3*z^3+b2*z^2+b1*z+b0], where a3-a0,b4-b0 are
>>> known.
>>> I want to obtain a factorized polynomial in the form of
>>> H(z)=3DA/(z-p0)+B/(z-p1)+C/(z-p2)+D/(z-p3)+E/(z-p4), I know
>>> I can use roots to get p0-p5, but is there a function that
>>> I can use to get the value of A,B,C,D,E? Or do I need to
>>> create my own symbolic equations to solve this?
>>>
>>> Thanks.
>>
>> residue is the function you want.
>
> But don't expect success. No matter what,
> it will need to factor a symbolic polynomial
> of the 5th degree with symbolic coefficients.
>
> Its not gonna work.
>
> John

Just to clarify, if numerical values are known for
a0-a3 and b0-b4, then residue will give you an
(approximate) numerical solution to the partial
fraction decomposition. That was my interpretation
of the OP's problem statement.

--Arthur
>

Re: polynomial factorization

John D'Errico — Sat, 23 Feb 2008 13:27:04 -0500

Arthur G <gorramfreak+news@gmail.com> wrote in message
<47c017cf$0$290$b45e6eb0@senator-bedfellow.mit.edu>...

> Just to clarify, if numerical values are known for
> a0-a3 and b0-b4, then residue will give you an
> (approximate) numerical solution to the partial
> fraction decomposition. That was my interpretation
> of the OP's problem statement.

You are probably correct.

John