Question : Root condition for polynomials
Chae Dong Suk
I have some question regarding to root condition of polynomials. I can't
find the solution as far as my mathematical knowledge is concerned.
Somebody suggests me reference or solution.
Let's suppose the polynomial of degree less than 5 like
P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
then what is range of coefficients satisfying the root condition that
the their magnitude are all confined in the unit circle on complex
plane?
I study this problem for zero-stability of numerical time integration
for ODE.
Thanks in advance.
Peter Spellucci
Chae Dong Suk <sch...@gong.snu.ac.kr> writes:
|> Hi everyone.
|>
|> I have some question regarding to root condition of polynomials. I can't
|> find the solution as far as my mathematical knowledge is concerned.
|> Somebody suggests me reference or solution.
|>
|> Let's suppose the polynomial of degree less than 5 like
|>
|> P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
|>
|> then what is range of coefficients satisfying the root condition that
|> the their magnitude are all confined in the unit circle on complex
|> plane?
the problem is not that simple. you also must assure that roots on the
boundary are simple. but, for a polynomial of degree four, it is possible
to express all roots in terms of 2nd, 3rd and 4th roots of polynomials
formed from the coefficients. then write down the absolute values of the roots
,set this <=1 and you are done. but the outcome will not be "ranges"
since the formulae itself depend on the actual values of the coefficients
(think about the simple case of degree 2), you will obtain a very
complicated set of conditions finally.
hope this helps
peter
Robin Becker
<sch...@gong.snu.ac.kr> writes
>Hi everyone.
>
>I have some question regarding to root condition of polynomials. I can't
>find the solution as far as my mathematical knowledge is concerned.
>Somebody suggests me reference or solution.
>
>Let's suppose the polynomial of degree less than 5 like
>
>P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
>
>then what is range of coefficients satisfying the root condition that
>the their magnitude are all confined in the unit circle on complex
>plane?
>
>for ODE.
>Thanks in advance.
>
>
problems the Hurwitz criterion is used. This gives structural conditions
on the coefficients such that re[zero(p(s)]<0; the required conditions
for the discrete time case can be determined by transforming the dt poly
into continuous time via
p(s) := pn s^n + pn-1 s(n-1)+.....p(0) has all zeros in |s|<1
<=>
q(s) := (1-s)^n * p[(1+s)/(1-s)] has all zeros in Re[s] < 0
R Brocket gives references to this and the Lienard-Chipart criterion.
--
Robin Becker
Dave Rusin
>I have some question regarding to root condition of polynomials. I can't
>find the solution as far as my mathematical knowledge is concerned.
>Somebody suggests me reference or solution.
>
>Let's suppose the polynomial of degree less than 5 like
>
>P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
>
>then what is range of coefficients satisfying the root condition that
>the their magnitude are all confined in the unit circle on complex
>plane?
>
>I study this problem for zero-stability of numerical time integration
>for ODE.
I don't really know the answer but this has come up before; see e.g.
http://www.math.niu.edu/~rusin/known-math/96/roots.ge1
For some reason this doesn't seem to be treated as an algebraic
problem, which is unusual since algebraic number theorists have shown a
lot of interest in the related (algebraic) numbers which are greater
than 1 but all of whose conjugates are at most equal to 1 in magnitude;
these are the Salem and Pisot numbers. There is interest in polynomials
all of whose roots have magnitude exactly equal to 1, too; these obviously
include products of cyclotomic polynomials but there are others.
So it's possible that there's more algebraic information about the
polynomials you seek (e.g. an easy characterization of the 4-tuples
[a1,a2,a3,a4] ) but I haven't seen it.
dave
Polynomials as analytic functions:
http://www.math.niu.edu/~rusin/known-math/index/26CXX.html
Polynomials as generators of algebraic extension fields:
http://www.math.niu.edu/~rusin/known-math/index/12FXX.html
Allan Hayes
Chae Dong Suk wrote in message <36B2679D...@gong.snu.ac.kr>...
>Hi everyone.
>
>I have some question regarding to root condition of polynomials. I can't
>find the solution as far as my mathematical knowledge is concerned.
>Somebody suggests me reference or solution.
>
>Let's suppose the polynomial of degree less than 5 like
>
>P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
>
>then what is range of coefficients satisfying the root condition that
>the their magnitude are all confined in the unit circle on complex
>plane?
>
>I study this problem for zero-stability of numerical time integration
>for ODE.
>
>
Chae:
Convert to classical stability problem of finding if the roots are all
inside L = {z: Re[z]<0} as follows (there is an excellent, up-to-date
discussion of the general problem in Mathematica in Education and Research
Vol 7, No2, Spring, 1998 by Janos Toth, Laszlo Szili and Andras Zachar, that
includes the general Mathematica code needed:
http://www.telospub.com/journal/MIER/)
f[z_] = (2z + 1)/(2z - 1);
Is a (1-1) mapping of the extended complex numbers onto themselve with
1/2 -> Infinity and takig L onto the interior of the unit circle, C.
So its inverse function invf maps C onto L
For any polynomial p[z] the following are equivalent
p[z]= 0 => z inside C
p[f[z]]=0 => f[z] inside C
p[f[z]]=0 => z inside invf[C] (= L)
For the polynomial
p = a0 x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4;
The Routh-Hurwitz Criterion for all roots to be inside L is
RH[{a0_, a1_, a2_, a3_, a4_}] := a0 > 0 && a1 > 0 &&
Det[{{a1, a0}, {a3, a2}}] > 0 &&
Det[{{a1, a0, 0}, {a3, a2, a1}, {0, a4, a3}}] > 0 &&
Det[{{a1, a0, 0, 0}, {a3, a2, a1, a0},
{0, a4, a3, a2}, {0, 0, 0, a4}}] > 0
We can easily define a Mathematica function to produce this for any degree 4
polynomial.
RHCriterion[poly_, x_] :=
RH[Reverse[CoefficientList[poly, x]]];
RHCriteria[p[x], x]
\!\(a0 > 0 && a1 > 0 && a1\ a2 - a0\ a3 > 0 &&
a1\ a2\ a3 - a0\ a3\^2 - a1\^2\ a4 > 0 &&
a1\ a2\ a3\ a4 - a0\ a3\^2\ a4 - a1\^2\ a4\^2 > 0\)
and from this to get the criterion for all the roots being inside the unit
circle C.
CircleCriterion[poly_, x_] :=
RHCriterion[poly /. x -> f[x] // Together // Numerator, x]
CircleCriterion[p, x]
16*a0 + 16*a1 + 16*a2 + 16*a3 + 16*a4 > 0 &&
32*a0 + 16*a1 - 16*a3 - 32*a4 > 0 &&
640*a0^2 + 320*a0*a1 + 64*a1^2 - 384*a0*a2 -
64*a1*a2 - 576*a0*a3 + 64*a2*a3 - 64*a3^2 +
576*a1*a4 + 384*a2*a4 - 320*a3*a4 - 640*a4^2 > 0 &&
4096*a0^3 - 4096*a0^2*a2 + 4096*a0*a1*a3 -
4096*a0*a3^2 - 4096*a0^2*a4 - 4096*a1^2*a4 +
8192*a0*a2*a4 + 4096*a1*a3*a4 - 4096*a0*a4^2 -
4096*a2*a4^2 + 4096*a4^3 > 0 &&
(a0 - a1 + a2 - a3 + a4)*(4096*a0^3 - 4096*a0^2*a2 +
4096*a0*a1*a3 - 4096*a0*a3^2 - 4096*a0^2*a4 -
4096*a1^2*a4 + 8192*a0*a2*a4 + 4096*a1*a3*a4 -
4096*a0*a4^2 - 4096*a2*a4^2 + 4096*a4^3) > 0
Lets test this ( we have to use Chop to get rid of small comples numbers
produced by multiplying out)
poly2 = Times @@ ( x - {-.3, .2 Exp[.5I], .2 Exp[-.5I], .7}) // Expand //
Chop;
CircleCriterion[poly2, x]
True
poly3 = Times @@ ( x - {-3, .2 Exp[.5I], .2 Exp[-.5I], .7}) // Expand //
Chop;
CircleCriterion[poly3, x]
False
---------------------
Allan Hayes
Mathematica Training and Consulting
www.haystack.demon.co.uk
h...@haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565
Andrzej Lewandowski
>
> In article <36B2679D...@gong.snu.ac.kr>, Chae Dong Suk
> <sch...@gong.snu.ac.kr> writes
> >
> >I have some question regarding to root condition of polynomials. I can't
> >find the solution as far as my mathematical knowledge is concerned.
> >Somebody suggests me reference or solution.
> >
> >Let's suppose the polynomial of degree less than 5 like
> >
> >P(x) = x^4 + a1 x^3 + a2 x^2 + a3 x^1 + a4
> >
> >then what is range of coefficients satisfying the root condition that
> >the their magnitude are all confined in the unit circle on complex
> >plane?
> >
> >I study this problem for zero-stability of numerical time integration
> >for ODE.
> >Thanks in advance.
> >
> >
> problems the Hurwitz criterion is used. This gives structural conditions
> on the coefficients such that re[zero(p(s)]<0; the required conditions
> for the discrete time case can be determined by transforming the dt poly
> into continuous time via
>
> p(s) := pn s^n + pn-1 s(n-1)+.....p(0) has all zeros in |s|<1
> <=>
> q(s) := (1-s)^n * p[(1+s)/(1-s)] has all zeros in Re[s] < 0
>
> R Brocket gives references to this and the Lienard-Chipart criterion.
> --
See the classical books on discrete control systems (that time called
"impulse control systems") written by E. Jury. Try to find the library
that
keeps old stuff; these books were published in late 60' - early 70'.
A.L.