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MathGroup Archive 2007

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Re: Approximate/asymptotic factorization

  • To: mathgroup at smc.vnet.net
  • Subject: [mg73652] Re: Approximate/asymptotic factorization
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Sat, 24 Feb 2007 02:07:59 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <200702210635.BAA17453@smc.vnet.net> <ermdo5$i0d$1@smc.vnet.net>

In article <ermdo5$i0d$1 at smc.vnet.net>,
 Daniel Lichtblau <danl at wolfram.com> wrote:

> Paul Abbott wrote:
> > Consider the monomial H[4,4]
> > 
> >   256*(23625 + 126000*z^4 - 7200*z^8 + 3840*z^12 + 256*z^16)
> > 
> > where H[m,n] is a Generalized Hermite Polynomial (Noumi & Yamada 1998), 
> > a rational solution to the Painlev� equation P_IV. A plot of the roots 
> > of H[4,4] displays an interesting quasi-rectangular structure.
> > 
> > Then consider the monomial Q[4,4]
> > 
> >   1528899609315374375625 + 6795109374734997225000*z^4 -  
> >   560866170613047390000*z^8 + 153399294526645440000*z^12 +  
> >   2734590598399296000*z^16 - 167891551278796800*z^20 +  
> >   2948686820352000*z^24 - 40649991782400*z^28 +  277762867200*z^32 - 
> >   920125440*z^36 + 1048576*z^40
> > 
> > where Q[m,n] is a Generalized Okamoto Polynomial, another rational 
> > solution to P_IV. A plot of the roots displays an even more interesting 
> > structure, with a quasi-rectangular core and 4 triangular "lobes" 
> > (related, in some way, to the the Yablonskii�Vorob�ev Polynomials, 
> > Q[n], 
> > which are Rational Solutions of P_II)
> > 
> > If you overlay plots of the roots of H[4,4] and Q[4,4] what is most 
> > striking is that the "common" roots are very similar. In other words, 
> > H[4,4] is "approximately" a factor of Q[4,4]. 
> > 
> > Is there a "standard" way of quantifying this approximate factorization?
> > 
> > This approximate factorization holds for any m and n and appears to 
> > improve as m and n increases. Perhaps there is a limit in which the 
> > factorization holds asymptotically?
> > 
> > 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
> 
> I'm not sure how to tackle the asymptotics you have in mind. For 
> particular examples this is very much the province of univariate 
> approximate gcd computation. There has been a good deal of work in this 
> area since around 1995, with some work even earlier.
> 
> A good place to start would be the "4M" article of Corless, Gianni, 
> Trager, and Watt, "The singular value decomposition for polynomial 
> systems", ISSAC 1995. The idea of their approach is to use the SVD of 
> the Sylvester matrix in order to determine the degree of a "good" approx 
> gcd, then derive that gcd also using the SVD.

OK, I've got that ref and also got

  N. Karmarkar and Y. N. Lakshman "Approximate polynomial greatest 
  common divisors and nearest singular polynomials", Proceedings of the 
  1996 international symposium on Symbolic and algebraic computation, 
  35 - 39, 1996

> For your example I show some quick-and-dirty code that, I think, does 
> more or less what they have in mind. For the Sylvester matrix I cribbed 
> and slightly modified code from MathWorld:
> 
> http://mathworld.wolfram.com/SylvesterMatrix.html
> 
> SylvesterMatrix[poly1_, poly2_, var_] := Module[
>    {coeffs1, coeffs2, l1, l2},
>    coeffs1 = Reverse[CoefficientList[poly1, var]];
>    coeffs2 = Reverse[CoefficientList[poly2, var]];
>    l1 = Length[coeffs1];
>    l2 = Length[coeffs2];
>    Join[Table[
>      Join[Table[0,{j}], coeffs1,
>        Table[0,{l2-2-j}]], {j,0,l2-2}],
>      Table[Join[Table[0,{j}], coeffs2,
>        Table[0,{l1-2-j}]], {j,0,l1-2}]]
>    ]

I note that

 Join[Table[
     Join[Table[0,{j}], coeffs1,
       Table[0,{l2-2-j}]], {j,0,l2-2}],
     Table[Join[Table[0,{j}], coeffs2,
       Table[0,{l1-2-j}]], {j,0,l1-2}]]

can be replaced by the more intelligible (to me)

 Join[
  NestList[RotateRight, PadRight[coeffs1, l1 + l2 - 2], l2 - 2],   
  NestList[RotateRight, PadRight[coeffs2, l1 + l2 - 2], l1 - 2]
 ]

which is a direct implementation of the description at MathWorld.

> h44 = 256*(23625 + 126000*z^4 - 7200*z^8 + 3840*z^12 + 256*z^16);
> 
> q44 = 1528899609315374375625 + 6795109374734997225000*z^4 -
>    560866170613047390000*z^8 + 153399294526645440000*z^12 +
>    2734590598399296000*z^16 - 167891551278796800*z^20 +
>    2948686820352000*z^24 - 40649991782400*z^28 +  277762867200*z^32 -
>    920125440*z^36 + 1048576*z^40;
> 
> I'll normalize so largest coefficient is unity.
> 
> normPoly[poly_, var_] := Expand[poly /
>    Max[Abs[CoefficientList[poly,var]]]]
> 
> p1 = normPoly[h44,z];
> p2 = normPoly[q44,z];

Perhaps Decompose will also be useful here, since both p1 are 
polynomials in z^4 (which explains the observed 4-fold symmetry).

> I'll use an epsilon of 10^(-3) as tolerance for what singular values to 
> remove.
> 
> eps = 10^(-3);
> 
> We count the number of singular values that were zero.
> 
> nulls = Select[S,Max[Abs[#]]<eps&];

What is S and where is sylmat defined? I assume that

  sylmat = SylvesterMatrix[p1, p2, z]

  S = SingularValueList[sylmat]

> We now get the degree of our approximate gcd as below (for the epsilon 
> we chose, it is 12).
> 
> k = Length[sylmat] - Length[nulls]
> 
> To form that gcd we'll row reduce a certain matrix and use the final row.
> 
> newmat = Take[newS.Transpose[V], k];

What is newS?

> coeffrow = Last[RowReduce[newmat]];
> 
> gcdapprox = coeffrow . z^Reverse[Range[0,Length[coeffrow]-1]]
> 
> So here is what I get.
> 
> In[150]:= InputForm[gcdapprox]
> Out[150]//InputForm=
> 6.980499621431014 + 0.000585041745605589*z + 0.0035109293315991083*z^2 +
>   0.002602929711965681*z^3 + 33.10928622044708*z^4 +
>   0.0019204205746833373*z^5 + 0.01152459048089943*z^6 +
>   0.008544087788492293*z^7 - 3.49084586854123*z^8 +
>   0.00011025782928825875*z^9 + 0.0006624721373616564*z^10 +
>   0.0004911544753497421*z^11 + z^12
> 
> I'll note that an epsilon of around 10^(-1) would have given a degree 16 
> approx gcd (not equal to a scalar multiple of p1, though). One can get 
> an idea of how close are the various roots by looking at them as below.
> 
> s1 = Sort[z /. NSolve[p1==0, z]];
> s2 = Sort[z /. NSolve[p2==0, z]];
> sg = Sort[z /. NSolve[gcdapprox==0, z]];
> 
> Playing tricks with Intersection and a comparison criterion might 
> automate this process.
> 
> Note that there are other approaches to approximate univariate gcds 
> using QR decomposition, remainder sequences, approximate factoring, and 
> probably some others I am forgetting.

Thanks for this -- all very useful and interesting.

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


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