Question: solving polynomial eqs, Macaulay2 vs. homotopy continuati​on

[Gradient]

Hello,

My research is on an engineering problem that requires solving a
polynomial equation in a super fast way. Since I have never worked
with professional math packages like Macaulay2 or PHCPack, I first
want to learn which one is the right choice for me to invest time on
learning its scripting language and details.

My problem is two equations in two variables, i.e. f_1(x,y)=0 and
f_2(x,y)=0. Each f_k is a polynomial with total degree of 6. The goal
is to find all the real roots (in the fastest way possible). If that
helps, we can restrict this discussion to isolted roots only.

I think there are two options here, homotopy continuation, and purely
symbolic approach like in Macaulay 2.

Now, as I understand, Macaulay2 is slow for solving large-sized system
of polynomial equations, compared to its numerical counter part of
homotopy continuation. Yet, I think my problem is what you would call
"small" and there might be a chance that Macaulay2 does better than
homotopy here. I guess so, because homotopy continuation is a
numerical procecure nd needs to iterate between predictor and
corrector until convergence, which may be a waste of CPU time for
small-sized problem, while symbolic methods I believe obtain the
solution in one shot.

I was wondering if you could provide any advice about what is the best
toolbox/package to solve the described equation in the fastest way.

Best Regards

--Hossein

[Douglas Leonard]

Since nobody seems to be responding, I'll try.
Here is a generic "small" eample over the ring A over the rationals QQ
with an ideal generated by two polynomials of total degree 6 in y and x,
G_(0,0) is a polynomial of degree 29, so has 29 complex roots,
and G_(0,1) can be used to solve for y given x.
Solving for any real values of x can probably be done using some other package;
at least I don't know how to do it in Macaulay2.

-------------------------------------------------------------------------------------------------------------------

Macaulay2, version 1.4
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : A=QQ[y,x,MonomialOrder=>{Lex}];
i2 : I=ideal(y^3*x^3+y*x^5-6*y^2*x-17,y^2*x^4+13*y*x^3+y^4-12*x^5);

o2 : Ideal of A

i3 : G=gens gb I;

1 2
o3 : Matrix A <--- A

i4 : toString(G_(0,0))

o4 = 12*x^29+288*x^28+1704*x^27-288*x^26+2040*x^25+936*x^24-11232*x^23-8213*x^
22-7524*x^21+18730*x^20+13260*x^19-32963*x^18-75990*x^17+29378*x^16+41310*
x^15-194869*x^14-81600*x^13-234651*x^12+22542*x^11-10404*x^10-552942*x^9+
249696*x^7-191607*x^6+58956*x^5-83521

i5 : toString(G_(0,1))

o5 = 26394566901900576875660089054044471481099146800287785393951489300*y-
738118645923549178705921578702279597801469690157175247552548*x^28-
17921388213786120980146707488606421459754851909131918454863796*x^27-
110105797665724486767022435524975376350545587361493402737846604*x^26-
19677851570459225684481538676790706808599497403283273602533060*x^25-
165774659876606110594862533999870291187632856451177529498436340*x^24-
25240914176276308631648375977869826720461431554074078977737164*x^23+
973140276334019528485127522131551359179732041694591235656119636*x^22+
650335860355466615951656587485196934492821223282968793436168635*x^21+
1414967050869264156205653884937007420980801660226981974444976451*x^20-
554603909227632252674927148652218451606992860053057821224055667*x^19-
3250590058286015869187342730053216888414978363274003308562131741*x^18-
537879835436715462698129091958267625456146295010131878578872646*x^17+
2596357304240608738360712218076671804885635905958097366628108472*x^16+
3524425188980867831335643979196197812558877656550819535038875954*x^15+
1935808993805133707618307296953246463455475184513951656981670172*x^14+
5800992030087614794730969853085050471949564802644553428357528267*x^13-
7055701091287540676080373992634620718460246212509304298418737649*x^12+
22989806287488129559944028878629986497225546239069636575116853732*x^11+
8980918498420804410316441246341066164395537611896180701720084628*x^10-
34601058864655395272730716809249340262422646828441234996656155550*x^9+
19176000811588467345886482908579190727139673593949255816640859018*x^8-
37641153445849619527505624938555419707354590987571832248087147096*x^7+
900644853023605707601136919650233794984664150018473228641751578*x^6+
928428596843416579143251022087756602961302314121023892143573837*x^5-
119842635226734957910483820731066084082913436290148096718379138913*x^4+
1474977004097777908866790817881667059182746263091474973548234161*x^3+
26551825976525275652069708236614751120743143674929902315433793133*x^2-
11665917642979157305721648529080433282551649492221631936753507851*x+
6503024944266406730458339289415116488232560839092373817601820347

________________________________________
From: maca...@googlegroups.com [maca...@googlegroups.com] on behalf of Gradient [hmo...@gmail.com]
Sent: Monday, October 17, 2011 4:20 PM
To: Macaulay2
Subject: [Macaulay2] Question: solving polynomial eqs, Macaulay2 vs. homotopy continuati​on

Hello,

Best Regards

--Hossein

--
You received this message because you are subscribed to the Google Groups "Macaulay2" group.
To post to this group, send email to maca...@googlegroups.com.
To unsubscribe from this group, send email to macaulay2+...@googlegroups.com.
For more options, visit this group at http://groups.google.com/group/macaulay2?hl=en.