 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%% You need to use Maple 12 for this package.
 %%% For the reference of the method, please read
 %%% J.S Cheng, X.S. Gao, and J Li,
 %%% Root Isolation for Bivariate Polynomial Systems with Local Generic Position  Method
 %%% MM-Preprints, Vol 27, 122-136, 2008
 %%% http://www.mmrc.iss.ac.cn/mmpreprints/
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

LGP(PS,VS,precision,type);

Here are some examples:


read "LGP.txt";

HELP();#show how to use the main functions.


f:=x^2+y^2-4;
g:=x^4-y^2-1/2;


LGP([f,g],[x,y],5,'interval');
# f,g are two bivariate polynomials.
# x,y are variables.
# 5 is precision, the output results is under precision 10^(-5).
# 'interval' is the output type.
#output:

The computing time is .31e-1 seconds!

      -695750209  -1391500417    -408917247  -204458559

   [[[----------, -----------], [----------, ----------]],

      536870912   1073741824     268435456   134217728



           -695750209  -1391500417    408917121  204458625

         [[----------, -----------], [---------, ---------]],

           536870912   1073741824     268435456  134217728



           1391500417  695750209    -204458625  -408917121

         [[----------, ---------], [----------, ----------]],

           1073741824  536870912    134217728   268435456



           1391500417  695750209    204458559  408917247

         [[----------, ---------], [---------, ---------]]]

           1073741824  536870912    134217728  268435456



LGP([f,g],[x,y],5); # =LGP([f,g],[x,y],5,'numeric');
# f,g are two bivariate polynomials.
# x,y are variables.
# 5 is precision, the output results is under precision 10^(-5).
# default for the fourth parameter is 'numeric', output result type.
#output:

The computing time is .34e-1 seconds!


   [[-1.2959357514046132564, -1.5233352128416299820],



         [-1.2959357514046132564, 1.5233352240175008774],



         [1.2959357514046132564, -1.5233352240175008774],



         [1.2959357514046132564, 1.5233352128416299820]]


> T:=LGP([f,g],[x,y],'symbolic');
# f,g are two bivariate polynomials.
# x,y are variables.
# 'symbolic' is to require the symbolic solution only [t(x),T(X),s,S], see the paper.
#output:

T := [(x^2-9/2+x^4)^2,
X^8+55/32*X^6-1849/256*X^4-3325655/262144*X^2+1404225729/67108864,
2,
1/8]



> LGP from symbolic solution(T,5);
#This function is to isolate the roots of the system from its symbolic solution.
# T is the output of LGP([f,g],[x,y],'symbolic').
# 5 is the required precision, the output results is under the precison 10^(-5).
# the third parameter is output type, 'interval' or 'numeric', default is 'numeric'
#output:

[[-1.295936, -1.523336], [-1.295936, 1.523336],
[1.295936,-1.523336],[1.295936, 1.523336]]
