function example
format long
intvalinit('DisplayInfsup')
%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--addvar1--')
syms x1 x2 x3
f1=x1^2+3*x1*x2+3*x1*x3-3*x3^2+2*x2+2*x3;
f2=-3*x1*x2+x1*x3-2*x2^2+x3^2+3*x1+x2;
f3=2*x2*x3+3*x1-3*x3+2;
f3=expand(f3^4)+f1;
sys=[f1,f2,f3];
var=[x1,x2,x3];
p0=[ -1.2966872160397030576,.47034450205110827769, -.91781263339251592021];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--addvar2--')
syms x1 x2 x3 x4
f1=-2*x1^2+3*x1*x2+x2*x3+x1+3*x4+2;
f2=-x1*x2+3*x2*x4-3*x1+x2+3*x3;
f3=-2*x1*x3+2*x2*x3-x3^2-3*x1-2*x2-2*x3;
f4=-x1*x4+3*x2^2-3*x4^2-3*x1-2*x2-3;
f1 = expand(f1^4)+f3;f2 = expand(f2^3)+2*f4;
sys=[f1,f2,f3,f4];
var=[x1,x2,x3,x4];
p0=[.31240278901615945790,-.89992427697989025263,.18705436134100288887,-.36848682028822223895];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%
disp('--addvar3--')
syms x1 x2 x3 x4 x5 x6
f1=2*x1*x5-3*x2*x3+3*x5^2-3*x5-1;
f2=-2*x1*x2+3*x3^2-x4*x5+3*x5*x6+2*x6^2+x4;
f3=2*x2*x6+2*x3^2+2*x3*x4-2*x3*x6+x1+3*x4;
f4=-3*x1*x3+3*x3*x6-x4*x5+2*x5*x6+2*x5;
f5=x1*x5-x2*x3+2*x3*x4+2*x1+3*x3;
f6=-3*x2^2+x3*x4+3*x4*x5-3*x5*x6+x3;
f1=expand(f1^2+f2^3+f4^3+f5^2+f6^4)+f3;
sys=[f1,f2,f3,f4,f5,f6];
var=[x1,x2,x3,x4,x5,x6];
p0=[.37241031193634718410,.31206263724655479831,-.62551476646003205293,-.79406652532081647853,-.15272613192316591654,.12471031335810081347];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%5
disp('--reduce1--')
syms x1 x2
f1=-x1^2+2*x1*x2+x2^2-2*x1-3;
f2=-x1*x2-2*x2^2-2*x1+2*x2+1;
f1=expand(f1^6);f2=expand(f2^6);
sys=[f1,f2];
var=[x1,x2];
p0=[-1.0750256832418609878, -.70298136758092416265];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--reduce2--')
syms x1 x2 x3 x4
f1=2*x1^2*x4+3*x1*x2^2+3*x3^3+2*x1^2+x1*x3+2*x2;
f2=-x1*x4+x2*x3-2*x2*x4+x4^2+3*x2+1;
f3=-2*x1*x4+3*x2*x3-x2*x4-2*x3*x4-2;
f4=-2*x1^2-x1*x4+x2*x3-x2*x4-2*x1+x3;
f1 = expand(f1^2);f4 = expand(f4^3);
sys=[f1,f2,f3,f4];
var=[x1,x2,x3,x4];
p0=[ -1.4171759958094385287,-.97984431143722620430, -0.29933953482788509423e-1,.49354064729672854381];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
disp('--reduce3--')
syms x1 x2 x3 x4
f1=5*x1*x2*x3+3*x2^2-5*x3^2-2*x3*x4-x2;
f2=-5*x1*x2-2*x2^2+5*x3*x4-5*x2-4*x3+1;
f3=x1^2+2*x1*x2-3*x1*x3-3*x1*x4-4*x2*x3-4*x3*x4;
f4=2*x2^2+5*x3^2+2*x2-3*x3+2*x4-5;
f1 = expand(f1^2);f2 = expand(f2^3);f3 = expand(f3^4);
sys=[f1,f2,f3,f4];
var=[x1,x2,x3,x4];
p0=[-.97047742962941707480,-1.1762839730111244305,-.64121478171624420917,.30292682454539563675];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--unreduce1--')
syms x1 x2
f1=-3*x1^2-x2^2-x1-3*x2-1;
f2=x1^2+2*x1*x2-x2^2-2*x1-3*x2-2;
f2=expand(f1^7+f2^7);
sys=[f1,f2];
var=[x1,x2];
p0=[-.14516215374081167951, -2.6540996495557905422];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,1);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--unreduce2--')

syms x1 x2 x3
f1=-x1^2*x2-4*x2^3-4*x1^2+x1*x3+5*x1+4*x3;
f2=-4*x1^2+3*x1*x3-x2^2-5*x2*x3-x3^2+3*x1;
f3=-2*x1*x2^2+5*x2^3-3*x1*x3+5*x2^2-4*x2;
f3=expand(f1^5+f3^5);
sys=[f1,f2,f3];
var=[x1,x2,x3];
p0=[.12962751084531194619, .53917347254472623714,  0.13343513581314737113e-1];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--unreduce3--')
syms x1 x2 x3 x4
f1=-2*x1^2-2*x3^2+3*x1-3*x4+2;
f2=-3*x1^2+3*x1*x2-x2*x3+3*x3*x4-x2+3*x4;
f3=-2*x1*x2+3*x1*x3+x3^2+2;
f4=x2^2-x3^2-2*x3*x4-3*x4^2+1;
f2 = expand(f1^4+f2^4);f3 = expand(f1^3+f3^3);f4 = expand(f1^3+f4^3);
sys=[f1,f2,f3,f4];
var=[x1,x2,x3,x4];
p0=[.70460378690978038392,.64704697541769570182,-1.2268265900925882383,0.36890467785906565175e-1];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
disp('--unreduce4--')
syms x1 x2 x3 x4 x5 x6
f1=3*x1*x6-x2*x3-2*x4-x6;
f2=-3*x1*x4-3*x3*x5+x3*x6-x4*x5-3*x2-2*x6;
f3=-x1*x3-2*x6^2-3*x3-3*x4+3*x5-3*x6;
f4=-3*x1^2-x1*x2-2*x2*x5-x6^2+3;
f5=-3*x1*x4-3*x1*x6-2*x5*x6-3*x1+3*x2;
f6=-2*x1*x5-x2*x4+x3^2+3*x3*x6+x4-2;
f2=expand(f2^4+f1^4);f3 = expand(f3^3+f1^3);f4 = expand(f4^3+f1^3);f5=expand(f5^2+f1^2);f6=expand(f6^2+f1^2);
sys=[f1,f2,f3,f4,f5,f6];
var=[x1,x2,x3,x4,x5,x6];
p0=[1.0327397320667456639, .43731874717150677018,  -.17632096143618344095, -.41198318451474020081,  -.95549939790050492652, -.42944742504340163449];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
disp('--unreduce5--')
syms x1 x2 x3 x4 x5 x6 x7 x8 
f1=3*x2*x4-2*x7^2+2*x7*x8-x1-2*x2-x3;
f2=x1^2+3*x2*x8-2*x5*x6+x3+2*x5;
f3=-2*x1*x7+x4^2+2*x4*x5-x7-2;
f4=-x2*x3+x2*x7-3*x3*x4-2*x4*x6-x4*x7+x3;
f5=3*x2*x4+3*x4*x5+x4*x8+2*x1+3*x2;
f6=2*x1*x5-x2*x5-3*x3^2+3*x3*x4+x4*x7-x3;
f7=-x2^2-x2*x6+x6*x7-2*x6*x8-3*x5+3*x8;
f8=2*x3*x8-3*x4*x8+x8^2-x4+x6;
f2=expand(f2^4+f1^4);f3 = expand(f3^3+f1^3);f4 = expand(f4^3+f1^3);f5=expand(f5^2+f1^2);f6=expand(f6^2+f1^2);f7=expand(f7^2+f1^2);f8=expand(f8^2+f1^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8];
var=[x1,x2,x3,x4,x5,x6,x7,x8];
p0=[ .65434591316798874358,-.64999465624183706794, -0.78435284149677283533e-1, -.25815052698461946076,  -.16708109717246412859,  -.23888592315307183019,-.80006094264236405708, -0.32951834335283076982e-1];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--unreduce6--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
f1=2*x1*x8-2*x10*x6+2*x3*x7+3*x5*x7;
f2=-3*x3^2+x4*x7-3*x6*x8+3*x4-2*x7;
f3=2*x10*x6+x3*x4-2*x4*x9-2*x5*x8-2*x6*x8;
f4=2*x10*x2+x10*x9+x2^2+x2*x9-x3*x8-x5*x8;
f5=-2*x1*x5+x10*x4-x2*x3+x5-3*x7;
f6=3*x1*x2-x1*x5+x10*x5+2*x2*x4+2*x6*x8+2*x5;
f7=-2*x1*x5-x10*x7+3*x4*x5+3*x5^2+x1+2*x5;
f8=x2*x7+x2*x8+2*x3*x7-2*x4*x9+3*x9^2-x6;
f9=-2*x10*x2-x10*x6-x5*x9+3*x7*x9+3*x1-x2;
f10=2*x1*x4+2*x2*x3+x2*x6-3*x7*x9+3*x8*x9+x4;
f2=expand(f1^5+f2^5);f3=expand(f1^3+f3^3);f4=expand(f1^3+f4^3);f5=expand(f1^2+f5^2);f6=expand(f1^2+f6^2);f7=expand(f1^2+f7^2);f8=expand(f1^2+f8^2);f9=expand(f1^2+f9^2);f10=expand(f1^2+f10^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10];
p0=[ 0.48051728217792837953e-1,1.3942182079924831596,  .46720010844425576977,.19161795989956626766,  -.90532915340232688639, -.89293135399348503480,  -.52205755138231030914, -.32257259630009203804,-.44756884341418940367, -.50344896522220257709];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('mul1')
syms x1 x2 x3 x4
f1=-x1^2+x2^2+x3;
f2=x1*x2+x1*x3+x1*x4-x2^2+x1;
f3=x1^2+x3^2+x4;
f4=x1*x4+x2*x4-x3^2+x4^2;
var=[x1,x2,x3,x4];
pstar=rand(1,4)*10^(-4);
sys=[f1,f2,f3,f4];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('mul2')
syms x1 x2 x3 x4
f1=-x1^2+x2^2+x3;
f2=x1*x2+x1*x3+x1*x4-x2^2+x1;
f3=x1^2+x3^2+x4;
f4=x1^3*x2*x3-2*x1*x2*x3*x4^2+2*x1*x3^3*x4-x3*x4^4;
var=[x1,x2,x3,x4];
pstar=rand(1,4)*10^(-4);
sys=[f1,f2,f3,f4];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('mul3')
syms x1 x2 x3 x4
f1=-x1^2+x2^2+x3;
f2=x1*x2+x1*x3+x1*x4-x2^2+x1;
f3=x1^2+x3^2+x4;
f4=-2*x1^2*x3^3*x4^5-x2^7*x3*x4^2-x2^6*x3^3*x4-3*x2^3*x3^5*x4^2-2*x3^3*x4^7;
var=[x1,x2,x3,x4];
pstar=rand(1,4)*10^(-4);
sys=[f1,f2,f3,f4];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('irmul1')
syms x1 x2 x3 x4
f1=-2*x2*x3-3*x1-3*x2+3;
f2=3*x1^2-3*x4^2-2*x1-x2+x4+1;
f3=-2*x2^2-2*x3*x4+2*x1-x3;
f4=-x1*x2-2*x2^2-3*x2*x3+2*x2-3*x4;
f1 = expand(f1^2);f2 = expand(f2^2);f3 = expand(f3^2);f4 = expand(f4^2);
var=[x1,x2,x3,x4];
sys=[f1,f2,f3,f4];
p0=[.39441198342169477001, .42063926408524209715, .65952742034888877806, -.17025693516051679394];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('irmul2')
syms x1 x2 x3 x4
f1=-2*x2*x3-3*x1-3*x2+3;
f2=3*x1^2-3*x4^2-2*x1-x2+x4+1;
f3=-2*x2^2-2*x3*x4+2*x1-x3;
f4=-x1*x2-2*x2^2-3*x2*x3+2*x2-3*x4;
f1 = expand(f1^3);f2 = expand(f2^3);f3 = expand(f3^3);f4 = expand(f4^3);
var=[x1,x2,x3,x4];
sys=[f1,f2,f3,f4];
p0=[.39441198342169477001, .42063926408524209715, .65952742034888877806, -.17025693516051679394];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,10^(-4),0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('irmul3')
syms x1 x2 x3 x4
f1=-2*x2*x3-3*x1-3*x2+3;
f2=3*x1^2-3*x4^2-2*x1-x2+x4+1;
f3=-2*x2^2-2*x3*x4+2*x1-x3;
f4=-x1*x2-2*x2^2-3*x2*x3+2*x2-3*x4;
f1 = expand(f1^3);f2 = expand(f2^3);f3 = expand(f3^5);f4 = expand(f4^5);
var=[x1,x2,x3,x4];
sys=[f1,f2,f3,f4];
p0=[.39441198342169477001, .42063926408524209715, .65952742034888877806, -.17025693516051679394];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,10^(-4),1);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('number1')
syms x1 x2 
f1=3*x1*x2-3*x2^2+3*x1+2*x2-3;
f2=3*x1^2+3*x1*x2-2*x2^2-3*x1+3*x2-2;
f1 = expand(f1^3);f2 = expand(f2^2);
var=[x1,x2];
sys=[f1,f2];
p0=[1.8585633828202561973,-.30351398767160391938];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--number2--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
f1=-2*x1*x10+x2*x3-x3*x7-x6*x7-2*x6*x8-x6;
f2=-3*x1^2+x2*x3-2*x3*x7+3*x6*x8+3*x6;
f3=-x1*x7*x9-3*x10*x6*x9-3*x2*x3*x8+3*x3*x4*x8-x9^3+x2*x4;
f4=x1*x2*x7+x1*x4*x7-3*x10*x3^2-3*x10*x4*x5-2*x2*x4*x9-2*x1*x10;
f5=x1*x2^2-2*x1*x5^2+2*x1*x7*x8-x10^2*x4+2*x6*x8^2;
f6=2*x1*x10*x2+3*x1*x7*x8-3*x4*x6*x8+3*x6*x7^2+3*x4*x8-3;
f7=x1*x2*x6+x1*x4*x8-2*x10^2*x4-3*x10*x5*x9-3*x5*x9^2-2*x10*x5;
f8=2*x1^2*x6-3*x3*x4*x5+3*x5^2*x6-3*x6*x9^2;
f9=x1*x7^2-3*x3*x4*x9-x4*x5^2+2*x6^2*x8-3*x10*x3+x2*x6;
f10=-3*x10^2*x3+2*x10^2*x5+2*x2*x3*x4+2*x2*x4*x9;
f1=expand(f1^3);f2=expand(f2^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10];
p0=[ .63236028616652175522, -.63471364276179181679, 0.59092950748716653817e-2, -.56886000893862479080, .41064969621478427374, .75527706568074848554, 1.1958835083551801729, -.46265945874240304287,-.66109547959707194596,-.76732035194890102388];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--number3--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20
f1=-x1*x8+x11*x12+2*x13*x2+2*x18*x3+2*x4*x5;
f2=-2*x14^2+x14*x18+3*x15*x9-2*x16*x3;
f3=2*x11*x17+x12*x5-2*x12*x8-2*x18*x9;
f4=2*x1*x6+x12*x16+2*x12*x20-3*x14*x8+3*x16*x8;
f5=x1*x19-x1*x8-x10*x12+2*x12*x19-x3*x4+3*x11;
f6=-x10*x11+3*x12*x9+3*x18*x4+2*x2*x5;
f7=-x14*x3-x19*x5-2*x20*x7-3*x3^2;
f8=3*x10*x19-3*x13*x4+3*x15*x17+x9^2+x3;
f9=x1*x18+x11*x6-x8^2-3*x18+3*x8;
f10=3*x13*x6-x17*x6+2*x18*x19-3*x18*x8+x2*x9-3*x8;
f11=-2*x1*x3-2*x16*x19-3*x16*x6-3*x17*x3+2*x2*x20+x4*x7;
f12=-3*x1*x7+3*x11*x7-3*x14^2+x17^2-x8;
f13=-x10*x19+x14*x2-x2^2-x20*x8-3*x3-x5;
f14=2*x10*x14-2*x15*x3+2*x16^2-2*x3*x9;
f15=-3*x1*x8+x10*x14+3*x11*x12+2*x14*x17+x18*x6+3*x19*x4;
f16=-x1*x11-3*x1*x3-x1*x8-2*x14*x17+3;
f17=-x12*x17-3*x12*x8-x13*x18-3*x17^2+2*x18*x7+x2;
f18=3*x10*x19-2*x11*x13+2*x15*x7-3*x3^2-2*x3*x8-2*x5^2;
f19=-3*x13*x17+3*x2*x3+2*x6;
f20=3*x11*x8-3*x13*x16-3*x14^2-x20*x3-2*x17+x18;
f1=expand(f1^3);f2=expand(f2^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20];
p0=[1.2330361429448079051, -2.5251768498783713379,-.48087601821939917434,1.4500955822751700991,-.77889907451782556657,-1.4838102216716242212, -1.6108734852221608057,3.3757184827641062822, -.18350170100168452324,-0.67704824657053934243,0.28403105318342077821,-0.062144863146355766978,-1.0928453523192901196,-0.64624179368755031992,-0.90123317021238789959,-0.28997605490177930072,-0.20596708760655823551,-0.95631158488139721451,2.3946400901175364145,-.26715181007094010647 ];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--number4--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30
f1=-x14*x29+x15*x9+x16*x17+3*x16*x9-2*x17*x27+2*x9;
f2=-x1*x14-x11*x2-x12*x15-2*x17*x19-2*x24*x5+2*x26*x9;
f3=-x1*x5+x14*x2-x15*x5-3*x7*x9;
f4=-2*x10*x22-x10*x9-3*x12*x19+2*x18*x25+2*x2*x28-3*x11;
f5=x11*x26-x11*x8+x12*x7+2*x13*x28+x13*x5;
f6=-x10*x16+x12*x3+2*x2*x5-3*x21*x6-3*x30^2;
f7=-x14*x23-3*x20*x9-2*x22*x28+3*x24*x27-2*x3*x8;
f8=3*x10*x17+x10*x23-3*x12*x16-3*x2*x30-2*x28*x30;
f9=-2*x1*x15-2*x11*x21+x20*x24+3*x20*x7+3*x30*x4-2*x15;
f10=-3*x17^2+3*x18*x30-2*x21*x22-2*x25*x7+x3*x4;
f11=-x13*x6+3*x14*x22+x14*x27-x19*x24-x23*x26+3*x18;
f12=x11*x14+3*x19*x3-x2*x3-2*x21*x4-2*x28*x9+2*x9;
f13=x1*x19+3*x11*x20+x15*x17+x30*x4;
f14=2*x11*x28+x14*x16+3*x15*x16+x17*x2-3*x3*x30+2;
f15=-x1*x9-3*x10*x28+3*x14*x24+x15*x6-x19*x26-2*x24*x3;
f16=2*x11*x3+2*x15*x20-x17*x3+x20^2+3*x27*x3;
f17=3*x11*x19-2*x13*x20-x16*x23-2*x16*x24-x20*x3+3*x22*x23;
f18=x15*x29+2*x22*x27-3*x1+x14+x18+2*x8;
f19=3*x10*x25+x13*x20+3*x3^2;
f20=-2*x1*x27-3*x13*x17+3*x17^2-x17*x7+x22*x30-2*x3*x9;
f21=x10*x19-3*x10*x6+3*x13*x25+x18*x23+2*x7^2+x12;
f22=x10*x7+x13*x26-2*x20*x22-3*x23*x5-2*x28*x29-2*x6*x9;
f23=-x12*x23+x15*x18-x15*x23-x16*x5+x19*x24-2*x21*x7;
f24=-3*x11*x25+2*x16*x29-3*x20*x9-3*x21*x4+2*x6^2;
f25=-3*x10*x24-x13*x7+x14*x7+3*x17*x2+2*x30*x9;
f26=x1*x14+2*x11*x13+x14*x27+x24*x6-3*x27*x28-x25;
f27=x11*x9+2*x12*x24+x18*x29+3*x23*x27-2*x27*x9;
f28=2*x11*x5-x18*x25-x5^2+3*x21+x26;
f29=-3*x1*x21-x17*x20+x22*x7-2*x23*x4+2*x29;
f30=x11*x14-x13*x2-3*x13*x6+3*x16*x8+3*x22*x29+3*x11;
f1=expand(f1^3);f2=expand(f2^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,f28,f29,f30];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20,x21 ,x22 ,x23 ,x24 ,x25 ,x26 ,x27 ,x28 ,x29 ,x30];
p0=[-0.82013408111505359244e-1,.53698383582577320570,-.98520969768254420802,-.34212374338185119603, .50546103026052401416, 0.74551660022723253649e-1, -.87684746829956597173, -.10111587428002111579, 0.14586155306016414501e-1,-2.3502904631613996801,-0.32121559389998833042,-1.2218841590089432337,-0.65279507929679076743,-0.27160339417446517049,-0.13061847829681723131,-0.17990494053050712165,-0.12181021331050581713,0.30970639494191478567,0.10854884887196472808,.25236687258632669468,  -.47353094784524376855, -.35310504380749803733, .57040219638504753893, 0.78091957198114694389e-1,  .38962142987297962094,  2.1214758751427046497, .17278295125890844362,0.21072791551662819433e-1,  -.30707436748622123269,-.69034764608789617172];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--big1--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20
f1=-x1*x8+x11*x12+2*x13*x2+2*x18*x3+2*x4*x5;
f2=-2*x14^2+x14*x18+3*x15*x9-2*x16*x3;
f3=2*x11*x17+x12*x5-2*x12*x8-2*x18*x9;
f4=2*x1*x6+x12*x16+2*x12*x20-3*x14*x8+3*x16*x8;
f5=x1*x19-x1*x8-x10*x12+2*x12*x19-x3*x4+3*x11;
f6=-x10*x11+3*x12*x9+3*x18*x4+2*x2*x5;
f7=-x14*x3-x19*x5-2*x20*x7-3*x3^2;
f8=3*x10*x19-3*x13*x4+3*x15*x17+x9^2+x3;
f9=x1*x18+x11*x6-x8^2-3*x18+3*x8;
f10=3*x13*x6-x17*x6+2*x18*x19-3*x18*x8+x2*x9-3*x8;
f11=-2*x1*x3-2*x16*x19-3*x16*x6-3*x17*x3+2*x2*x20+x4*x7;
f12=-3*x1*x7+3*x11*x7-3*x14^2+x17^2-x8;
f13=-x10*x19+x14*x2-x2^2-x20*x8-3*x3-x5;
f14=2*x10*x14-2*x15*x3+2*x16^2-2*x3*x9;
f15=-3*x1*x8+x10*x14+3*x11*x12+2*x14*x17+x18*x6+3*x19*x4;
f16=-x1*x11-3*x1*x3-x1*x8-2*x14*x17+3;
f17=-x12*x17-3*x12*x8-x13*x18-3*x17^2+2*x18*x7+x2;
f18=3*x10*x19-2*x11*x13+2*x15*x7-3*x3^2-2*x3*x8-2*x5^2;
f19=-3*x13*x17+3*x2*x3+2*x6;
f20=3*x11*x8-3*x13*x16-3*x14^2-x20*x3-2*x17+x18;
f1=expand(f1^4);f2=expand(f2^2);f3=expand(f3^2);f4=expand(f4^2);f5=expand(f5^2);f6=expand(f6^2);f7=expand(f7^2);f8=expand(f8^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20];
p0=[1.2330361429448079051, -2.5251768498783713379,-.48087601821939917434,1.4500955822751700991,-.77889907451782556657,-1.4838102216716242212, -1.6108734852221608057,3.3757184827641062822, -.18350170100168452324,-0.67704824657053934243,0.28403105318342077821,-0.062144863146355766978,-1.0928453523192901196,-0.64624179368755031992,-0.90123317021238789959,-0.28997605490177930072,-0.20596708760655823551,-0.95631158488139721451,2.3946400901175364145,-.26715181007094010647 ];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--big2--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20
f1=-x1*x8+x11*x12+2*x13*x2+2*x18*x3+2*x4*x5;
f2=-2*x14^2+x14*x18+3*x15*x9-2*x16*x3;
f3=2*x11*x17+x12*x5-2*x12*x8-2*x18*x9;
f4=2*x1*x6+x12*x16+2*x12*x20-3*x14*x8+3*x16*x8;
f5=x1*x19-x1*x8-x10*x12+2*x12*x19-x3*x4+3*x11;
f6=-x10*x11+3*x12*x9+3*x18*x4+2*x2*x5;
f7=-x14*x3-x19*x5-2*x20*x7-3*x3^2;
f8=3*x10*x19-3*x13*x4+3*x15*x17+x9^2+x3;
f9=x1*x18+x11*x6-x8^2-3*x18+3*x8;
f10=3*x13*x6-x17*x6+2*x18*x19-3*x18*x8+x2*x9-3*x8;
f11=-2*x1*x3-2*x16*x19-3*x16*x6-3*x17*x3+2*x2*x20+x4*x7;
f12=-3*x1*x7+3*x11*x7-3*x14^2+x17^2-x8;
f13=-x10*x19+x14*x2-x2^2-x20*x8-3*x3-x5;
f14=2*x10*x14-2*x15*x3+2*x16^2-2*x3*x9;
f15=-3*x1*x8+x10*x14+3*x11*x12+2*x14*x17+x18*x6+3*x19*x4;
f16=-x1*x11-3*x1*x3-x1*x8-2*x14*x17+3;
f17=-x12*x17-3*x12*x8-x13*x18-3*x17^2+2*x18*x7+x2;
f18=3*x10*x19-2*x11*x13+2*x15*x7-3*x3^2-2*x3*x8-2*x5^2;
f19=-3*x13*x17+3*x2*x3+2*x6;
f20=3*x11*x8-3*x13*x16-3*x14^2-x20*x3-2*x17+x18;
f1=expand(f1^4);f2=expand(f2^2);f3=expand(f3^2);f4=expand(f4^2);f5=expand(f5^2);f6=expand(f6^2);f7=expand(f7^2);f8=expand(f8^2);f9=expand(f9^2);f10=expand(f10^2);f11=expand(f11^2);f12=expand(f12^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20];
p0=[1.2330361429448079051, -2.5251768498783713379,-.48087601821939917434,1.4500955822751700991,-.77889907451782556657,-1.4838102216716242212, -1.6108734852221608057,3.3757184827641062822, -.18350170100168452324,-0.67704824657053934243,0.28403105318342077821,-0.062144863146355766978,-1.0928453523192901196,-0.64624179368755031992,-0.90123317021238789959,-0.28997605490177930072,-0.20596708760655823551,-0.95631158488139721451,2.3946400901175364145,-.26715181007094010647 ];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--big3--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30
f1=-x14*x29+x15*x9+x16*x17+3*x16*x9-2*x17*x27+2*x9;
f2=-x1*x14-x11*x2-x12*x15-2*x17*x19-2*x24*x5+2*x26*x9;
f3=-x1*x5+x14*x2-x15*x5-3*x7*x9;
f4=-2*x10*x22-x10*x9-3*x12*x19+2*x18*x25+2*x2*x28-3*x11;
f5=x11*x26-x11*x8+x12*x7+2*x13*x28+x13*x5;
f6=-x10*x16+x12*x3+2*x2*x5-3*x21*x6-3*x30^2;
f7=-x14*x23-3*x20*x9-2*x22*x28+3*x24*x27-2*x3*x8;
f8=3*x10*x17+x10*x23-3*x12*x16-3*x2*x30-2*x28*x30;
f9=-2*x1*x15-2*x11*x21+x20*x24+3*x20*x7+3*x30*x4-2*x15;
f10=-3*x17^2+3*x18*x30-2*x21*x22-2*x25*x7+x3*x4;
f11=-x13*x6+3*x14*x22+x14*x27-x19*x24-x23*x26+3*x18;
f12=x11*x14+3*x19*x3-x2*x3-2*x21*x4-2*x28*x9+2*x9;
f13=x1*x19+3*x11*x20+x15*x17+x30*x4;
f14=2*x11*x28+x14*x16+3*x15*x16+x17*x2-3*x3*x30+2;
f15=-x1*x9-3*x10*x28+3*x14*x24+x15*x6-x19*x26-2*x24*x3;
f16=2*x11*x3+2*x15*x20-x17*x3+x20^2+3*x27*x3;
f17=3*x11*x19-2*x13*x20-x16*x23-2*x16*x24-x20*x3+3*x22*x23;
f18=x15*x29+2*x22*x27-3*x1+x14+x18+2*x8;
f19=3*x10*x25+x13*x20+3*x3^2;
f20=-2*x1*x27-3*x13*x17+3*x17^2-x17*x7+x22*x30-2*x3*x9;
f21=x10*x19-3*x10*x6+3*x13*x25+x18*x23+2*x7^2+x12;
f22=x10*x7+x13*x26-2*x20*x22-3*x23*x5-2*x28*x29-2*x6*x9;
f23=-x12*x23+x15*x18-x15*x23-x16*x5+x19*x24-2*x21*x7;
f24=-3*x11*x25+2*x16*x29-3*x20*x9-3*x21*x4+2*x6^2;
f25=-3*x10*x24-x13*x7+x14*x7+3*x17*x2+2*x30*x9;
f26=x1*x14+2*x11*x13+x14*x27+x24*x6-3*x27*x28-x25;
f27=x11*x9+2*x12*x24+x18*x29+3*x23*x27-2*x27*x9;
f28=2*x11*x5-x18*x25-x5^2+3*x21+x26;
f29=-3*x1*x21-x17*x20+x22*x7-2*x23*x4+2*x29;
f30=x11*x14-x13*x2-3*x13*x6+3*x16*x8+3*x22*x29+3*x11;
f1=expand(f1^4);f2=expand(f2^3);f3=expand(f3^2);f4=expand(f4^2);f5=expand(f5^2);f6=expand(f6^2);f7=expand(f7^2);f8=expand(f8^2);f9=expand(f9^2);f10=expand(f10^2);f11=expand(f11^2);f12=expand(f12^2);f13=expand(f13^2);f14=expand(f14^2);f15=expand(f15^2);f16=expand(f16^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,f28,f29,f30];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20,x21 ,x22 ,x23 ,x24 ,x25 ,x26 ,x27 ,x28 ,x29 ,x30];
p0=[-0.82013408111505359244e-1,.53698383582577320570,-.98520969768254420802,-.34212374338185119603, .50546103026052401416, 0.74551660022723253649e-1, -.87684746829956597173, -.10111587428002111579, 0.14586155306016414501e-1,-2.3502904631613996801,-0.32121559389998833042,-1.2218841590089432337,-0.65279507929679076743,-0.27160339417446517049,-0.13061847829681723131,-0.17990494053050712165,-0.12181021331050581713,0.30970639494191478567,0.10854884887196472808,.25236687258632669468,  -.47353094784524376855, -.35310504380749803733, .57040219638504753893, 0.78091957198114694389e-1,  .38962142987297962094,  2.1214758751427046497, .17278295125890844362,0.21072791551662819433e-1,  -.30707436748622123269,-.69034764608789617172];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--big4--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30
f1=-x14*x29+x15*x9+x16*x17+3*x16*x9-2*x17*x27+2*x9;
f2=-x1*x14-x11*x2-x12*x15-2*x17*x19-2*x24*x5+2*x26*x9;
f3=-x1*x5+x14*x2-x15*x5-3*x7*x9;
f4=-2*x10*x22-x10*x9-3*x12*x19+2*x18*x25+2*x2*x28-3*x11;
f5=x11*x26-x11*x8+x12*x7+2*x13*x28+x13*x5;
f6=-x10*x16+x12*x3+2*x2*x5-3*x21*x6-3*x30^2;
f7=-x14*x23-3*x20*x9-2*x22*x28+3*x24*x27-2*x3*x8;
f8=3*x10*x17+x10*x23-3*x12*x16-3*x2*x30-2*x28*x30;
f9=-2*x1*x15-2*x11*x21+x20*x24+3*x20*x7+3*x30*x4-2*x15;
f10=-3*x17^2+3*x18*x30-2*x21*x22-2*x25*x7+x3*x4;
f11=-x13*x6+3*x14*x22+x14*x27-x19*x24-x23*x26+3*x18;
f12=x11*x14+3*x19*x3-x2*x3-2*x21*x4-2*x28*x9+2*x9;
f13=x1*x19+3*x11*x20+x15*x17+x30*x4;
f14=2*x11*x28+x14*x16+3*x15*x16+x17*x2-3*x3*x30+2;
f15=-x1*x9-3*x10*x28+3*x14*x24+x15*x6-x19*x26-2*x24*x3;
f16=2*x11*x3+2*x15*x20-x17*x3+x20^2+3*x27*x3;
f17=3*x11*x19-2*x13*x20-x16*x23-2*x16*x24-x20*x3+3*x22*x23;
f18=x15*x29+2*x22*x27-3*x1+x14+x18+2*x8;
f19=3*x10*x25+x13*x20+3*x3^2;
f20=-2*x1*x27-3*x13*x17+3*x17^2-x17*x7+x22*x30-2*x3*x9;
f21=x10*x19-3*x10*x6+3*x13*x25+x18*x23+2*x7^2+x12;
f22=x10*x7+x13*x26-2*x20*x22-3*x23*x5-2*x28*x29-2*x6*x9;
f23=-x12*x23+x15*x18-x15*x23-x16*x5+x19*x24-2*x21*x7;
f24=-3*x11*x25+2*x16*x29-3*x20*x9-3*x21*x4+2*x6^2;
f25=-3*x10*x24-x13*x7+x14*x7+3*x17*x2+2*x30*x9;
f26=x1*x14+2*x11*x13+x14*x27+x24*x6-3*x27*x28-x25;
f27=x11*x9+2*x12*x24+x18*x29+3*x23*x27-2*x27*x9;
f28=2*x11*x5-x18*x25-x5^2+3*x21+x26;
f29=-3*x1*x21-x17*x20+x22*x7-2*x23*x4+2*x29;
f30=x11*x14-x13*x2-3*x13*x6+3*x16*x8+3*x22*x29+3*x11;
f1=expand(f1^4);f2=expand(f2^3);f3=expand(f3^2);f4=expand(f4^2);f5=expand(f5^2);f6=expand(f6^2);f7=expand(f7^2);f8=expand(f8^2);f9=expand(f9^2);f10=expand(f10^2);f11=expand(f11^2);f12=expand(f12^2);f13=expand(f13^2);f14=expand(f14^2);f15=expand(f15^2);f16=expand(f16^2);f17=expand(f17^2);f18=expand(f18^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,f28,f29,f30];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20,x21 ,x22 ,x23 ,x24 ,x25 ,x26 ,x27 ,x28 ,x29 ,x30];
p0=[-0.82013408111505359244e-1,.53698383582577320570,-.98520969768254420802,-.34212374338185119603, .50546103026052401416, 0.74551660022723253649e-1, -.87684746829956597173, -.10111587428002111579, 0.14586155306016414501e-1,-2.3502904631613996801,-0.32121559389998833042,-1.2218841590089432337,-0.65279507929679076743,-0.27160339417446517049,-0.13061847829681723131,-0.17990494053050712165,-0.12181021331050581713,0.30970639494191478567,0.10854884887196472808,.25236687258632669468,  -.47353094784524376855, -.35310504380749803733, .57040219638504753893, 0.78091957198114694389e-1,  .38962142987297962094,  2.1214758751427046497, .17278295125890844362,0.21072791551662819433e-1,  -.30707436748622123269,-.69034764608789617172];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--big5--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 x21 x22 x23 x24 x25 x26 x27 x28 x29 x30
f1=-x14*x29+x15*x9+x16*x17+3*x16*x9-2*x17*x27+2*x9;
f2=-x1*x14-x11*x2-x12*x15-2*x17*x19-2*x24*x5+2*x26*x9;
f3=-x1*x5+x14*x2-x15*x5-3*x7*x9;
f4=-2*x10*x22-x10*x9-3*x12*x19+2*x18*x25+2*x2*x28-3*x11;
f5=x11*x26-x11*x8+x12*x7+2*x13*x28+x13*x5;
f6=-x10*x16+x12*x3+2*x2*x5-3*x21*x6-3*x30^2;
f7=-x14*x23-3*x20*x9-2*x22*x28+3*x24*x27-2*x3*x8;
f8=3*x10*x17+x10*x23-3*x12*x16-3*x2*x30-2*x28*x30;
f9=-2*x1*x15-2*x11*x21+x20*x24+3*x20*x7+3*x30*x4-2*x15;
f10=-3*x17^2+3*x18*x30-2*x21*x22-2*x25*x7+x3*x4;
f11=-x13*x6+3*x14*x22+x14*x27-x19*x24-x23*x26+3*x18;
f12=x11*x14+3*x19*x3-x2*x3-2*x21*x4-2*x28*x9+2*x9;
f13=x1*x19+3*x11*x20+x15*x17+x30*x4;
f14=2*x11*x28+x14*x16+3*x15*x16+x17*x2-3*x3*x30+2;
f15=-x1*x9-3*x10*x28+3*x14*x24+x15*x6-x19*x26-2*x24*x3;
f16=2*x11*x3+2*x15*x20-x17*x3+x20^2+3*x27*x3;
f17=3*x11*x19-2*x13*x20-x16*x23-2*x16*x24-x20*x3+3*x22*x23;
f18=x15*x29+2*x22*x27-3*x1+x14+x18+2*x8;
f19=3*x10*x25+x13*x20+3*x3^2;
f20=-2*x1*x27-3*x13*x17+3*x17^2-x17*x7+x22*x30-2*x3*x9;
f21=x10*x19-3*x10*x6+3*x13*x25+x18*x23+2*x7^2+x12;
f22=x10*x7+x13*x26-2*x20*x22-3*x23*x5-2*x28*x29-2*x6*x9;
f23=-x12*x23+x15*x18-x15*x23-x16*x5+x19*x24-2*x21*x7;
f24=-3*x11*x25+2*x16*x29-3*x20*x9-3*x21*x4+2*x6^2;
f25=-3*x10*x24-x13*x7+x14*x7+3*x17*x2+2*x30*x9;
f26=x1*x14+2*x11*x13+x14*x27+x24*x6-3*x27*x28-x25;
f27=x11*x9+2*x12*x24+x18*x29+3*x23*x27-2*x27*x9;
f28=2*x11*x5-x18*x25-x5^2+3*x21+x26;
f29=-3*x1*x21-x17*x20+x22*x7-2*x23*x4+2*x29;
f30=x11*x14-x13*x2-3*x13*x6+3*x16*x8+3*x22*x29+3*x11;
f1=expand(f1^4);f2=expand(f2^3);f3=expand(f3^3);f4=expand(f4^3);f5=expand(f5^3);f6=expand(f6^2);f7=expand(f7^2);f8=expand(f8^2);f9=expand(f9^2);f10=expand(f10^2);f11=expand(f11^2);f12=expand(f12^2);f13=expand(f13^2);f14=expand(f14^2);f15=expand(f15^2);f16=expand(f16^2);f17=expand(f17^2);f18=expand(f18^2);f19=expand(f19^2);f20=expand(f20^2);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14,f15,f16,f17,f18,f19,f20,f21,f22,f23,f24,f25,f26,f27,f28,f29,f30];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10,x11 ,x12 ,x13 ,x14 ,x15 ,x16 ,x17 ,x18 ,x19 ,x20,x21 ,x22 ,x23 ,x24 ,x25 ,x26 ,x27 ,x28 ,x29 ,x30];
p0=[-0.82013408111505359244e-1,.53698383582577320570,-.98520969768254420802,-.34212374338185119603, .50546103026052401416, 0.74551660022723253649e-1, -.87684746829956597173, -.10111587428002111579, 0.14586155306016414501e-1,-2.3502904631613996801,-0.32121559389998833042,-1.2218841590089432337,-0.65279507929679076743,-0.27160339417446517049,-0.13061847829681723131,-0.17990494053050712165,-0.12181021331050581713,0.30970639494191478567,0.10854884887196472808,.25236687258632669468,  -.47353094784524376855, -.35310504380749803733, .57040219638504753893, 0.78091957198114694389e-1,  .38962142987297962094,  2.1214758751427046497, .17278295125890844362,0.21072791551662819433e-1,  -.30707436748622123269,-.69034764608789617172];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone1--')
syms x1 x2 x3 x4
f1=-2*x1^2-2*x3^2+3*x1-3*x4+2;
f2=-3*x1^2+3*x1*x2-x2*x3+3*x3*x4-x2+3*x4;
f3=-2*x1*x2+3*x1*x3+x3^2+2;
f4=x2^2-x3^2-2*x3*x4-3*x4^2+1;
f2 = expand(f2^2+f1);f3 = expand(f1+f3^2);f4 = expand(f1+f4^2);
sys=[f1,f2,f3,f4];
var=[x1,x2,x3,x4];
p0=[.70460378690978038392,.64704697541769570182,-1.2268265900925882383,0.36890467785906565175e-1];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone2--')
syms x1 x2 x3 x4 x5 x6
f1=3*x1*x6-x2*x3-2*x4-x6;
f2=-3*x1*x4-3*x3*x5+x3*x6-x4*x5-3*x2-2*x6;
f3=-x1*x3-2*x6^2-3*x3-3*x4+3*x5-3*x6;
f4=-3*x1^2-x1*x2-2*x2*x5-x6^2+3;
f5=-3*x1*x4-3*x1*x6-2*x5*x6-3*x1+3*x2;
f6=-2*x1*x5-x2*x4+x3^2+3*x3*x6+x4-2;
f2=expand(f2^2+f1);f3 = expand(f3^2+f1);f4 = expand(f4^2+f1);f5=expand(f5^2+f1);f6=expand(f6^2+f1);
sys=[f1,f2,f3,f4,f5,f6];
var=[x1,x2,x3,x4,x5,x6];
p0=[1.0327397320667456639, .43731874717150677018,  -.17632096143618344095, -.41198318451474020081,  -.95549939790050492652, -.42944742504340163449];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone3--')
syms x1 x2 x3 x4 x5 x6
f1=3*x1*x6-x2*x3-2*x4-x6;
f2=-3*x1*x4-3*x3*x5+x3*x6-x4*x5-3*x2-2*x6;
f3=-x1*x3-2*x6^2-3*x3-3*x4+3*x5-3*x6;
f4=-3*x1^2-x1*x2-2*x2*x5-x6^2+3;
f5=-3*x1*x4-3*x1*x6-2*x5*x6-3*x1+3*x2;
f6=-2*x1*x5-x2*x4+x3^2+3*x3*x6+x4-2;
f2=expand(f2^4+f1);f3 = expand(f3^3+f1);f4 = expand(f4^2+f1);f5=expand(f5^2+f1);f6=expand(f6^2+f1);
sys=[f1,f2,f3,f4,f5,f6];
var=[x1,x2,x3,x4,x5,x6];
p0=[1.0327397320667456639, .43731874717150677018,  -.17632096143618344095, -.41198318451474020081,  -.95549939790050492652, -.42944742504340163449];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone4--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
f1=-2*x1*x10+x2*x3-x3*x7-x6*x7-2*x6*x8-x6;
f2=-3*x1^2+x2*x3-2*x3*x7+3*x6*x8+3*x6;
f3=-x1*x7*x9-3*x10*x6*x9-3*x2*x3*x8+3*x3*x4*x8-x9^3+x2*x4;
f4=x1*x2*x7+x1*x4*x7-3*x10*x3^2-3*x10*x4*x5-2*x2*x4*x9-2*x1*x10;
f5=x1*x2^2-2*x1*x5^2+2*x1*x7*x8-x10^2*x4+2*x6*x8^2;
f6=2*x1*x10*x2+3*x1*x7*x8-3*x4*x6*x8+3*x6*x7^2+3*x4*x8-3;
f7=x1*x2*x6+x1*x4*x8-2*x10^2*x4-3*x10*x5*x9-3*x5*x9^2-2*x10*x5;
f8=2*x1^2*x6-3*x3*x4*x5+3*x5^2*x6-3*x6*x9^2;
f9=x1*x7^2-3*x3*x4*x9-x4*x5^2+2*x6^2*x8-3*x10*x3+x2*x6;
f10=-3*x10^2*x3+2*x10^2*x5+2*x2*x3*x4+2*x2*x4*x9;
f1=expand(f1^2+f10);f2=expand(f2^2+f10);f3=expand(f3^2+f10);f4=expand(f4^2+f10);f5=expand(f5^2+f10);f6=expand(f6^2+f10);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10];
p0=[ .63236028616652175522, -.63471364276179181679, 0.59092950748716653817e-2, -.56886000893862479080, .41064969621478427374, .75527706568074848554, 1.1958835083551801729, -.46265945874240304287,-.66109547959707194596,-.76732035194890102388];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone5--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
f1=-2*x1*x10+x2*x3-x3*x7-x6*x7-2*x6*x8-x6;
f2=-3*x1^2+x2*x3-2*x3*x7+3*x6*x8+3*x6;
f3=-x1*x7*x9-3*x10*x6*x9-3*x2*x3*x8+3*x3*x4*x8-x9^3+x2*x4;
f4=x1*x2*x7+x1*x4*x7-3*x10*x3^2-3*x10*x4*x5-2*x2*x4*x9-2*x1*x10;
f5=x1*x2^2-2*x1*x5^2+2*x1*x7*x8-x10^2*x4+2*x6*x8^2;
f6=2*x1*x10*x2+3*x1*x7*x8-3*x4*x6*x8+3*x6*x7^2+3*x4*x8-3;
f7=x1*x2*x6+x1*x4*x8-2*x10^2*x4-3*x10*x5*x9-3*x5*x9^2-2*x10*x5;
f8=2*x1^2*x6-3*x3*x4*x5+3*x5^2*x6-3*x6*x9^2;
f9=x1*x7^2-3*x3*x4*x9-x4*x5^2+2*x6^2*x8-3*x10*x3+x2*x6;
f10=-3*x10^2*x3+2*x10^2*x5+2*x2*x3*x4+2*x2*x4*x9;
f1=expand(f1^4+f10);f2=expand(f2^2+f10);f3=expand(f3^2+f10);f4=expand(f4^2+f10);f5=expand(f5^2+f10);f6=expand(f6^2+f10);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10];
p0=[ .63236028616652175522, -.63471364276179181679, 0.59092950748716653817e-2, -.56886000893862479080, .41064969621478427374, .75527706568074848554, 1.1958835083551801729, -.46265945874240304287,-.66109547959707194596,-.76732035194890102388];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('--rankone6--')
syms x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
f1=-2*x1*x10+x2*x3-x3*x7-x6*x7-2*x6*x8-x6;
f2=-3*x1^2+x2*x3-2*x3*x7+3*x6*x8+3*x6;
f3=-x1*x7*x9-3*x10*x6*x9-3*x2*x3*x8+3*x3*x4*x8-x9^3+x2*x4;
f4=x1*x2*x7+x1*x4*x7-3*x10*x3^2-3*x10*x4*x5-2*x2*x4*x9-2*x1*x10;
f5=x1*x2^2-2*x1*x5^2+2*x1*x7*x8-x10^2*x4+2*x6*x8^2;
f6=2*x1*x10*x2+3*x1*x7*x8-3*x4*x6*x8+3*x6*x7^2+3*x4*x8-3;
f7=x1*x2*x6+x1*x4*x8-2*x10^2*x4-3*x10*x5*x9-3*x5*x9^2-2*x10*x5;
f8=2*x1^2*x6-3*x3*x4*x5+3*x5^2*x6-3*x6*x9^2;
f9=x1*x7^2-3*x3*x4*x9-x4*x5^2+2*x6^2*x8-3*x10*x3+x2*x6;
f10=-3*x10^2*x3+2*x10^2*x5+2*x2*x3*x4+2*x2*x4*x9;
f1=expand(f1^4+f10);f2=expand(f2^2+f10);f3=expand(f3^3+f10);f4=expand(f4^2+f10);f5=expand(f5^2+f10);f6=expand(f6^2+f10);f7=expand(f7^2+f10);f8=expand(f8^2+f10);
sys=[f1,f2,f3,f4,f5,f6,f7,f8,f9,f10];
var=[x1 ,x2 ,x3 ,x4 ,x5 ,x6 ,x7 ,x8 ,x9 ,x10];
p0=[ .63236028616652175522, -.63471364276179181679, 0.59092950748716653817e-2, -.56886000893862479080, .41064969621478427374, .75527706568074848554, 1.1958835083551801729, -.46265945874240304287,-.66109547959707194596,-.76732035194890102388];
pstar=roundn(p0,-6);
tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
abs(p(1:n)-p0)
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('new1')
syms x1 x2 x3 
f1=2*x1+2*x1^2+2*x2+2*x2^2+x3^2-1;
f2=(x1+x2-x3-1)^3-x1^3;
f3=(2*x1^3+5*x2^2+10*x3+5*x3^2+5)^3-1000*x1^5;
var=[x1,x2,x3];
pstar=rand(1,3)*10^(-4)+[0,0,-1];
sys=[f1,f2,f3];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,1);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('breadthtwo 5')
sys=[myseq('x[i]^3+x[i]^2','i',1)+myseq('-x[i]^2','i',2);myseq('x[i]^3+x[i]^2','i',2:4)+myseq('-x[i]','i',3:5);myseq('x[j]^2','j',5)].';
var=[myseq('x[i]','i',1:5)].';
pstar = rand(1,5)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('breadthtwo 100')
sys=[myseq('x[i]^3+x[i]^2','i',1)+myseq('-x[i]^2','i',2);myseq('x[i]^3+x[i]^2','i',2:99)+myseq('-x[i]','i',3:100);myseq('x[j]^2','j',100)].';
var=[myseq('x[i]','i',1:100)].';
pstar = rand(1,100)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('breadthtwo 1000')
sys=[myseq('x[i]^3+x[i]^2','i',1)+myseq('-x[i]^2','i',2);myseq('x[i]^3+x[i]^2','i',2:999)+myseq('-x[i]','i',3:1000);myseq('x[j]^2','j',1000)].';
var=[myseq('x[i]','i',1:1000)].';
pstar = rand(1,1000)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('nonpoly1')
syms x y 
f1=1-cos(x^2);
f2=sin(y)+x^2*exp(x+y);
var=[x,y];
pstar=rand(1,2)*10^(-4)+[0,0];
sys=[f1,f2];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('nonpoly2')
syms x y z
f1=x^2*sin(y);
f2=y-z^2;
f3=z+sin(x^2);
var=[x,y,z];
pstar=rand(1,3)*10^(-4)+[0,0,0];
sys=[f1,f2,f3];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('nonpoly3')
syms x y z
f1=x^2*sin(y);
f2=y-z^2;
f3=z+sin(x^5);
var=[x,y,z];
pstar=rand(1,3)*10^(-4)+[0,0,0];
sys=[f1,f2,f3];

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large1')
for i=1:50 syms (['x',num2str(i)]);
end
sys=[-3*x21*x35*x50-x21*x22*x28-3*x26*x36*x46+2*x33*x46*x49+x38*x43*x49-2*x5*x8*x49+2*x15*x33*x37-3*x5*x33*x48-x3*x9*x10-3*x18*x23*x50+2*x1*x9*x24+3*x6*x38*x47-2*x2*x10*x12+x9*x28*x30+3*x6*x18^2-x7*x15*x37-3*x10*x31*x48+3*x17^2*x26-2*x4*x9*x37-x8^2*x47+x9*x23*x45-3*x7*x13*x26+x16*x30*x46-2*x18*x36^2-2*x11*x12*x36, -2*x21*x25*x32+x31*x40*x44+2*x29*x45*x50-3*x35*x42*x47-3*x25*x46*x49+3*x40*x44*x47-3*x34^2*x50-2*x23*x35*x36+3*x5*x15*x19+2*x8^2*x10-2*x9*x13*x21+2*x16*x40*x49-x10*x15*x36+x15*x31*x41-3*x5*x17*x42-2*x8*x23*x49-3*x10*x15*x26-3*x4*x11*x50+2*x1*x10*x42+2*x16*x34*x39-2*x16*x35*x41+x16*x34*x47-x6*x35*x44-2*x8*x26*x33+x4*x12*x40-2*x13*x19*x42, -2*x45*x47*x49+x21*x31*x49-2*x27*x40*x46+x20*x42*x49+3*x20*x21^2-x20*x29*x42+3*x20^2*x21-x23*x32*x39+x31*x35*x41+x5*x7*x48+3*x7*x27*x47-x4*x26*x43-3*x4*x14*x47+3*x5*x32*x38+3*x11*x18*x33+2*x3^2*x5-3*x9*x38^2-2*x14*x20*x30+2*x16*x43^2-2*x15*x18*x39-2*x5*x29*x32-3*x10*x13*x24+3*x9*x17*x50-2*x6*x27*x45-3*x1*x23*x44, -2*x37*x38*x39+3*x32*x34*x42-x28*x31*x45-x43^2*x44-x31*x35*x43+x20*x41^2+3*x30*x31*x46+x22*x25*x34+2*x33*x39*x46-x23*x26*x28-2*x39*x40*x49-2*x38*x45*x46+3*x13*x38*x46+x8*x33*x35+2*x7*x39*x49+x15*x30*x37+3*x9*x27*x49-3*x19*x35*x44+2*x19*x39*x45+3*x16*x17*x23+2*x13*x19*x31+2*x1*x3*x29+2*x4^2*x20+x6*x7*x17-2*x6*x17*x46-2*x9*x21*x49-3*x3*x12*x29, x36*x45*x49+3*x20*x34*x37+2*x34*x37*x44+3*x34*x36*x47+3*x25*x38*x46-x35*x41*x44+3*x48^3+2*x19*x26*x49-3*x7*x26*x28+x11*x22*x24-2*x4*x23*x28+3*x6*x25*x44-2*x9*x26*x39+2*x2*x19*x21-x4*x21*x44+x8*x16*x32+3*x2*x5*x9+2*x5*x13*x42+2*x3*x19*x38-2*x1*x10*x35+2*x17*x28*x41+x12*x33*x36+2*x5*x27*x44+2*x12*x19*x20-3*x19*x31*x45-x3*x34*x38-3*x8*x10*x47, -2*x8*x41*x45+3*x11*x25*x41-3*x1*x35*x43+2*x34^2*x43+3*x28*x32*x45+2*x26*x34*x37+x20*x22*x36-3*x20*x33^2-2*x24*x34*x47-3*x11*x32*x44-3*x12*x23*x46-x14*x25*x36+3*x2*x10*x31-3*x18*x24*x37-2*x3*x24*x39+2*x2*x16*x32-x6*x15*x36+x16*x33*x39-3*x18*x37*x38-x19*x37*x41-x8*x36*x46-2*x19*x34*x36-x5*x17*x29-x17*x36^2-3*x18*x19*x30, 3*x9*x15*x16-3*x6*x9*x30+x10*x17*x24+x2*x10*x32-x4*x12*x13+2*x6*x9*x12-x18*x22*x25-3*x4*x42*x43-3*x9^2*x12-x17*x45*x47+2*x2*x34*x46+2*x12*x19*x49-x8*x22*x46-x3*x5*x6-x5*x19*x20+3*x5*x24^2+x1*x28*x47+x9*x16*x29-2*x1*x3*x12+3*x12*x39*x44-2*x3*x32*x42-3*x10*x27*x38-2*x22*x27*x33+2*x22*x34*x49+3*x23*x46^2-x23*x26*x27, x8*x21*x32-3*x15*x21*x31+2*x42*x47*x50-x17*x43*x50+x8*x17*x43+2*x33*x44*x49+2*x42*x46*x49-x2*x10*x14-x3*x9*x19+3*x3*x8*x13+x3*x18*x46-x2*x11*x45+2*x23*x47*x49-3*x15*x32*x35+3*x8*x24*x38-3*x4^2*x13-3*x3*x5*x30+x15*x37*x41+2*x25*x26*x41+2*x3*x19*x26-3*x10*x43*x45-x21*x36*x47+x21*x30*x47-3*x8*x24*x32, 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var=[myseq('x[i]','i',1:50)].';
pstar = rand(1,50)*10^(-4);


tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large2')
for i=1:50 syms (['x',num2str(i)]);
end
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var=[myseq('x[i]','i',1:50)].';
pstar = rand(1,50)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large3')
for i=1:100 syms (['x',num2str(i)]);
end
sys=[-x13*x24*x94+x21*x49*x78-3*x29*x52*x72+2*x4*x48*x85+x5*x10*x15+2*x4*x43*x90-2*x53*x79*x90-x24^2*x27-x3*x66*x94+x19*x49*x82-2*x14*x67*x72-2*x5*x10*x65-x28*x83*x91-x61*x81*x89+x48*x49*x95+3*x9*x16*x65-2*x13*x43*x85-2*x15*x17*x68-x34*x35*x92-3*x23*x40*x94-x36*x43*x55+3*x30*x34*x75+2*x26*x64*x78, x5*x30*x55-3*x17*x39*x55-3*x41*x50*x85+2*x8*x56*x92+x26*x38*x86+x10*x24*x59-2*x1*x30*x83+x32*x50*x53-3*x8*x97*x99-3*x25*x34*x53-x49*x55*x67+x24*x44*x62+2*x73*x75*x88-3*x17*x25*x50-3*x37*x69^2+3*x28*x35*x71-2*x25*x37*x68-x29*x58*x92-3*x40*x70*x78-3*x36*x58*x67+x43*x64*x79-2*x31*x36*x49+3*x86*x93*x94+3*x7^2*x29+3*x77*x79*x80, 3*x10*x31*x43+2*x71*x81*x99-3*x28^2*x31+x43*x48*x87-3*x50*x86*x100-3*x33*x42*x96-2*x29*x38*x91-2*x37*x77*x83-x65*x71*x85+x1*x29*x76-3*x24*x50*x84-3*x14*x31*x65-2*x4*x68*x79+3*x29*x38*x61-2*x7*x72*x94+x14*x64*x90+3*x2*x56*x63+3*x1*x19*x74-x6*x18*x56-x12*x57*x90-x1*x8*x78+2*x20*x48*x74-2*x33*x72*x86-3*x46^2*x91-2*x7^2*x91-x39^2*x70-2*x51*x54*x56+2*x7*x18*x50, 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var=[myseq('x[i]','i',1:100)].';
pstar = rand(1,100)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large4')
for i=1:100 syms (['x',num2str(i)]);
end
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var=[myseq('x[i]','i',1:100)].';
pstar = rand(1,100)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large5')
for i=1:300 syms (['x',num2str(i)]);
end
sys=[x23*x102*x149*x270+3*x31*x180*x229*x232+x45*x48*x75*x130-3*x47*x179*x198*x218+x104*x221*x259*x273-2*x112*x150*x154*x212+3*x147*x176*x201*x235, -3*x16*x20*x54*x255-3*x25*x152*x210*x264-2*x32*x124*x210*x275-2*x40*x88*x265*x292+3*x55^2*x60*x259+3*x79*x163*x253*x282, -x3*x13*x167*x228+3*x8*x39*x130*x293-3*x12*x30*x227*x292-2*x13*x132*x147*x148-2*x20*x48*x111*x249-3*x67*x90*x100*x296+3*x87*x124*x139*x196, -3*x22*x38*x87*x167+2*x23*x38*x253*x258-2*x27*x33*x154*x288-x32*x80*x99*x161-x65*x105*x217*x255-3*x127*x265*x281*x292-x159*x183*x186*x218, 2*x15*x84*x118*x207+2*x16*x67*x184*x256+3*x22*x186*x245*x292-3*x28*x50*x55*x107-2*x39*x77*x290^2-x45*x86*x146*x229+2*x104*x156*x223*x278, -x4*x65*x68*x173-2*x15*x19*x106*x292-3*x23*x109*x205*x294-2*x74*x80*x212*x296-3*x136*x229*x230*x250+2*x154*x199*x224*x238, -2*x15*x71*x156*x176+2*x23*x37*x54*x77-x47*x177*x195*x207+x66*x210*x264*x294+3*x68*x116*x138*x289+2*x73*x203*x226*x286+3*x196*x224*x234*x245, 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var=[myseq('x[i]','i',1:300)].';
pstar = rand(1,300)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large6')
for i=1:500 syms (['x',num2str(i)]);
end
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-2*x7*x23*x91*x94+3*x14*x51*x88*x92+3*x37*x38*x51*x94-x116*x240*x267*x305+x384, -3*x14*x23*x32*x51+x20*x57*x66*x94+x282*x317*x348*x413+x385, -2*x2*x8*x76*x81-3*x5*x76*x79*x100+x65*x74*x91*x99-x247*x288*x303*x396+x386, 2*x28*x31*x47*x86+2*x79*x83*x95*x99+2*x144*x172*x400*x435+3*x181*x189*x332*x469+x387, 3*x3*x12*x46*x64+x5*x21*x36*x86-x111*x172*x434*x488-2*x261*x303*x378*x451+x388, -x17*x65*x68*x93-x23*x26*x27*x93-3*x273*x351*x355*x415+x389, -3*x12*x21*x65*x94+2*x15*x35*x82*x96-x25*x38*x92*x96+3*x122*x310*x321*x424+x390, -3*x9*x55*x67*x94+3*x24*x39*x51*x99-3*x242*x244*x296*x426+x391, 3*x7*x11*x57*x91-x9*x74*x77*x79-2*x73*x74*x93*x99+x238*x266*x422*x466+x392, -x8*x52*x63*x98-x65*x76*x94*x96-2*x157*x250*x264*x339-x171*x215*x349*x498+x393, -3*x10*x14*x35*x96-2*x11*x20*x35*x75+x17*x47*x59*x89+3*x155*x270*x308*x395-2*x196*x231*x405*x450+x394, 2*x19*x35*x37*x58-3*x45*x55*x86*x99-3*x125*x185*x260*x269+2*x205*x276*x377*x486+x395, 3*x7*x81*x83*x96-3*x18^2*x43*x99+3*x132*x354*x488*x496+3*x140*x297*x365*x444+x396, x21*x22*x29*x77+3*x21*x47*x55*x60+2*x27*x30^2*x55+x180*x197*x344*x413+x397, -3*x12*x15*x35*x97-x21*x31*x36*x100+x153*x163*x270*x335+x398, -3*x5*x47*x60*x62-3*x20*x49*x83*x90-2*x135*x286*x330*x354+2*x240*x317*x327*x471+x399, -3*x11*x37*x84*x86+x32*x37*x71*x72-3*x42^2*x65*x85+3*x113*x180*x337*x450+x400, -3*x4*x22*x45*x60+2*x16*x19*x94*x98-x24*x33*x52*x87-x213*x299*x425*x467-x254*x329*x404*x415+x401, -2*x7*x18*x26*x62-2*x8*x37*x38*x85+2*x19*x71*x88*x97-3*x133*x186*x267*x336+x402, -2*x3*x44*x89*x93-x5*x14*x64*x93+x34*x48*x63*x73-3*x132*x210*x484*x495-2*x144*x295*x306*x392+x403, 2*x17*x36*x44*x59-3*x33*x51*x54*x88-x183*x228^2*x278+x194*x234*x450*x481+x404, -2*x12*x13*x38*x51+x20^2*x33*x81-2*x39*x65^2*x70+3*x104*x152*x219*x352+2*x123*x163*x176*x481+x405, -2*x12*x41*x88*x92-x13*x42*x44*x49-2*x57*x81*x92*x95-x140*x154*x348*x483+2*x171*x255*x287*x329+x406, 3*x13*x44*x53*x72+x15*x67*x89*x90+3*x32*x44*x81*x86-3*x226*x325*x497*x498+x407, -2*x8*x36*x46*x61+x23*x47*x95^2+2*x60*x73*x75*x97-x101*x131*x246*x375-2*x120*x322*x370*x408+x408, 2*x11*x65*x83*x92-3*x26*x33*x58*x73+2*x45*x69*x88*x91+3*x134*x239*x345*x430+3*x256*x277*x295*x500+x409, 3*x4*x15*x20*x65-2*x9*x28*x44*x97+x15*x83*x84*x86-3*x156*x316*x381*x419+x410, -3*x1*x17*x18*x67-2*x7*x46*x47*x88-2*x203*x235*x306*x482-2*x238*x292*x317*x338+x411, 2*x7*x19*x64*x77+2*x37*x46*x85*x94-2*x168*x283*x311*x412-x311*x389*x474*x498+x412, x8*x77*x86*x100-2*x47*x49*x71*x77-2*x59*x67*x88*x89-2*x139*x208*x237*x475-2*x322*x361*x362*x368+x413, 2*x21*x32*x39*x99-x25*x59*x60*x61+x43*x54*x78*x87+x165*x246*x396*x451+x414, 3*x8*x39*x99*x100-2*x16*x58*x74*x92+2*x25*x76*x83*x92+3*x118*x291^2*x427+3*x398*x399*x416*x452+x415, -x2*x20*x30*x44-x7*x13*x42*x57+2*x151*x352*x392*x437+2*x272*x384*x435*x486+x416, 3*x20*x54*x76*x95+2*x35*x45*x75*x83-3*x106*x146*x266*x329+x417, 3*x6*x51*x79*x91-3*x12*x46*x52*x92-3*x45*x75*x84*x89-3*x191*x417*x453*x481+2*x228*x391*x449*x481+x418, -2*x17*x18*x42*x99+x18*x50*x72*x93+x28*x37*x63*x87+x199*x259*x300*x354-3*x199*x326*x398*x462+x419, x16*x46*x53*x79+3*x40*x51*x53*x73+3*x44*x72*x80*x89+x420, -3*x1*x5*x10*x28-2*x60*x65*x92*x94+3*x122*x218*x264*x450+x213*x237*x365*x431+x421, -x19*x21*x58*x85+3*x43*x45*x79*x87+3*x46*x84*x94*x96+x135*x137*x489*x496-x268*x287*x291*x332+x422, x31*x60*x61*x94-2*x40*x61*x75*x86-x197*x228*x381*x407-3*x289*x380*x477*x493+x423, -3*x34*x70*x72*x93-3*x49*x51^2*x70-x85*x87^2*x97+3*x103*x257*x276*x339+2*x132*x245*x261*x464+x424, 2*x4*x7*x52*x82-2*x12*x30*x67*x82-x60*x66*x74*x89+x115*x128*x197*x372+x425, 2*x3*x61*x66*x86+x6*x30*x72*x78-3*x18*x37^2*x53+3*x188*x278*x435*x484+x307*x443*x450*x473+x426, 3*x25*x45*x74*x96-x25*x64*x79*x87-x27*x34*x38*x79+x129*x219*x431*x450-3*x132*x142*x242*x480+x427, -2*x1*x31*x34*x47-x35*x80^2*x91+2*x73*x83*x91*x99-x262*x295*x406*x464+x428, x3*x17*x32*x94-x6*x9*x74*x89-x26*x28*x62*x79-2*x117*x135*x178*x287+3*x123*x144*x402*x458+x429, -x2*x58*x83*x99-3*x11*x20*x27*x57+x214*x247*x290*x436+x430, -2*x7*x16*x57*x72-2*x16*x23*x42*x78-3*x101*x269*x311*x470+2*x110*x134*x199*x451+x431, 2*x2*x31*x42*x76+3*x22*x37*x43*x89+2*x22*x40*x71*x91-x152*x155*x216*x226-2*x172*x175*x314*x489+x432, -2*x1*x38*x48*x55+x5^2*x54*x65+3*x39*x77*x80*x85+3*x104*x125*x169*x357+3*x108*x209*x319*x462+x433, 2*x15*x37*x60*x67-3*x44*x47*x87*x98+2*x187*x208*x228*x451+x228*x382*x398*x468+x434, -2*x9^2*x22*x66+x9*x22*x42*x51-x35*x38*x53*x70+x105*x169*x176*x288-x120*x135*x429*x477+x435, x7*x17*x50*x84-2*x16*x73*x83*x88-3*x22*x57*x62*x70+2*x107*x210^2*x361+2*x129*x312*x320*x473+x436, -2*x17*x27*x58*x61-x21*x40*x59*x92-2*x54*x87^2*x99+2*x150*x211*x222*x299+x437, -2*x14*x42*x51*x71+x64*x80*x88*x94-x77*x78^2*x97+x438, -3*x2*x16*x33*x71+2*x36*x44*x45*x84-3*x243*x253*x325*x372+x439, -3*x13*x78*x80*x87-3*x26*x31*x33*x71+x40*x45*x69*x73+x114*x195*x407*x411-x146*x157*x168*x491+x440, 2*x5*x60*x82*x90-x19*x61*x70*x85+2*x173*x187*x323*x407-x250*x274*x474*x475+x441, x13*x15*x39*x56-x34*x46*x99^2-x41*x56*x66*x71+3*x235*x240*x479*x484-2*x241*x256*x300*x370+x442, 3*x18*x52*x71*x78-3*x20*x24*x56*x76+3*x244*x291*x358*x451+x443, -2*x15*x26*x42*x60-3*x30*x68*x75*x100-2*x264*x277*x290*x456+x444, -2*x8*x28*x76*x87-x18*x40*x74*x76-2*x23*x63*x67*x100-2*x142*x310*x422*x431+x445, 2*x9*x11*x60*x99-3*x11*x22*x53^2+x24*x41*x59*x62-3*x135*x259*x309*x329+3*x199*x284*x361*x469+x446, 2*x16*x21*x44^2+2*x23*x30*x54*x60+x31*x41*x47*x92-3*x185*x309*x409*x429+x447, x3*x7*x36*x49-2*x7*x9*x93*x95+x448, -2*x20*x31*x60*x65+x27*x47*x52*x97+2*x49*x52*x53*x57+x144*x190*x388*x394-x293*x308*x446*x485+x449, -x4*x14*x20*x34-x11*x17*x40*x86-x134*x171*x180*x403+x450, 3*x1*x9*x28*x58-3*x11*x57*x99*x100-3*x16*x18*x25*x72+x185*x380*x457*x496+x451, -3*x12*x31*x42*x91+2*x71*x76*x80*x98+x212*x229*x364*x437+x452, x5*x7*x11*x25-3*x7*x39*x67*x78+x9*x67*x82*x96+3*x137*x152*x184*x358-2*x141*x190*x196*x215+x453, -x31*x49*x83*x98-2*x36*x39*x76*x92-x47*x53*x56*x71+x111*x240*x322*x417-2*x352*x355*x475*x494+x454, -2*x5*x42*x59*x74-2*x14*x54*x61*x99-x143*x240*x263*x454+x170*x201*x253*x376+x455, -3*x6*x25*x32*x46-2*x16*x32*x35*x97+3*x26*x27*x30*x69-3*x105*x203*x293*x425+3*x220*x223*x404*x466+x456, x11*x22*x45*x81+x14*x52*x95*x98+2*x30^2*x39*x48+x145*x289*x357*x434-3*x167*x195*x248*x418+x457, 3*x6*x12*x72*x73+x12*x46*x84^2-x256*x257*x332*x436+x458, -2*x14*x16*x66*x74+3*x25*x43*x50*x62-x41^2*x47*x71+3*x115*x268*x337*x485+3*x174*x386*x410*x439+x459, x7*x12*x83*x89-x11*x34*x64*x92+2*x64*x66*x80*x82-x234*x243*x386*x429+x460, 3*x33*x38*x58*x81-3*x177*x236*x454*x461+x461, -3*x5*x9*x36*x63+x15*x23*x39*x56+x115*x385*x400*x416-x239*x297*x392*x486+x462, -x9*x16*x78*x95-3*x11*x46*x78*x90-x51*x72*x77*x99+3*x102*x249*x295*x440-3*x111*x125*x132*x409+x463, 3*x9*x53*x63*x92-x26*x35*x68*x76-2*x213*x376*x404*x433+x464, 2*x12*x24*x60*x78+3*x188*x193*x246*x429-x223*x328*x332*x421+x465, x4*x71*x74*x99+x13*x40*x97*x98+2*x114*x268*x439*x477+2*x202*x274*x369*x400+x466, -3*x1*x56*x75*x92+x6*x10*x40*x91-2*x31*x41*x49*x86-x174*x180*x186*x288+x467, 3*x1*x7*x35*x76-x8*x17*x47*x58-2*x37*x38*x44*x52+x147*x284*x381*x460-x182*x188*x453*x468+x468, -2*x17*x63*x95*x100+2*x33*x46*x79^2-2*x44*x67*x75^2-3*x168*x170*x413*x463-x196*x238*x391*x496+x469, -x5^2*x7*x24+x5*x15*x81*x100+x42^2*x53*x89-2*x117*x174*x459*x462-3*x204*x314*x318*x478+x470, 3*x3*x29*x64*x90-3*x17*x21*x74*x82+3*x18*x29^2*x87-x126*x269*x319*x406-2*x127*x197*x208*x334+x471, -2*x5*x17*x41*x96-3*x18*x31*x41*x59+2*x21*x22*x23*x60+x122*x282*x394*x493-x209*x252*x263*x388+x472, -3*x4*x23*x54*x95-3*x200*x224*x282*x466+x473, -3*x9*x28*x38*x89+3*x17*x53*x69*x100+3*x108*x170*x299*x397-3*x305*x392*x486*x490+x474, 3*x34*x44*x70*x80-3*x119*x204*x299*x475-x246*x275*x450*x490+x475, 2*x8*x12*x54*x91+2*x9*x61*x69*x82+2*x54^2*x63*x82-3*x158*x296*x413*x456-3*x204*x288*x299*x359+x476, 2*x4*x35*x70*x97-2*x13*x42*x58*x71-3*x45*x63*x67*x86+2*x108*x118*x312*x480-3*x206*x263*x341*x371+x477, -x1*x14*x39*x56-2*x27*x48*x73*x81-3*x114*x391*x423*x430+3*x258*x322*x369*x440+x478, x3*x64*x66*x95+3*x216*x242*x298*x473-x231*x256*x357*x375+x479, 3*x11*x39*x61*x81+3*x23*x58*x76*x92-2*x44*x69*x91*x94-2*x101*x326*x447*x472-x124*x146*x400*x470+x480, x8*x16*x87*x89+3*x8*x27*x67*x76+3*x32*x42*x63*x98+3*x101*x180*x479*x491+3*x167*x390*x407*x478+x481, -2*x7*x63*x85*x99+2*x29*x59*x73*x79-2*x156*x190*x197*x381+2*x213*x353*x426*x470+x482, -2*x37*x40*x74*x96+x59*x74*x87*x94-3*x190*x218*x291*x386-2*x287*x370*x400*x439+x483, -2*x3*x24*x45*x63+x23*x32*x38*x99+3*x289*x350*x425*x448+x484, 3*x31*x43*x63*x74-2*x144*x262*x287*x448+x485, 2*x15*x22*x56*x76+x50*x57*x74*x81+3*x104*x129*x153*x185+x486, 3*x11*x30*x67*x72-x31*x32*x37*x84-3*x34*x48^2*x86+x189*x272*x386*x425+3*x256*x259*x453*x464+x487, 3*x6*x40^2*x76-2*x8*x42*x79*x100-2*x26*x46*x59*x71-3*x114*x123*x249*x355-3*x141*x293*x327*x495+x488, x1*x14*x71*x99-x22*x34*x39*x85+2*x22*x43*x81^2-x125*x220*x338*x375-x174*x277*x348*x397+x489, 2*x6*x18*x40*x84+3*x7*x51*x56*x63-3*x18*x19*x36*x62+3*x196*x219*x309*x400+x234*x246*x268*x346+x490, 2*x2*x48*x76*x88+2*x3^2*x35*x88+3*x11*x76*x88*x93-x313*x457*x468*x473+x491, x6*x80*x82*x92-2*x7*x20*x22*x85+2*x62*x63*x84*x91+x118*x360^2*x452-x202*x300*x428*x481+x492, -x1*x10*x28*x84+x4*x11*x40*x99-2*x27*x57*x66*x67-x132*x169*x206*x402-x262*x296*x380*x382+x493, -3*x2*x8*x33*x35-x3*x50*x73*x75-x9*x14*x95*x98+x117*x284*x475*x481+x494, -2*x18*x27*x28*x94+x25*x70*x74*x84+2*x104*x219*x337*x457-3*x121*x190*x199*x495+x495, -x22*x87*x89*x97-x39*x46*x81*x97+3*x43*x85*x97^2+x237*x264*x351*x495-3*x240*x294*x448*x479+x496, -x19*x51*x100^2-3*x57*x63*x67*x75+x168*x341*x458*x468+2*x244*x303*x322*x420+x497, 3*x13*x74*x77*x96+3*x15*x31*x43*x45+2*x16*x43*x49*x55+2*x191*x271*x278*x301+x208*x289*x324*x330+x498, -3*x1*x15*x44*x99-2*x9*x36*x63*x89-2*x61*x63*x67*x81+x141*x146*x307*x450+x147*x167*x338*x373+x499, -2*x16*x37*x54*x83+x52*x57*x65*x98+3*x174*x181*x241*x284+x500];
var=[myseq('x[i]','i',1:500)].';
pstar = rand(1,500)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large7')
for i=1:500 syms (['x',num2str(i)]);
end
sys=[-2*x7*x85*x110-x11*x95*x166+2*x68*x122*x164+x97*x113*x165+2*x214*x260*x405+2*x389*x438*x444-x413*x457*x479+x201, -x2*x46*x135+x15*x20*x27-2*x49*x111*x182-2*x60*x117*x171-2*x238*x387*x399-3*x254*x328*x376+2*x282*x465*x496+x202, x6*x33*x57+x40*x135*x137-2*x85*x87*x109-x244*x309*x314-x260*x286*x422+x203, -3*x29*x48*x50+2*x43*x63*x71+x64*x160*x166-3*x210*x257*x436-3*x351*x367*x424-x415*x436*x491+x204, 3*x3*x24*x184+3*x16*x61*x170+2*x18*x155*x158+2*x328*x371*x467+x205, -x35*x140*x152-2*x60*x74*x114+3*x63*x139^2-2*x231*x364*x418-3*x239*x255*x283-x313*x318*x481+x206, -2*x16*x73*x137+x23*x80*x136-2*x38*x44*x123-2*x99*x109*x162+3*x228*x415*x486+x284*x309*x333+x288*x435*x499+x207, -2*x1*x152*x190+3*x32*x73*x195-3*x51*x92*x129+x60*x92*x110-x314*x323*x363+3*x375*x443*x487+x208, -x7*x136*x186+3*x62*x172*x174-3*x82*x84*x167-2*x95*x104*x124+x224*x367*x408+3*x252*x390*x456+3*x397*x415*x494+x209, 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-x7*x42*x85*x122-3*x20*x44*x76*x99-3*x25*x70*x80*x98+3*x288*x299*x310*x485, 3*x19^2*x158*x163+3*x31*x133*x167*x200+3*x61*x64*x92*x185-2*x201*x247*x327*x382+2*x268*x336*x344*x364-2*x452*x460*x492*x494, -2*x3*x60*x123*x137+x29*x190*x191^2-2*x73*x99*x102*x142+x223*x268*x292*x398-3*x288*x324*x382*x446, 3*x11*x24*x28*x31+3*x119^2*x162*x179+x239*x411*x440*x449-x245*x291*x410*x479-2*x314*x338*x455*x495, -x20*x50*x153*x163+x32*x39*x82*x93+3*x145*x146*x175*x188-3*x218*x338^2*x429-3*x237*x314*x456*x483+x266*x401*x444*x447, -x5*x155*x157*x186-x67*x129*x134*x156+3*x243*x244*x393*x497+3*x266*x287*x366*x413-3*x331*x347*x362*x453, 2*x12*x81*x178*x186-2*x17*x48*x127*x150-x83*x141*x170*x196-x229*x310*x336*x490-2*x252*x324*x336*x461-3*x278*x282*x479*x488, -x3*x37*x70*x91-3*x47*x96*x104*x108+2*x143*x156*x175*x199+x201*x393*x437*x448-2*x274*x296*x446*x490, -2*x35*x85*x89*x187-2*x49*x123*x141*x195+3*x62*x118*x124*x133+3*x212*x369*x395*x461+3*x216*x232*x374*x376+x276*x404*x443*x472, -2*x4*x77*x97*x185+x52*x53*x143*x184+2*x136*x138*x144*x197-3*x201*x226*x229*x330+2*x281*x350*x432*x494, 3*x1*x72*x121*x154-2*x87*x108*x140*x147-2*x98^2*x108*x143-2*x209*x400^2*x425+x210*x243*x388*x482-x257*x326*x414*x458, -2*x11*x37*x63*x126+3*x30*x65*x85*x149-x36*x114*x136*x137+2*x202*x372*x390*x416+2*x281*x286*x396*x475, 2*x20*x37*x90*x164+x39*x46*x60*x62-x227*x272*x346*x407+x231*x335*x355*x479-x239*x242*x251*x436, 3*x62*x99*x112*x192+2*x64*x68*x110*x140-x229*x241*x377*x398-2*x299*x380*x399*x450, 3*x10*x63*x64*x89-x17*x38*x73*x161+2*x93*x112*x120*x165-3*x250*x261*x350*x483-2*x257*x404*x424*x427-x294*x355*x361*x497, 3*x4*x23*x126*x156+2*x10*x90*x166*x187-2*x24*x62*x104*x115-2*x261*x383*x413*x419+x272*x280*x438*x460-3*x301*x308*x371*x375, -x24*x60*x87*x154+3*x89*x117*x142*x198-x216*x236*x279*x280-2*x225*x332*x424*x495, -2*x1*x76*x92*x126+x31*x100*x183*x186-2*x51*x67*x198*x199+2*x201*x327*x364*x422-x209*x294*x453*x483+2*x213*x348*x499^2, -x1*x40*x164*x188+3*x52*x61*x178*x187-3*x64*x92*x124*x186+x250*x285*x390*x395-x281*x405*x441^2+3*x320*x369*x482*x486, -x1*x34*x71*x198-x7*x85*x109*x129-x23*x82*x149*x200+2*x245*x289*x302*x304-2*x388*x434*x451*x458, x10*x105*x112*x184-2*x49*x59*x114*x196+2*x51*x86*x151*x198+3*x266*x308*x336*x437-x323*x423*x494^2, -x10*x28*x48*x166-3*x13*x32*x75*x172-2*x74*x137*x182*x193-3*x207*x320*x340*x372+2*x365*x372*x411*x454, -3*x55*x91*x96*x112+x84*x90*x116*x197+x294*x409*x422*x464+3*x322*x360*x442*x495-x366*x394*x423*x443, -3*x6*x32*x60*x140+2*x7*x41*x99*x110+3*x229*x311*x330*x421-3*x263*x464*x483*x488-3*x387^2*x388*x438, -2*x13*x80*x101*x181+3*x19*x41*x96*x152-3*x240*x306*x445*x496-3*x297*x414*x421*x454+x315*x382*x436*x474, -x7*x68*x150*x168-x54*x106*x126*x175-x208*x225*x318*x325-3*x211*x216*x348*x445+3*x229*x347*x425*x489, x15*x66*x92*x172+3*x213*x217*x321*x435-2*x308*x311*x364*x465, x18*x126*x169*x187+2*x49*x115*x168*x195-x221*x359*x391*x482-3*x245*x250*x329*x472+3*x273*x443*x449*x469, -2*x5*x106*x142*x165-3*x44*x72*x94*x109+2*x115*x141*x149*x175-2*x202*x321*x331*x337-x275*x418*x441*x451+2*x281*x403*x415*x472, -2*x4*x53*x54*x132+2*x18*x159*x162*x191+2*x34*x79*x145*x200+3*x208*x313*x316*x429+3*x289*x370*x389*x401-3*x363*x401*x420*x497, -x8*x97*x178*x196+2*x69*x154*x175*x199+2*x107*x141*x150*x151+3*x219*x226*x305*x387+x295*x306*x375*x474-3*x319*x402*x407*x475, -x6*x29*x79*x181-x59*x72*x92*x189-x71*x109*x159^2+x210*x243*x245*x476-3*x370*x410*x477*x484, x24*x78*x83*x128-3*x66*x70*x139*x160-3*x236*x350*x367*x414+2*x237*x258*x326*x377, -x9*x78*x94*x100-2*x43*x80*x192*x195-2*x93*x116*x139*x198-2*x277*x322*x426*x472-2*x279*x282*x332*x497-3*x394*x402*x419*x454, x31*x42*x46*x65-3*x97*x116*x129*x143+x112*x122*x168*x178+2*x242*x323*x371*x454+3*x246*x270*x287*x458, -2*x25*x61*x178*x192+2*x211*x294*x389*x458-2*x238*x272*x304*x454-x444*x469^2*x481, -x35*x65*x124*x145+x46*x59*x61*x176-2*x229*x240*x279*x387+3*x240*x369*x479*x499+x260*x363*x401*x480, 3*x32*x107*x149*x189+3*x69*x79*x94*x198-2*x211*x263*x349*x461-2*x250*x357*x428*x461+x264*x294*x398*x405, -3*x4*x80*x152*x166-3*x35*x70*x73*x120-2*x88*x89*x167*x175-x252*x277*x310*x469-2*x262*x360*x363*x423-2*x272*x338*x403*x433, -x214*x215*x281*x298-2*x216*x222*x260*x451+3*x330*x339*x457*x483, 3*x1*x15*x82*x200-2*x82*x99*x137^2-x131*x139*x146*x196+x205*x329*x337*x457, 3*x10*x34*x84*x137+3*x41*x68*x72*x104+2*x255*x304*x308*x347-x259*x299*x431*x432, 2*x22*x55*x102*x194-2*x63*x146*x156*x159+2*x237^2*x264*x307-x255*x273*x283*x354, 3*x12*x19*x107*x117-3*x42*x150*x174*x193+x56*x58*x69*x165+2*x232*x261*x262*x425+2*x264*x299*x344*x360+x277*x347*x354*x491, 2*x27*x67*x90*x199-2*x40*x41*x130*x169-2*x43*x61*x100*x162+2*x251*x260*x292*x341-x262*x302*x431*x477+2*x277*x310*x370*x438, -3*x5*x17*x84*x107-3*x82*x127*x146*x191-x232*x298*x456*x499-3*x254*x296*x309*x480, -2*x20*x47*x58*x151-3*x73*x103*x108*x179-2*x92*x98*x99*x166-2*x209*x311*x413*x482+x228*x351*x400*x487-x297*x329*x431*x454, 3*x9*x98*x141*x196-x63*x132*x173*x186+x207*x328*x405*x457-3*x217*x295*x374*x425+2*x303*x318*x426*x494, x1*x49*x132*x135-2*x24*x58*x69*x131+2*x214*x308^2*x321+x238*x266*x328*x451, -x31*x168*x182*x198-2*x46*x92*x96*x120-x251*x322*x333*x388-2*x258*x270*x279*x323+3*x352*x413*x446*x466, 3*x9*x29*x82*x187+3*x32^2*x59*x102+3*x64*x68*x86*x115-x203*x208*x220*x500-2*x252*x382*x453*x456, 2*x31*x39*x149*x156+x79*x83*x88*x158-2*x232*x428*x477*x496+x281*x303*x344*x447, -x32*x101*x104*x182-3*x38*x84*x118*x121-2*x232*x310*x371*x455-2*x283*x297*x300*x417+x348*x355*x430*x486, x10*x49*x51*x123+x70*x79*x133*x163+3*x78*x95*x133*x167-3*x210*x289*x373*x375-x232*x237*x243*x370-x363*x442*x476*x481, -x10*x69*x114*x164-x47*x59*x152*x177-x237*x259*x438*x442-2*x295*x316*x456*x490+2*x354*x415*x441^2, -2*x32*x42*x66*x69-x44*x89*x154*x195-3*x158^2*x175*x190+2*x206*x268^2*x310+3*x261*x434*x484*x497+x317*x407*x462*x466, -2*x1*x68*x165*x191-x15*x27*x115*x164-x87*x125*x144*x194-x210*x306*x361*x422+3*x235*x242*x284*x472, 3*x8*x48*x61*x142+x52*x87*x155*x192+2*x53*x112*x147*x185-2*x272*x311*x468*x491+3*x385*x402*x410*x464, 3*x1*x124*x158*x170+3*x80*x108*x138*x170-2*x207*x284*x412*x486+2*x242^2*x406*x497-x271*x351*x354*x389, 2*x39*x62^2*x153-2*x112*x140*x147*x184-x234*x238*x351*x462-3*x240*x400*x421*x458-2*x277*x289*x292*x388, 2*x4*x64*x65*x178-x51*x77*x99*x195-3*x208*x299*x382*x411-x384*x390*x443*x474, x20*x32*x131*x183-2*x206*x277*x309*x449+x226*x247*x332*x346, 3*x5*x89*x151*x152-2*x45*x92*x121*x131-x49*x90*x167*x173+2*x233*x430*x466*x490-x247*x356*x434*x488+2*x268*x345*x363*x479, -x25*x35*x85*x199+3*x43*x45*x73*x128+x127*x150*x154*x188-2*x213*x278*x280*x299, 2*x7*x163*x178*x190+x81*x142*x177*x192+2*x229*x231*x390*x479+2*x242*x300*x317*x335-3*x247*x276*x310*x373, -2*x10*x19*x126*x146+3*x41*x106*x110*x170-3*x146*x159*x160*x191+x219*x247*x307*x334+3*x268*x338*x449*x465, -3*x8*x19*x55*x91-3*x10*x50*x146*x184+x23*x30*x45*x108+x206*x261*x291*x479+2*x241*x311*x382*x496+2*x281*x311*x353*x450, -2*x2*x35*x95*x103-2*x29*x42*x92*x134+2*x60*x130*x138*x146+x212*x224*x411*x486-3*x221*x236*x241*x340, 2*x2*x27*x72*x148-3*x4*x88*x192*x197+3*x91*x109*x118*x185-x216*x410*x416*x486-3*x253*x254*x383*x430-x260*x316*x344*x416, -2*x16*x117*x122*x147+2*x93*x132*x179*x196+3*x202*x318*x405*x498+3*x228*x257*x327*x406-x254*x377*x425*x477, 2*x79*x109*x133*x185+2*x116*x118*x157^2+3*x212*x280*x323*x395-3*x217*x265*x297*x299+3*x219*x359*x399*x467, 3*x11*x109*x155*x180+x109*x127*x135*x188-2*x212*x221*x380*x389+x214*x232*x439*x488+2*x256*x330*x477*x482, -2*x5*x71*x110*x159-3*x12*x116*x134*x152+x220*x371*x374*x429+3*x236*x310*x418*x443+2*x311*x331*x430*x497, -3*x23*x25*x47*x197+2*x39*x144*x180*x199+3*x55*x71*x122*x133-x202*x233*x308*x463+x253*x346*x379*x476+3*x307*x414*x483*x484, -x11*x17*x91*x114+x57*x102*x109*x154+3*x234*x333*x358*x431+2*x276*x330*x344*x350+x319*x412*x455*x489, -2*x56*x99*x158*x168+x67*x85*x92*x156+x201*x266*x268*x389+x253*x312*x399*x499, x12*x14*x137*x179+3*x17*x40*x109*x159-x34*x48^2*x164+3*x218*x309*x326*x364-2*x221*x259*x287*x476+3*x341*x418*x419*x473, x43*x47*x55*x127+2*x49*x91*x175*x191+x209*x216*x305*x437+2*x324*x398*x422*x427, x14*x51*x196*x197-x47*x149*x162*x199-x115*x122*x192*x199+3*x202*x228*x369*x407-2*x212*x293*x399*x494-2*x317*x331*x432*x476, 2*x4*x72*x85*x125-2*x12*x52*x73*x113-x108*x117*x187*x199-x207*x255*x264*x477-3*x217*x317*x334*x492-x282*x348*x419*x487, -3*x12*x54*x119*x142+3*x39*x47*x149*x168+x252*x341*x390*x497-x286*x403*x425*x445-x295*x367*x374*x464, x2*x35*x124*x143+3*x60*x125*x146*x168+3*x63*x85*x92*x151+x201*x208*x248*x498-3*x250*x268*x307*x446, -2*x19*x67*x104*x144-x48*x59*x92*x186-2*x85*x121*x139*x163+x201*x209*x272*x325+3*x206*x356*x357*x454+2*x246*x255*x299*x445, -x5*x44*x119*x195-2*x19*x88*x135*x155-x225*x235*x304*x454-2*x230*x272*x308*x476+3*x305*x314*x356*x470, -x20*x33*x69*x193-3*x20*x115*x167*x190+3*x67*x85*x118*x154-3*x216*x313*x459*x494+3*x247*x316*x411*x459+3*x297*x355*x383*x409, 3*x1*x21*x71*x102-x15*x38*x135*x198+x37*x43*x86*x107+x239*x250*x357*x409+x275*x327*x470*x478-3*x305*x324*x356*x434, -2*x2*x7*x153*x191+x46*x91*x116*x119+x58*x94*x107*x130-x207*x309*x372*x446-x218*x221*x295*x329+3*x224*x227*x322*x391, -x47*x98*x138*x198+3*x56*x132*x141*x165-3*x63*x109*x152*x187-3*x287*x351*x381*x404];
var=[myseq('x[i]','i',1:500)].';
pstar = rand(1,500)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
disp('large8')
for i=1:500 syms (['x',num2str(i)]);
end
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-3*x2*x27*x60*x141-2*x94*x99*x145*x187+3*x97*x136*x154*x178+3*x222*x227*x272*x484-x360*x395*x454*x485-x434*x442*x480*x498, -x20*x24*x71*x188-x33*x139*x191*x198+2*x84*x102*x103*x134-3*x218*x293*x415*x447-3*x220*x234*x307*x361+3*x248*x258*x321*x461, -3*x1*x59*x73*x183-2*x21*x144*x159*x183+3*x25*x124*x129*x178+x217*x261*x469*x484-2*x355*x372*x396*x408+2*x369*x374*x408*x481, -x4*x89*x106*x186+3*x40*x78*x117*x131+2*x237*x250*x305*x334-x425*x442*x457*x460, 2*x13*x88*x103*x188+x30*x96*x176*x188-2*x52*x96*x171*x182-x265*x356*x357*x394+2*x280*x282*x430*x487, -3*x2^2*x4*x72+3*x11*x50*x107*x154-x100^2*x119*x185-3*x218*x307*x413*x436-2*x245*x312*x385*x435+x286*x329*x398*x450, 3*x2*x11*x37*x51+x4*x22*x103*x112-x254*x329*x352*x354-3*x296*x396*x417*x429+x305*x366*x448*x477, 2*x39*x86*x140*x172+3*x100*x136*x169*x196+2*x254*x259*x374*x438+2*x279*x300*x381*x395-3*x313*x392*x428*x483, 2*x1*x31*x81*x170+x31*x51*x112*x138+x83*x97*x118*x162+x204*x356*x360*x431+2*x215*x217*x339*x464-2*x257*x326*x338*x489, -3*x7*x70*x125*x146-2*x14*x37*x179*x192-x85*x86*x109*x116+x261*x287*x290*x411+2*x301*x341*x434*x452-x364*x370*x436*x490, -x8*x22*x138*x145-x17*x104*x108*x136-x208*x220*x331*x372-x270*x385^2*x476, x5*x21*x24*x84+3*x6*x38*x56*x107+3*x38*x59*x90*x174-3*x220*x245*x371*x460-x295*x337*x390*x447+3*x316*x333*x336*x378, -2*x5*x16*x22*x53-x14*x37*x193*x199-x58*x149*x155*x178+x234*x257*x294*x380-3*x243*x312*x323*x498-3*x375*x384*x459*x498, x5*x12*x21*x177+3*x11*x34*x97*x133-3*x42*x43*x170*x174-3*x244*x245*x359*x391+2*x264*x289*x440*x478, -2*x64*x72*x167*x178-x101*x119*x185*x195-2*x237*x373*x464*x493+x249^2*x376*x464-2*x250*x393*x406*x419, 2*x3*x74*x156*x197-x37*x73*x147*x153+2*x68^2*x80*x108+3*x227*x285*x298*x476+x283*x383*x406*x430, 2*x25*x107*x168*x187-2*x28*x42*x127*x179+3*x42*x146*x166*x177-x209*x339*x380*x390+2*x238*x270*x407*x476-2*x333*x403*x411*x412, -x31*x116*x142*x183+3*x32^2*x54*x89+3*x50*x66*x127*x137-x240*x334*x396*x468-3*x255*x379*x399*x423+x274*x312*x336*x434, -3*x27*x99*x168*x176+2*x234*x243*x270*x383-3*x310*x327*x418*x448, -x30*x56*x163*x196-3*x153*x159*x188*x198+3*x202*x270*x277*x474-2*x209*x240*x254*x452+x343*x394*x423*x488, 2*x12*x23*x38*x185-3*x30*x116*x147*x192-2*x46*x171*x173^2-3*x202*x205*x312*x421-3*x318^2*x325*x355, -3*x2*x4*x47*x50-3*x113*x143*x154^2+x252*x262*x412*x432-2*x356*x369*x391*x430, -3*x45*x185^2*x192-x66*x92*x119*x175+3*x70*x95*x114*x144+2*x206*x338*x347*x429+3*x239*x286*x289*x350, -2*x14*x26*x121*x124+3*x60*x101*x159*x186-2*x226*x236*x351*x472-x271*x278*x337*x492-2*x310*x324*x339*x444, x24*x86*x115*x184-x77*x86*x130*x147-2*x209*x216^2*x326+x258*x280*x401*x476-3*x278*x314*x352*x418, 3*x7*x105^2*x128+3*x32*x79*x114*x147+3*x83*x113*x168*x173+2*x256*x263*x405*x412+2*x288*x317*x385*x418, -2*x7*x109*x121*x140-3*x27*x50*x73*x125+x50*x56*x92*x175+3*x280*x324*x381*x419-x405*x424*x432*x435, 3*x34*x38*x132*x190+x41*x59*x162*x169-3*x95*x133*x156*x181+3*x214*x225*x247*x259+x341*x357*x437^2, -3*x2*x91*x121*x145+2*x3*x85*x163*x168-x239*x283*x386*x424, 2*x96*x98*x103*x185+2*x205*x214*x302*x442-2*x254*x310*x336*x460-3*x310*x434*x435*x442, -2*x6*x46*x56*x170+3*x15*x85*x147*x161+x61*x184*x188*x196-2*x273*x312*x314^2+x286*x291*x299*x445-x300*x428*x456*x471, -2*x20*x47*x53*x195+x37*x109*x125*x183-2*x49*x96*x157*x198-3*x210*x348*x407*x440-3*x286*x371*x390*x469, -2*x27*x32*x70*x100+2*x27*x88*x139*x178+x44*x101*x118*x119+x211*x226*x277*x408-3*x224*x226*x332*x444-2*x243*x341*x393*x403, -x6*x90*x165*x193-2*x9*x45*x86*x161-x50*x161*x189*x197+3*x215*x226*x373*x494+2*x223*x354*x387*x402+x228*x378*x385*x460, x18*x145*x151*x185+x39*x54*x178*x190+2*x69*x108*x130*x192+3*x228*x238*x282*x333-x248*x371*x402*x471, 2*x8*x34*x35*x73-2*x57*x65*x183*x187+x207*x302*x397*x480+2*x296*x343*x459*x470, 2*x3*x23*x68*x95-x66*x68*x114*x159-3*x79*x89*x147*x187-3*x212*x290*x299*x487+2*x260*x271*x307*x468-x261*x337*x368*x482, -x2*x65*x139*x161-x35*x44*x55*x101+x208*x286*x288*x363-2*x231*x320*x440*x494-3*x283*x311*x369*x456, x26*x62*x113*x117-3*x61*x100*x191*x195+x222*x328*x427*x500+3*x229*x269*x290*x444-3*x257*x344*x422*x489, -x2*x53*x67*x182+3*x14*x21*x50*x111-2*x37*x153*x184*x198-2*x229*x294*x319*x485-3*x235*x291*x396*x432-x253*x311*x336*x348, 2*x8*x110*x121*x164+x27*x138^2*x155-3*x91*x95*x127*x185+3*x211*x257*x293*x355-2*x251*x329*x454*x488, -x6*x34*x47*x52-2*x23*x32*x150*x164, -x2*x13*x124*x162+3*x45*x67*x99*x152-x87*x168*x179*x194+x205*x316*x449*x452-2*x223*x284*x323*x407-x257*x275*x398*x474, -2*x8*x19*x134*x178+x79*x80*x111*x121+2*x217*x312*x316*x388+x264*x319*x372*x476, -3*x30*x50*x64*x170-x61*x88*x97*x150-3*x66*x71^2*x104-x201*x205*x273*x352-x236*x304*x347*x472-x375*x457*x467*x475, 3*x27*x54*x60*x69+2*x54*x59*x150*x172-3*x217*x240^2*x492+2*x227*x245*x286*x399, -3*x5*x89*x102*x149-x68*x78*x109*x162-2*x110*x143*x170*x176-3*x249*x334*x356*x363-x343*x348*x378*x450+2*x392*x400*x416*x424, -2*x2*x35*x98*x107+2*x22^2*x42*x169+x25*x32*x71*x124+x207*x247*x335*x405+3*x230*x266*x329*x489-x334*x458*x466*x492, -x17*x77*x101*x105+3*x43*x99*x107*x184+2*x242*x249*x302*x399+2*x299*x479*x489*x493+3*x327*x335*x389*x398, -2*x56*x76*x159*x170+3*x139*x143*x157*x184-2*x320*x379*x387*x422+2*x349*x379*x442*x483, -x27^2*x100*x144-2*x27*x100*x104*x168-2*x33*x98*x142*x164+3*x223*x374*x419*x439+x268*x290*x336*x349+3*x284*x430*x495*x498, -3*x12*x111*x112*x145-2*x13*x19*x52^2-x30*x32*x87*x100+3*x245*x275*x372*x449, -3*x32*x51*x179*x185-x132*x146*x182*x195+2*x228*x298*x321*x454+2*x236*x251*x288*x485-3*x384*x396*x456*x461, -x22*x38*x125*x177-2*x61*x117*x134*x135+2*x73*x158*x179*x190+3*x228*x400*x444*x462+2*x306*x319*x384*x467, x5*x18*x90*x196+2*x50*x59*x115*x153+x79*x94*x167*x188+2*x203*x349*x424*x451-x213*x285*x354*x437+x219*x236*x335*x358, 2*x17*x45*x52*x107-3*x31*x114*x132^2+3*x222*x258*x435*x461-x258*x431*x482*x494-x377*x392*x461*x467, -3*x14*x108*x150*x162+x24*x58*x99*x137-3*x27*x68*x99*x188+x263*x316*x340*x358-3*x263*x341*x354*x450, -3*x45*x94*x192^2+x53*x74*x125*x161-x206*x310*x465*x491+3*x240*x342*x386*x428, -3*x54*x81*x104*x165-x126*x135*x176*x197-x127*x130*x141*x151-x267*x271*x272*x484, -3*x7*x17*x149*x185+3*x56*x95*x129*x138-x64*x70*x88*x112+2*x276*x316*x350*x378+2*x375*x451*x453*x479, 3*x19*x27*x145*x154+2*x71*x92*x146^2-3*x75*x111*x116*x153+x216*x278*x334*x363+x238*x448*x469*x476+3*x296*x334*x433*x488, -2*x12*x48*x123*x196+2*x29*x119*x156*x180+3*x61*x155*x159*x187-2*x208*x215*x411*x430-3*x241*x255*x397*x416+2*x256*x375*x405*x500, 2*x17*x88*x120*x180+3*x36*x57*x105*x127-x51*x70*x91*x182+3*x231*x327*x427*x487+3*x250*x432*x478^2, x9*x14*x22*x34+x18*x57*x83*x166+3*x39*x47*x161*x187+x214*x400*x436*x496+3*x221*x356*x370*x376, 2*x13*x129*x158*x164+x13*x142*x162*x169+2*x141^2*x186^2+2*x211*x316*x334*x467-x306*x446*x474*x475, x57*x61*x119*x156+x78*x138*x149*x189-2*x203*x297*x380*x495+x223*x225*x379*x461-x299*x394*x451*x464, -x19*x90*x108*x117-2*x30*x82*x110*x136+3*x53*x78*x108*x144-2*x371*x401*x440*x471, 2*x33*x34*x62^2+x34*x61*x108*x125-2*x209*x330*x389*x396-x225*x297*x315*x462, 3*x5*x25*x33*x120-2*x24*x51*x73*x93-2*x57*x68*x81*x98-3*x203*x281*x313*x498-x402^2*x441*x487, 2*x9*x154*x166*x196-x10*x22*x41*x172-3*x26*x33*x141*x175+2*x204*x329*x406*x441+3*x222*x247*x334*x336-3*x245*x274*x325*x329, 2*x18*x56*x131*x194+3*x46*x155*x158*x174+x53*x58*x72*x124-2*x213*x232*x234*x236-3*x261*x277*x321*x424+3*x265*x365*x470*x477, 3*x23*x79*x104*x145+2*x27*x140*x174*x187-3*x38*x41*x74*x146-x221*x299*x337*x402+x293*x334*x457*x470, x9*x11*x43*x62+3*x10*x157*x167*x169+x20*x128*x173*x196-x219*x425*x451*x469-x298*x322*x443*x500+2*x367*x459*x469*x499, 3*x2*x91*x130*x145+3*x23*x61*x93*x99-3*x116*x118*x132*x174+3*x231*x327*x407*x440-x245*x344*x371*x377+x270*x284*x432*x495, x8*x42*x82*x195-3*x11*x19*x94*x164-3*x50*x95*x156*x164+2*x256*x352*x359*x448+3*x268*x330*x485*x487, -x142*x146*x160*x170+3*x214*x245*x349*x396-3*x242*x292*x380*x484, -x6*x10*x67*x171-3*x22*x66*x69*x121+x31*x32*x115*x183+2*x216*x270*x275*x394-x351*x385*x423*x481-3*x421*x447*x453*x459, -x41*x74*x127*x137-2*x48*x69*x96*x100+2*x109*x117*x148*x194-x201*x302*x327*x479-2*x225*x364*x460*x489, -2*x1*x8*x93*x139+3*x7*x45*x49*x139-x18*x45*x77*x106-x203*x328*x423*x432+x205*x257*x259*x411+x253*x261*x360*x434, 3*x38*x143*x146*x199+x42*x52*x145*x185-2*x52*x151*x159*x195+2*x242*x273*x274*x312-x268*x395*x408*x424, 2*x13*x64*x127*x173-3*x110*x126*x140*x173-3*x231*x308*x357*x498+x240*x330*x363*x439, 3*x17*x161*x182*x189+x42*x69*x100*x110-x201*x332*x433*x454-2*x212*x240*x474*x482+3*x367*x371*x415*x423, 3*x2*x44*x111*x160+3*x58*x90*x103*x111+3*x71*x104*x109*x137+2*x213*x364*x419*x438-2*x242*x302*x397*x460+x403*x428*x454*x478, -x40*x51*x99*x122+2*x259*x352*x377*x401-3*x294*x336*x345*x435, 3*x5*x13*x17*x130+3*x17*x61*x84*x152-3*x259*x300^2*x443, -3*x15*x126*x127*x170+x45*x57*x58*x66+x58*x69*x155*x187+2*x232*x236*x365*x464-x243*x358*x408*x430+2*x268*x358*x412*x485, -x33*x41*x136*x170-x88*x98*x131*x189-2*x181*x187*x192*x194+2*x268*x376*x380*x488+x279*x340*x378*x393-x317*x332*x369*x455, -2*x2*x4*x6*x187-2*x6*x30*x50*x62-x22*x71*x148*x183-3*x211*x223*x225*x381-3*x234*x290*x347*x447+3*x272*x411*x465*x478, -3*x7*x151*x175*x186-3*x48*x82*x140*x165-3*x232*x276*x296*x440, 3*x10*x96*x105*x158+3*x29*x52*x94*x107-2*x250*x424*x459*x484-3*x267*x389*x414*x443, -x42*x95*x101*x184+x75*x137*x161*x177-x116*x170*x176*x182+2*x219*x280*x368*x405+2*x228*x412*x440*x492-x423*x430*x442*x447, -x13*x31*x75*x77+3*x215*x319*x433*x488-2*x264*x326*x394*x457+3*x278*x361*x394*x465, -2*x51*x139*x179*x195+3*x57*x92*x142*x184-x71*x113*x135*x183+2*x205*x239*x247*x324-3*x256*x354*x391*x443+3*x308*x400*x403*x420, -3*x5*x56*x73*x95-x72*x83*x171*x177-3*x207*x295*x315*x344+x338*x367*x464*x496, 3*x41*x53*x80*x194-2*x140*x144*x165*x176+x204*x207*x349*x427+2*x249*x291*x349*x476, -3*x4*x74*x89*x137-x27*x50*x121*x131+2*x120*x162*x175*x199+2*x272*x287*x349*x423+2*x288*x341*x387^2, 2*x9*x80*x137*x157+3*x28*x88*x144*x182-3*x81*x96*x166*x173-2*x248*x303*x349*x372+x275*x304*x351*x424, -2*x8*x19*x87*x137+3*x8*x115*x146*x194-x44*x61*x169*x173+3*x204*x388^2*x423, x25*x66*x152*x193+x57*x82*x107*x135+2*x204*x236*x264*x317-x239*x411^2*x443+3*x270*x373*x412*x493, 2*x20*x64*x80*x152+2*x35*x36*x89*x92+3*x47*x89*x111*x120+x206*x306*x426*x478+x341*x344*x386*x441-x348*x374*x388*x420, -x71*x89^2*x183+3*x104^2*x181*x193-3*x263*x288*x371*x406-2*x289*x358*x442*x468-x325*x460*x467*x490];
var=[myseq('x[i]','i',1:500)].';
pstar = rand(1,500)*10^(-4);

tic;[X,xs,g,G,newvar]=refine(sys,var,pstar,0.01,0);toc
m=max(size(xs))
n=max(size(var));
p=xs';
if ~isempty(G)
Gstar=subs(G,newvar(n+1:m),p(n+1:m));
sys=[sys,Gstar];
end
err=maxerr(sys,var,p(1:n))


