Software

Proineq, a Maple Package

A package using Maple language, which provides several functions for proving inequalities and solving global optimization problems efficiently. The latest version is proineq0126 (update in 2013). You can download it here (the file "proineq0126"). The following are some examples running by proineq0126.

LL5:=(x[1]-x[2])*(x[1]-x[3])*(x[1]-x[4])*(x[1]-x[5])+(x[2]-x[3])*(x[2]-x[4])*(x[2]-x[5])*(x[2]-x[1])+(x[3]-x[4])*(x[3]-x[5])*(x[3]-x[1])*(x[3]-x[2])+(x[4]-x[5])*(x[4]-x[1])*(x[4]-x[2])*(x[4]-x[3])+(x[5]-x[1])*(x[5]-x[2])*(x[5]-x[3])*(x[5]-x[4]);

proineq(LL5);

about 24 seconds

findkmax(x1^2*y1^2+x2^2*y2^2+x3^2*y3^2-2*x1*x2*y1*y2-2*x2*x3*y2*y3-2*x1*x3*y1*y3-k* (x1^2*y2^2+x2^2*y3^2+x3^2*y1^2),[x1,x2,x3,y1,y2,y3],k);

`The global supremum of`, k, `is the real root of`, k+1, `=0`, `which is between`, [-2, -1/2]

Vor1:=16*(c^2+d^2+1)*a^2*z^4+16*a*(2*a*d^2+a*y*d+a*c*x-c*d+2*a+2*a*c^2-a^2*c*d)*z^3+z^2*((24*a^2+4*a^4)*c^2+(-24*d*a^3-24*c*d-8*y*a^3+24*a^2*x-8*a*y)*c+24*a^2*d^2+4*d^2-8*d*x*a^3+4*y^2*a^2+24*y*d*a^2-8*a*d*x+16*a^2+4*x^2*a^2)+(-4*d*a^3+4*y*a^2-4*x*a-8*a*c+8*d*a^2+4*d)*(d-a*c+y-x*a)*z+(1+a^2)*(x*a-y-d+a*c)^2;

proineq(%);

`Not psd`

63.938

The following example once studied by Safey El Din etc., (H. Everett, D. Lazard, S. Lazard, M. Safey El Din, The Voronoi diagram of three lines in $R^3$, Proc. of Symposium on Computational Geometry, 2007])http://www-spiral.lip6.fr/~safey/RAGLib/examples.html

Voronoi:=16*a^2*(alpha^2+1+beta^2)*u^4+16*a*(-alpha*beta*a^2+a*x*alpha+2*a*alpha^2+2*a+2*a*beta^2+a*y*beta-alpha*beta)*u^3+ 4*(6*a^2*beta^2+beta^2-2*x*beta*a^3-6*a*alpha*beta+6*y*beta*a^2-6*alpha*beta*a^3-2*a*x*beta+6*a^2*alpha^2+4*a^2+a^4*alpha^2+ 6*x*alpha*a^2+y^2*a^2+x^2*a^2-2*y*a^3*alpha-2*a*y*alpha)*u^2+4*(-alpha*a^3+y*a^2-a*x-2*a*alpha+2*a^2*beta+beta)*(beta-a*alpha+y-a*x)*u+ (a^2+1)*(beta-a*alpha+y-a*x)^2

proineq(%);

`psd`

.125

IT:=1-8*c^2*d^2-196608*e^3*a^2*d^2*c+1536*e*a*d^4*c^2+21504*e^2*a*d^2*c-4096*e^2*a*c^3*d^2-384*e*a*d^2+1024*e^2*a*c+16*c^4*d^4-72*c^2*d^4+1024*c^2*e^2+36864*e^2*a^2*d^4-3456*e*a*d^4+262144*e^4*a^2*c^2-32768*e^3*a*c^2+256*c^3*d^2*e-576*c*e*d^2+81*d^4+64*c*e-18*d^2;

proineq(%);

`Not psd`

.125

For more examples, please download the file Examples proine0126.mws.

psdgcdv53 (update in 2014) is an improved version of proineq0126, please download the file ISSAC2014ex.mw and psdgcdv53guide2018.mw for more examples.