%% The best constants/formulae which belongs to the unit square @var=3 %% %% (1) (u0,v0)=(0,0) (u1,v1) (u2,1) %% %% input formula QEPCAD %% (Ex1)(Ex2)(Ex3)[ L>1 /\ 0 <= x1 /\ x1<= 1 /\ 0 <= x2 /\ x2<= 1 /\ 0 <= x3 /\ x3<= 1 /\ x1 - x2 x3 > 0 /\ x1 + 2 x2 - x2 x3 <= L (x1 - x2 x3) /\ 2 - x1 + x2 x3 <= L (x1 - x2 x3) /\ x1 + 2 x3 - x2 x3 <= L (x1 - x2 x3) /\ 2 + x1 - 2 x2 - 2 x3 + x2 x3 <= L (x1 - x2 x3) /\ 2 - x1 - 2 x3 + x2 x3 <= L (x1 - x2 x3)] %% (EL)(Ex1)(Ex2)(Ex3)[ L>1 /\ 5L^2-10L+1<0 /\ 0 <= x1 /\ x1<= 1 /\ 0 <= x2 /\ x2<= 1 /\ 0 <= x3 /\ x3<= 1 /\ x1 - x2 x3 > 0 /\ x1 + 2 x2 - x2 x3 <= L (x1 - x2 x3) /\ 2 - x1 + x2 x3 <= L (x1 - x2 x3) /\ x1 + 2 x3 - x2 x3 <= L (x1 - x2 x3) /\ 2 + x1 - 2 x2 - 2 x3 + x2 x3 <= L (x1 - x2 x3) /\ 2 - x1 - 2 x3 + x2 x3 <= L (x1 - x2 x3)] %% %% input formula MMA %% Exists[{x1,x2,x3}, L > 1 \[And] 0 <= x1 <= 1 \[And] 0 <= x2 <= 1 \[And] 0 <= x3 <= 1 \[And] x1 - x2 x3 > 0 \[And] x1 + 2 x2 - x2 x3 <= L (x1 - x2 x3) \[And] 2 - x1 + x2 x3 <= L (x1 - x2 x3) \[And] x1 + 2 x3 - x2 x3 <= L (x1 - x2 x3) \[And] 2 + x1 - 2 x2 - 2 x3 + x2 x3 <= L (x1 - x2 x3) \[And] 2 - x1 - 2 x3 + x2 x3 <= L (x1 - x2 x3)] %% Exists[{L,x1,x2,x3},L>1 \[And] 5L^2-10L+1<0 \[And] 0 <= x1 <= 1 \[And] 0 <= x2 <= 1 \[And] 0 <= x3 <= 1 \[And] x1 - x2 x3 > 0 \[And] x1 + 2 x2 - x2 x3 <= L (x1 - x2 x3) \[And] 2 - x1 + x2 x3 <= L (x1 - x2 x3) \[And] x1 + 2 x3 - x2 x3 <= L (x1 - x2 x3) \[And] 2 + x1 - 2 x2 - 2 x3 + x2 x3 <= L (x1 - x2 x3) \[And] 2 - x1 - 2 x3 + x2 x3 <= L (x1 - x2 x3)] %% %% input formula for pari/gp tst12([ex,ex,ex],[l,x1,x2,x3],(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14)->(f1*f2*f3*f4*f5*f6*f7*f8*f9*f10*f11*f12*f13*f14),"l>1,l<2, 0<=x1, x1<=1, 0<=x2, x2<=1, 0<=x3, x3<=1, x1 - x2*x3 > 0 , x1 + 2*x2 - x2*x3 <= l*(x1 - x2*x3), 2 - x1 + x2*x3 <= l*(x1 - x2*x3), x1 + 2*x3 - x2*x3 <= l*(x1 - x2*x3), 2 + x1 - 2*x2 - 2*x3 + x2*x3 <= l*(x1 - x2*x3), 2 - x1 - 2*x3 + x2*x3 <= l*(x1 - x2*x3)",1) tst12([ex,ex,ex,ex],[l,x1,x2,x3],(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14)->(f1*f2*f3*f4*f5*f6*f7*f8*f9*f10*f11*f12*f13*f14),"l>1,5*l^2-10*l+1<0, 0<=x1, x1<=1, 0<=x2, x2<=1, 0<=x3, x3<=1, x1 - x2*x3 > 0 , x1 + 2*x2 - x2*x3 <= l*(x1 - x2*x3), 2 - x1 + x2*x3 <= l*(x1 - x2*x3), x1 + 2*x3 - x2*x3 <= l*(x1 - x2*x3), 2 + x1 - 2*x2 - 2*x3 + x2*x3 <= l*(x1 - x2*x3), 2 - x1 - 2*x3 + x2*x3 <= l*(x1 - x2*x3)",1) %% input formula INTLAB %% f03a=@(x) max(max((1+x(2,:)+abs(-1+x(1,:)+x(2,:)-x(2,:).*x(3,:)))./abs(x(1,:)-x(2,:).*x(3,:)),(x(1,:)+2*x(3,:)-x(2,:).*x(3,:))./abs(x(1,:)-x(2,:).*x(3,:))),(2-2*x(3,:)-x(2,:)+x(2,:).*x(3,:)+abs(x(1,:)-x(2,:)))./abs(x(1,:)-x(2,:).*x(3,:))) [mu,L,Ls]=verifyglobalmin(f03a,infsup(0,1)*ones(3,1)) %%OUTPUTS: L - 1 > 0 /\ 5 L^2 - 10 L + 1 >= 0 Counts for Projection Phase: parisize = 8000000, primelimit = 500000 tst12, 08/06/2021 17:43:16 ? tst12([ex,ex,ex],[l,x1,x2,x3],(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10,f11,f12,f13,f14)->(f1*f2*f3*f4*f5*f6*f7*f8*f9*f10*f11*f12*f13*f14),"l>1,l<2, 0<=x1, x1<=1, 0<=x2, x2<=1, 0<=x3, x3<=1, x1 - x2*x3 > 0 , x1 + 2*x2 - x2*x3 <= l*(x1 - x2*x3), 2 - x1 + x2*x3 <= l*(x1 - x2*x3), x1 + 2*x3 - x2*x3 <= l*(x1 - x2*x3), 2 + x1 - 2*x2 - 2*x3 + x2*x3 <= l*(x1 - x2*x3), 2 - x1 - 2*x3 + x2*x3 <= l*(x1 - x2*x3)",1) *** using Lazard's method (MPP17). [x3,8] [x2,18] [x1,61] [l,660] proj = 13,803 ms. 171 8683(6,234) 263149(336,15781) 20(0,271566) *** combined adjacent 19 cells. time = 11min, 47,579 ms. ? Ans(); 1[[5*l^2 - 10*l + 1, 2] <= l < [l - 2, 1],true,true,true]