Ferraris

Here we deal with a non polynomial system of 2 equations with a very large initial search domain. An additional problem is that the second equation involves the interval evaluation of the exponential of a variable that may leads to an overflow. Hence we use the mechanism described in section 2.1.4 to avoid the evaluation of the exponential if the upper bound of the first variable is larger than 1e4.

 
eq1:= -0.25/Pi*x2 - 0.5*x1  + 0.5*sin(x1*x2);
eq2:= exp(1)/Pi*x2 - 2*exp(1)*x1  + (1 - 0.25/Pi)*(exp(2*x1) - exp(1));

EQ:=[evalf(eq1),evalf(eq2)]:
VAR:=[x1,x2]:
INIT:=[[-10^8,10^8],[-10^8,10^8]]:

with(ALIAS):
`ALIAS/debug`:=1:
`ALIAS/single_bisection`:=2:
`ALIAS/storage_mode`:=40:
`ALIAS/lib`:="../Test/Lib":
`ALIAS/profil`:="../Profil":
`ALIAS/3B`:=1:
`ALIAS/Max3B`:=5*10^9:
`ALIAS/Delta3B`:=0.0001:
`ALIAS/Full3B`:=0:
`ALIAS/Full3B_Change`:=0.0001:
`ALIAS/maxgradient`:=20:
`ALIAS/maxkraw`:=1:
`ALIAS/maxnewton`:=1:

`ALIAS/user_FINIT`:="INTERVAL U;":
ALIAS_F:=proc(fid,i)
if i=2 then
fprintf(fid,"if(Sup(v_IS(1))>1.e4)V(2)=INTERVAL(-1.e6,1.e12);else\n"):
fi:
RETURN(0):
end:

HullConsistency(EQ,VAR,"Simp",0.1);
HessianSolve(EQ,VAR,INIT,"Simp"):

To manage the possible overflow problem instead of designing the ALIAS_F procedure by hand we may have also used the procedures described in the section 2.1.5 that create automatically the ALIAS_F and ALIAS_J procedures.

 
with(ALIAS):
with(linalg):
`ALIAS/gpp_linux`:="/usr/local/gcc-2.95/bin/g++":
`ALIAS/debug`:=1:
`ALIAS/single_bisection`:=1:
`ALIAS/storage_mode`:=40:
`ALIAS/lib`:="../Test/Lib":
`ALIAS/profil`:="../Profil":
`ALIAS/3B`:=1:
`ALIAS/Max3B`:=5*10^9:
`ALIAS/Delta3B`:=0.0001:
`ALIAS/Full3B`:=0:
`ALIAS/Full3B_Change`:=0.0001:
`ALIAS/maxgradient`:=20:
`ALIAS/maxkraw`:=1:
`ALIAS/maxnewton`:=1:
`ALIAS/optimized`:=0:

J:=jacobian(EQ,VAR):

Verify_Problem_Expression(EQ,VAR):
Verify_Problem_ExpressionJ(eval(J),VAR):

HullConsistency(EQ,VAR,"Simp1",0.1);
HessianSolve(EQ,VAR,INIT,"Simp1"):