-- This files contains examples of execution of the robust certified homotopy tracking algorithm (implemented in robustCHT.m2).
load "robustCHT.m2"
DBG = 1 -- determines the verbosity level (default=0)

-- This is a family of examples that are constructed to have a near-singularity in the middle of the path (note that if a=0, then t=0.5 gives a system with a singular solution)
K = makeQI(); R = K[x,y];
a = 1/1000
S = {x^2-y^2}
solsS = {(1,1),(-1,1)};
T = {(a+2*I)*x*y} 
solsT = trackProjectiveCertified(S,T,{first solsS});

-- This example gives the family featured in subsection 9.1 of the paper "Robust certified numerical homotopy tracking" by Carlos Beltran and Anton Leykin.
K = makeQI(); R = K[x,y]; 
a=9999;
S = {x^2-y^2};
T = {x^2-(1+a)*y^2}; -- solving this computes the square roots of 10000 
solsS = {(1,1)};
solsT = trackProjectiveCertified(S,T,{first solsS});    

-- This example illustrates the discussion in subsection 9.2 of the paper "Robust certified numerical homotopy tracking" by Carlos Beltran and Anton Leykin.
K = makeQI(); I = K_0; K[x1, x2, x3, x4, x5, x6, h]
(S,T,hsols) = ({-298*x1*x4-104*x2*x4+3*x1*x5+1727*x2*x5+458*x3*x5+1049*x1*x6-1797*x2*x6-574*x3*x6-1123*x1*h+104*x3*h+698*x4*h-1484*x5*h-606*x6*h+2295*h^2, 2487*x1*x4-1965*x2*x4-1122*x1*x5+690*x2*x5+543*x3*x5+1173*x1*x6-1335*x2*x6+1128*x3*x6-900*x1*h+1965*x3*h+3018*x4*h-1524*x5*h+1086*x6*h-1680*h^2, 1176*x1*x4-756*x2*x4+304*x1*x5-78*x2*x5-168*x3*x5+468*x1*x6-477*x2*x6+252*x3*x6-822*x1*h+756*x3*h-840*x4*h-128*x5*h-468*x6*h+186*h^2, 152*x1*x4+616*x2*x4-56*x1*x5+56*x2*x5+384*x3*x5-76*x1*x6-168*x2*x6+190*x3*x6+112*x1*h-616*x3*h-8*x4*h-280*x5*h-136*x6*h+448*h^2, 332*x1*x4+192*x2*x4+2137*x1*x5-823*x2*x5+2860*x3*x5-524*x1*x6+198*x2*x6-696*x3*x6-239*x1*h-192*x3*h-448*x4*h-3323*x5*h+814*x6*h+352*h^2, -218*x1*x4-33*x2*x4-470*x1*x5-495*x2*x5+150*x3*x5+322*x1*x6+1001*x2*x6-337*x3*x6+616*x1*h+33*x3*h-203*x4*h+55*x5*h-807*x6*h+682*h^2},{-298*x1*x4-104*x2*x4+3*x1*x5+1727*x2*x5+458*x3*x5+1049*x1*x6-1797*x2*x6-574*x3*x6-1123*x1*h+104*x3*h+698*x4*h-1484*x5*h-606*x6*h+2295*h^2, 2487*x1*x4-1965*x2*x4-1122*x1*x5+690*x2*x5+543*x3*x5+1173*x1*x6-1335*x2*x6+1128*x3*x6-900*x1*h+1965*x3*h+3018*x4*h-1524*x5*h+1086*x6*h-1680*h^2, 1176*x1*x4-756*x2*x4+304*x1*x5-78*x2*x5-168*x3*x5+468*x1*x6-477*x2*x6+252*x3*x6-822*x1*h+756*x3*h-840*x4*h-128*x5*h-468*x6*h+186*h^2, 152*x1*x4+616*x2*x4-56*x1*x5+56*x2*x5+384*x3*x5-76*x1*x6-168*x2*x6+190*x3*x6+112*x1*h-616*x3*h-8*x4*h-280*x5*h-136*x6*h+448*h^2, 332*x1*x4+192*x2*x4+2137*x1*x5-823*x2*x5+2860*x3*x5-524*x1*x6+198*x2*x6-696*x3*x6-239*x1*h-192*x3*h-448*x4*h-3323*x5*h+814*x6*h+352*h^2, -282*x1*x4-192*x2*x4-338*x1*x5-198*x2*x5-15*x3*x5-162*x1*x6-136*x2*x6+12*x3*x6+604*x1*h+192*x3*h+693*x4*h+847*x5*h+385*x6*h-1694*h^2},{{-807541*I+787644, 134284*I-471283, 608635*I+469202, -267074*I+328770, 17385*I+1026125, -168834*I+875325, 1000000}, {2061130, 1172754, -124379, 1189047, 2284281, 2172445, 1000000}, {311796, 614317, 1084451, -153749, 2018542, 1059622, 1000000}, {-203981*I+2452936, 562711*I+21574, -2112029*I+3418482, -409478*I-2185407, 406942*I+546639, 1464768*I+1774512, 1000000}, {749305*I+143513, -396570*I+175646, 35649*I+420865, -216945*I-141136, 143756*I+401804, 375506*I+1177243, 1000000}, {-749305*I+143513, 396570*I+175646, -35649*I+420865, 216945*I-141136, -143756*I+401804, -375506*I+1177243, 1000000}, {203981*I+2452936, -562711*I+21574, 2112029*I+3418482, 409478*I-2185407, -406942*I+546639, -1464768*I+1774512, 1000000}, {-262863*I+712457, 141229*I-30921, 215104*I+616840, -219329*I+175388, -59334*I+1224099, 38024*I+1084292, 1000000}, {807541*I+787644, -134284*I-471283, -608635*I+469202, 267074*I+328770, -17385*I+1026125, 168834*I+875325, 1000000}, {262863*I+712457, -141229*I-30921, -215104*I+616840, 219329*I+175388, 59334*I+1224099, -38024*I+1084292, 1000000}, {5002229, 2907859, 5374649, 1233875, -482654, -1912615, 1000000}, {3061118, 4329248, -450203, -1530669, -2069263, -2474512, 1000000}, {435855*I+607557, -16033*I+163096, -325897*I+765357, 96891*I+432134, -439972*I+1471608, 114449*I+735518, 1000000}, {-435855*I+607557, 16033*I+163096, 325897*I+765357, -96891*I+432134, 439972*I+1471608, -114449*I+735518, 1000000}})
solsT = trackProjectiveCertified(S,I*T,hsols)