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load "quadratic_tp_experiment.m2"
randomPolynomial = (d, R) -> sum(d+1, i -> random(i,R))
findTurningPoints = L -> (
temp := sortTurningPoints L;
if #temp === 0 then null
else {(temp#0).LastT, (temp#1).LastT}
)
sortTurningPoints = L -> apply(sort apply(L, sol -> {sol.LastT, sol}), last)
-- will replace with better method (deflation)
getXo = tp -> (1/2) * (matrix first tp + matrix last tp)
gett0 = tp -> (1/2) * ((first tp).LastT + (last tp).LastT)
getTangent = (tp, x0, H) -> (
t0 := gett0 tp;
J := evaluate(transpose jacobian transpose H, x0 | matrix{{t0}});
n := numRows H;
transpose submatrix'(((SVD J)#2)^{n}, ,{n} )
)
-- tp is a pair of points approaching a turning point
-- this function is currently producing some sort of fatal error
jump = (tp, H, epsilon) -> (
if tp === null then error("no turning points inputted");
x0 := getXo(tp);
t0 := gett0(tp);
v := getTangent(tp, x0, H);
if any((first tp).Coordinates | (last tp).Coordinates, c -> not areEqual(imaginaryPart c, 0)) then
l1 := {transpose x0 + epsilon * v, transpose x0 - epsilon * v }
else (
print "two conjugate nonreal";
x1 := transpose x0 + epsilon * ii * v;
l1 = {x1, conjugate x1};
);
-- following two lines from prev implementation give toR = zero mapping, generates fatal error in solveRealSystem
-- R := RR[drop(gens ring H, -1)];
-- toR := map(R, ring H, gens R | {sub(t0+ epsilon^2, R)});
-- coefficient corrector: c := solve(x0, xprev, ClosestFit=>true), xprev := trackSegment(H, t0, t0- epsilon, x0)
-- needs a better stopping criterion (pass number of corrections as option?)
l1 = apply(l1, l -> l || matrix {{t0 + epsilon^2}});
for i from 1 to 5 do l1 = apply(l1, l -> newton(polySystem H, l));
l1
)
-- first example seems to generate valid output, but gives weird "wrong # of inputs" error message
solveRealSystem = method()
solveRealSystem (PolySystem, PolySystem, List) := (f, g, startSols) -> (
if ring f =!= ring g then error "expected systems from same ring";
R := ring f;
t := local t;
S := CC (monoid[gens R,t]);
t = last gens S;
H := transpose (t*sub(gens ideal g,S) + (1-t)*sub(gens ideal f,S));
temp := trackSegment(H,0,1,startSols);
sols := select(temp, s -> status s != Regular);
finishedSols := select(temp, s -> status s == Regular);
while sols != {} do (
sols' := sortTurningPoints(sols);
if #sols' === 1 or not areEqual((sols'#0).LastT, (sols'#1).LastT) then error "expected two failed paths";
epsilon := 0.01;
tp := take(sols', 2);
jumped := jump(tp, H, epsilon);
t0 := (tp#0).LastT;
tp' := trackSegment(H, t0 +epsilon^2, 1, {point(jumped#0), point(jumped#1)});
sols = drop(sols,2);
for s in tp' do
if status s == Regular then finishedSols = finishedSols | {s}
else sols = sols | {s};
);
finishedSols
)
end
restart
needs "example_2d.m2"
setRandomSeed 1
R = CC[x,y]
(d1,d2) = (3,3)
f = polySystem {sub(randomPolynomial(d1, RR[gens R]), R), sub(randomPolynomial(d2, RR[gens R]),R)}
startSols = solveSystem polySystem f
g = polySystem {sub(randomPolynomial(d1, RR[gens R]), R), sub(randomPolynomial(d2, RR[gens R]),R)}
sols' = solveRealSystem(f,g,startSols)
apply(sols', p -> evaluate(g, matrix p))
findTurningPoints track(f,g,startSols)
(sols#1).Coordinates
epsilon = 0.01
tp = findTurningPoints sols
predictor = getPredictor(tp, epsilon)
sols2 = trackSegment(H, t0 +epsilon^2, 1, {point(predictor#0), point(predictor#1)})
epsilon = 0.01
tp = findTurningPoints sols2
predictor = getPredictor(tp, epsilon)
trackSegment(H, t0 +epsilon^2, 1, {point(predictor#0), point(predictor#1)})
solveSystem polySystem g
solutions = oo
x1
H = matrix {{t*sub(g,S) + (1-t)*sub(f,S)}}
(t0, t1) = (0, 1)
sols = trackSegment(H, 0, 1, {point{{3}},point{{2}},point{{0}},point{{-1}}})
x0 = (1/2) * (matrix sols#2 + matrix sols#3)
t0 = (sols#2).LastT
J
J = evaluate(transpose jacobian H, x0 | matrix{{t0}})
n = numRows H
v = submatrix'(((SVD J)#2)_{n}, {n}, )
epsilon = 0.1
x1 = x0 + epsilon * ii * v
toR = map(R, S, gens R | {t0+epsilon^2})
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
trackSegment(H, t0 +epsilon^2, 1, {point(x1)})
solutions = oo
H = matrix {{t*sub(g,S) + (1-t)*sub(f,S)}}
(t0, t1) = (0, 1)
sols = trackSegment(H, 0, 1, {point{{3}},point{{2}},point{{0}},point{{-1}}})
x0 = (1/2) * (matrix sols#2 + matrix sols#3)
t0 = (sols#2).LastT
J
J = evaluate(transpose jacobian H, x0 | matrix{{t0}})
n = numRows H
v = submatrix'(((SVD J)#2)_{n}, {n}, )
epsilon = 0.1
x1 = x0 + epsilon * ii * v
toR = map(R, S, gens R | {t0+epsilon^2})
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
trackSegment(H, t0 +epsilon^2, 1, {point(x1)})
solutions = oo
H = matrix {{t*sub(g,S) + (1-t)*sub(f,S)}}
(t0, t1) = (0, 1)
sols = trackSegment(H, 0, 1, {point{{3}},point{{2}},point{{0}},point{{-1}}})
x0 = (1/2) * (matrix sols#2 + matrix sols#3)
t0 = (sols#2).LastT
J
J = evaluate(transpose jacobian H, x0 | matrix{{t0}})
n = numRows H
v = submatrix'(((SVD J)#2)_{n}, {n}, )
epsilon = 0.1
x1 = x0 + epsilon * ii * v
toR = map(R, S, gens R | {t0+epsilon^2})
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
x1 = newton(polySystem toR H, x1)
trackSegment(H, t0 +epsilon^2, 1, {point(x1)})
solutions = oo
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