algebraic intersection is way to slow

[videlec]

The algebraic intersection is an anti-symmetric bilinear map $H_1(S; \mathbb{Z}) \times H_1(S; \mathbb{Z}) \to \mathbb{Z}$. The values should be computed once on a basis (or alternatively generators) and stored as a matrix so that algebraic_intersection is as fast as vector x matrix x vector multiplication. Right now, it is extremely slow.

With

S = translation_surfaces.mcmullen_L(1,1,1,1)
H = S.homology()
gens = H.gens()
intersection_matrix = matrix(ZZ, len(gens))
for i, gi in enumerate(gens):
    for j, gj in enumerate(gens):
        intersection_matrix[i, j] = gi.algebraic_intersection(gj)
B = set(H(sc) for sc in S.saddle_connections(10))

One can compare the 50ms of

for h1 in B:
    for h2 in B:
        _ = vector(ZZ, h1.coefficients()) * intersection_matrix * vector(ZZ, h2.coefficients())

with the 3s of

%%time
for h1 in B:
    for h2 in B:
        _ = h1.algebraic_intersection(h2)