Reference

index

The indices that span a simplex. coord::Vector{Int} The indices of the simplex refering to vertices in the vector points. active When the simplex is active, it is still part of the Picard Lefschetz flow.

parameters

Contains the parameters of the Picard Lefschetz integration method.

dim         The dimension of the integral.
δ           The maximal edge length of the simplices.
τ           Threshold on the real part of the exponent.
ϵ           The steps size of the downward flow.
N           The number of steps in the downward flow.
n           The order of the simplex quadrature integration scheme.

point

A point with two boolian attributes. The first determines whether the point is active. The second determines whether the point flows or stays fixed.

coord::Vector{Complex}  The coordinates of the point.
active                  When the point is active, it is still part of the Picard Lefschetz flow.
frozen                  When a point is fozen, it does not flow. This is particularly relevant for points on the boundary of the integration domain.

quadrature

The simplex quadrature rule on a thimble. Building it once and reusing it avoids repeating the setup in every call of PL_integrate, for instance when evaluating the integral for many values of the parameters on the same thimble.

nodes::Matrix{ComplexF64}       The quadrature nodes, one per column.
weights::Vector{ComplexF64}     The quadrature weights multiplied by the Jacobian of their simplex.
vertices::Matrix{ComplexF64}    The vertices of the thimble, one per column.
simplices::Vector{Vector{Int}}  The indices of the vertices spanning each simplex.
npts::Int                       The number of quadrature nodes per simplex.

quadrature(thim::thimble, pars::parameters)

Construct the simplex quadrature rule of order pars.n on the thimble.

thimble

The thimble consisting of a set of points and a set of simplices.

PL_integrate(S, Q::quadrature, pars::parameters)

Evaluate the integral ∫exp(im * S(x))dx with the quadrature rule Q on the thimble. Simplices on which the real part of im * S is below the threshold pars.τ at all vertices are skipped.

PL_integrate(S, thim::thimble, pars::parameters)

Evaluate the integral ∫exp(im * S(x))dx along the thimble.

divide(thim, pars::parameters, dictionary = Dict{Vector{Int}, Int}())

Subdivide the simplices that have an edge longer than the threshold pars.δ by bisecting their longest edge. A simplex whose edges are all shorter than pars.δ, but which has an edge that was bisected in a neighbouring simplex, is bisected along that edge, so that the triangulation stays conforming. The dictionary maps the bisected edges to the index of their midpoint.

divide_rep(thim::thimble, pars::parameters)

Repeat subdivision till all edges of the simplices are shorter than par.δ. The bisected edges are remembered across the repetitions, so that no hanging vertices remain.

flow!(S, thim::thimble, pars::parameters)

Flow the thimble pars.N steps forward in the downward flow.

flow_point(S, p, pars::parameters)

Push a point one step in the downward flow.

flow_points(S, thim::thimble, pars::parameters)

Flow the thimble one step in the downward flow.

hypercube_triangulation_indices(dim::Int)

Triangulate a dim dimensional hypercube and output the simplices.

initialGrid(min, max, pars::parameters, frozen = false)

Initialize a cubic grid with the lower left corner min and the upper right corner max. When forzen = true, the boundary points do not flow.

normalize(p, thres = 1.)

Normalize a vector when the norm exceeds the threshold.

remove_inactive_points(thim::thimble)

Remove inactive points.

volume_simplex(vertices)

The volume of a simplex.