HarmonicPolya.jl

This is a package for homogeneous polynomial minimization on the sphere using harmonic hierarchies found in Cristancho & Velasco also implementing two required features: polynomial cubature/quadrature rules on the sphere using FastGaussQuadrature and harmonic polynomial analysis on the sphere based on Axler & Ramey. The implementation of polynomials uses DynamicPolynomials although FixedPolynomials is suggested for calculations using quadratures.

Quadratures on the Sphere

The method sphericalquadrature(n,deg) defines the nodes and weights of a quadrature on the n-dimensional sphere $S^{n-1}$ for polynomials of degree $\leq$ deg. Example of usage:

julia> using HarmonicPolya

julia> z,w = sphericalquadrature(3,4) #this defines a quadrature on the 3-dimensional sphere for polynomials of degree =< 4

julia> poly(x) = x[1]^2*x[2]^2*x[3]^2 #defines a polynomial function on RR^3 

julia> I = sum(w*.(poly.(z))) #integrates f on the sphere using the quadrature rule

Harmonic Analysis on the Sphere

Features three functions: laplacian(poly,vars;power=1) is the laplacian for a polynomial poly from DynamicPolynomials using variables vars to the power of power (default is 1);generatebasissphere(deg,vars) generates a basis for the space of homogeneous harmonic polynomials of degree deg in the variables vars; and harmonicdecomposition(poly,vars) produces a vector representing the decomposition in harmonic components (as a tuple (degree,component) in ascending degree) of the polynomial poly in the variables vars.

julia> using HarmonicPolya, DynamicPolynomials

julia> @polyvar x[1:3]

julia> u = x[1]^2 + x[2]*x[3] # defines a polynomial in DynamicPolynomials

julia> ∇u = laplacian(u,x) # calculates the laplacian of u

julia> basis = generatebasissphere(3,x) # produces the basis of harmonic polynomials of degree 3 in the variables x

julia> desc = harmonicdecomposition(u,x) # produces a vector with the harmonic decomposition of u in ascending degree

julia> norm = sum(x .* x)

julia> u' = sum( map(v->norm^(floor(Int,v[1]/2))*v[2],desc)) #reconstructs u from its decomposition (working on a more elegant way)

Polynomial Minimization on the Sphere

The primary function of the package. There are three different methods (as in Cristancho & Velasco): upperbound(poly,vars,m) produces an upper bound on the minimum of the homogeneous polynomial poly in variables vars using a quadrature of degree m; lowerboundsquares(poly,vars,m) produces a lower bound of poly in vars using the hierarchy of pure square powers of degree m; and lowerboundfawzi(poly,vars,m) produces a lower bound of poly in vars using the Fang-Fawzi hierarchy of degree m.

julia> using HarmonicPolya, DynamicPolynomials

julia> @polyvar y[1:3]

julia> motzkin = y[1]^2*y[2]^4+ y[1]^4*y[2]^2+y[3]^6-3*y[1]^2*y[2]^2*y[3]^2 #defines the Motzkin polynomial (which is non-negative and homogeneous)

julia> u = upperbound(motzkin,y,20) #upper bound with quadrature of degree 20

julia> ls = lowerboundsquares(motzkin,y,20) #lower bound with squares hierarchy degree 20

julia> lf = lowerboundfawzi(motzkin,y,20) #lower bound with Fang-fawzi hierarchy degree 20