Tw/interface

[tweisser]

I unified the interface for Monomial and Polynomial Bases (now the field is called elements).
The idea is, that <:AbstractPolynomialBasis is a wrapper for a familiy of polynomials that we label as a basis in some sense (similar to MP.MonomialVector).
I added iterate, getindex, and a coefficients(poly, basis_type) function which returns a vector of coefficients which should yield

p == polynomial(coefficients(p, basis), basis_covering_monomials(basis, monomials(p))

I'm working on changes to depended packages.

[codecov-commenter]

Codecov Report

Attention: 6 lines in your changes are missing coverage. Please review.

Comparison is base (417aec8) 96.87% compared to head (2aeb23f) 96.08%.

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Files	Patch %	Lines
src/orthogonal.jl	68.42%	6 Missing ⚠️

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@@            Coverage Diff             @@
##           master       #5      +/-   ##
==========================================
- Coverage   96.87%   96.08%   -0.79%     
==========================================
  Files          10       10              
  Lines         224      281      +57     
==========================================
+ Hits          217      270      +53     
- Misses          7       11       +4     

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[blegat]

Define

function MP.coefficients(p, Basis::Type{<:AbstractPolynomialBasis})
    return MP.coefficients(p, maxdegree_basis(Basis, variables(p), maxdegree(p)))
end
function MP.coefficients(p, basis::AbstractPolynomialBasis)
    # ...
end

It is needed for https://github.com/JuliaAlgebra/MultivariateMoments.jl/pull/16/files#r438604571

[tweisser]

Specifying an instantiated basis would kind of imply that we return a vector of coefficients of length length(basis). So far I am deleting zero coefficients to be consistent with coefficients(p).

[tweisser]

I introduced inner products for the orthogonal bases and defined MP.coefficients(p, basis::AbstractMultipleOrthogonalBasis).
The inner products are numerically quite instable (dividing huge numbers). As a result coefficient function is numerically unstable, too. I had to introduce isapprox in place of == at some points to make the tests pass. I imagine, going to higher degrees, the tolerances will not be sufficient.

[blegat]

n = div(i, 2)

[blegat]

elseif isodd(i)

[blegat]

This will very quickly exceed the capacity of Int64. We can simplify factorial(i) with factorial(n), do pi / 2^i so that it transforms to Float64 before multiplying with factorial(n)

[blegat]

Similar comments than for the other one

[blegat]

function scalar_product_function(::Type{<:LegendreBasis}, i::Int)

[blegat]

struct PolynomialInBasis{C, B}
    coefficients::Vector{C}
    basis::B
end

and instead of MP.coefficient, we could have in_basis(p, basis)::PolynomialInBasis

[blegat]

in_basis(x^2 + 2, MonomialBasis([x^2, x, 1])) -> PolynomialInBasis([1, 0, 2], MonomialBasis([x^2, x, 1])
in_basis(x^2 + 2, MonomialBasis) -> PolynomialInBasis([1, 2], MonomialBasis([x^2, 1])