Add orthonormal basis (#22)

* Add orthonormal basis

* Fix format

docs/src/index.md

@@ -15,6 +15,7 @@ AbstractPolynomialBasis
 maxdegree_basis
 basis_covering_monomials
 FixedPolynomialBasis
+OrthonormalCoefficientsBasis
 ```
 
 ## Monomial basis

src/MultivariateBases.jl

@@ -13,6 +13,7 @@ include("monomial.jl")
 include("scaled.jl")
 
 export FixedPolynomialBasis,
+    OrthonormalCoefficientsBasis,
     AbstractMultipleOrthogonalBasis,
     ProbabilistsHermiteBasis,
     PhysicistsHermiteBasis,
@@ -31,6 +32,7 @@ export generators,
 include("fixed.jl")
 
 import LinearAlgebra
+include("orthonormal.jl")
 include("orthogonal.jl")
 include("hermite.jl")
 include("laguerre.jl")

src/orthogonal.jl

@@ -209,16 +209,10 @@ function _integral(
     return sum([_integral(t, basis_type) for t in MP.terms(p)])
 end
 
-function MP.coefficients(
-    p,
-    basis::AbstractMultipleOrthogonalBasis;
-    check = true,
-)
+function MP.coefficients(p, basis::AbstractMultipleOrthogonalBasis)
     B = typeof(basis)
-    coeffs = [
+    return [
         LinearAlgebra.dot(p, el, B) / LinearAlgebra.dot(el, el, B) for
         el in basis
     ]
-    idx = findall(c -> !isapprox(c, 0; atol = 1e-10), coeffs)
-    return coeffs[idx]
 end

src/orthonormal.jl

@@ -0,0 +1,48 @@
+"""
+    struct OrthonormalCoefficientsBasis{PT<:MP.AbstractPolynomialLike, PV<:AbstractVector{PT}} <: AbstractPolynomialBasis
+        polynomials::PV
+    end
+
+Polynomial basis with the polynomials of the vector `polynomials` that are
+orthonormal with respect to the inner produce derived from the inner product
+of their coefficients.
+For instance, `FixedPolynomialBasis([1, x, 2x^2-1, 4x^3-3x])` is the Chebyshev
+polynomial basis for cubic polynomials in the variable `x`.
+"""
+struct OrthonormalCoefficientsBasis{
+    PT<:MP.AbstractPolynomialLike,
+    PV<:AbstractVector{PT},
+} <: AbstractPolynomialVectorBasis{PT,PV}
+    polynomials::PV
+end
+
+function LinearAlgebra.dot(
+    p::MP.AbstractPolynomialLike{S},
+    q::MP.AbstractPolynomialLike{T},
+    ::Type{<:OrthonormalCoefficientsBasis},
+) where {S,T}
+    s = zero(MA.promote_operation(*, S, T))
+    terms_p = MP.terms(p)
+    terms_q = MP.terms(q)
+    tsp = iterate(terms_p)
+    tsq = iterate(terms_q)
+    while !isnothing(tsp) && !isnothing(tsq)
+        tp, sp = tsp
+        tq, sq = tsq
+        cmp = MP.compare(MP.monomial(tp), MP.monomial(tq))
+        if iszero(cmp)
+            s += conj(MP.coefficient(tp)) * MP.coefficient(tq)
+            tsp = iterate(terms_p, sp)
+            tsq = iterate(terms_q, sq)
+        elseif cmp < 0
+            tsp = iterate(terms_p, sp)
+        else
+            tsq = iterate(terms_q, sq)
+        end
+    end
+    return s
+end
+
+function MP.coefficients(p, basis::OrthonormalCoefficientsBasis)
+    return [LinearAlgebra.dot(q, p, typeof(basis)) for q in generators(basis)]
+end

test/fixed.jl

@@ -14,8 +14,6 @@ using DynamicPolynomials
     @test collect(basis) == gens
     @test generators(basis) == gens
     @test length(basis) == 2
-    @test firstindex(basis) == 1
-    @test lastindex(basis) == 2
     @test mindegree(basis) == 0
     @test mindegree(basis, x) == 0
     @test maxdegree(basis) == 2

test/monomial.jl

@@ -12,9 +12,6 @@ using DynamicPolynomials
     @test basis[2] == x
     @test generators(basis) == [y, x]
     @test collect(basis) == [y, x]
-    @test length(basis) == 2
-    @test firstindex(basis) == 1
-    @test lastindex(basis) == 2
     @test variables(basis) == [x, y]
     @test nvariables(basis) == 2
     @test sprint(show, basis) == "MonomialBasis([y, x])"

test/orthonormal.jl

@@ -0,0 +1,82 @@
+using Test
+using MultivariateBases
+using DynamicPolynomials
+
+@polyvar x y
+
+@testset "Polynomials" begin
+    gens = [1 + x + y + x * y, 1 - x + y - x * y] / 2
+    basis = OrthonormalCoefficientsBasis(gens)
+    @test iszero(dot(gens[1], gens[2], OrthonormalCoefficientsBasis))
+    coefficient_test(basis, [2, -3])
+    coefficient_test(basis, [-2im, 1 + 5im])
+    coefficient_test(basis, [1im, 2im])
+    @test polynomial_type(basis, Int) == polynomial_type(x, Float64)
+    @test polynomial(one, basis) == 1 + y
+    @test basis[1] == gens[1]
+    @test basis[2] == gens[2]
+    @test collect(basis) == gens
+    @test generators(basis) == gens
+    @test length(basis) == 2
+    @test mindegree(basis) == 0
+    @test mindegree(basis, x) == 0
+    @test mindegree(basis, y) == 0
+    @test maxdegree(basis) == 2
+    @test maxdegree(basis, x) == 1
+    @test maxdegree(basis, y) == 1
+    @test extdegree(basis) == (0, 2)
+    @test extdegree(basis, x) == (0, 1)
+    @test extdegree(basis, y) == (0, 1)
+    @test variables(basis) == [x, y]
+    @test nvariables(basis) == 2
+    @test sprint(show, basis) ==
+          "OrthonormalCoefficientsBasis([0.5 + 0.5y + 0.5x + 0.5xy, 0.5 + 0.5y - 0.5x - 0.5xy])"
+    @test sprint(print, basis) ==
+          "OrthonormalCoefficientsBasis([0.5 + 0.5*y + 0.5*x + 0.5*x*y, 0.5 + 0.5*y - 0.5*x - 0.5*x*y])"
+    b2 = basis[2:2]
+    @test length(b2) == 1
+    @test b2[1] == gens[2]
+    b3 = basis[2:1]
+    @test isempty(b3)
+end
+@testset "Linear" begin
+    basis = OrthonormalCoefficientsBasis([x, y])
+    @test polynomial_type(basis, Int) == polynomial_type(x, Int)
+    @test polynomial(identity, basis) == x + 2y
+end
+@testset "One variable" begin
+    basis = OrthonormalCoefficientsBasis([x])
+    @test polynomial_type(basis, Int) == polynomial_type(x, Int)
+    @test polynomial(i -> 5, basis) == 5x
+    @test typeof(polynomial(i -> 5, basis)) == polynomial_type(basis, Int)
+    @test typeof(polynomial(ones(Int, 1, 1), basis, Int)) <:
+          AbstractPolynomial{Int}
+    @test typeof(polynomial(ones(Int, 1, 1), basis, Float64)) <:
+          AbstractPolynomial{Float64}
+end
+@testset "Complex" begin
+    for a in [1, -1, im, -im]
+        basis = OrthonormalCoefficientsBasis([a * x])
+        @test 5x^2 ==
+              @inferred polynomial(5ones(Int, 1, 1), basis, Complex{Int})
+        @test 5x^2 == @inferred polynomial(5ones(Int, 1, 1), basis, Int)
+        coefficient_test(basis, [2])
+        coefficient_test(basis, [-2im])
+        coefficient_test(basis, [1 + 5im])
+    end
+end
+@testset "Empty" begin
+    basis = OrthonormalCoefficientsBasis(typeof(x + 1)[])
+    @test isempty(basis)
+    @test isempty(eachindex(basis))
+    p = @inferred polynomial(zeros(Int, 0, 0), basis, Int)
+    @test iszero(p)
+end
+
+@testset "Enumerate" begin
+    monos = [1, x, y^2]
+    basis = OrthonormalCoefficientsBasis(monos)
+    for (i, e) in enumerate(basis)
+        @test e == monos[i]
+    end
+end

test/runtests.jl

@@ -12,6 +12,8 @@ function api_test(B::Type{<:AbstractPolynomialBasis}, degree)
     ]
         n = binomial(2 + degree, 2)
         @test length(basis) == n
+        @test firstindex(basis) == 1
+        @test lastindex(basis) == n
         @test typeof(copy(basis)) == typeof(basis)
         @test nvariables(basis) == 2
         @test variables(basis) == x
@@ -87,13 +89,31 @@ function orthogonal_test(
     end
 end
 
+function coefficient_test(basis::AbstractPolynomialBasis, p, coefs; kwargs...)
+    cc = coefficients(p, basis)
+    @test isapprox(coefs, cc; kwargs...)
+    @test isapprox(p, polynomial(cc, basis); kwargs...)
+end
+
+function coefficient_test(
+    basis::AbstractPolynomialBasis,
+    coefs::AbstractVector;
+    kwargs...,
+)
+    return coefficient_test(
+        basis,
+        sum(generators(basis) .* coefs),
+        coefs;
+        kwargs...,
+    )
+end
+
 function coefficient_test(B::Type{<:AbstractPolynomialBasis}, coefs; kwargs...)
     @polyvar x y
     p = x^4 * y^2 + x^2 * y^4 - 3 * x^2 * y^2 + 1
     basis = basis_covering_monomials(B, monomials(p))
-    cc = coefficients(p, basis)
-    @test isapprox(coefs, cc; kwargs...)
-    @test isapprox(p, polynomial(cc, basis); kwargs...)
+    coefficient_test(basis, p, coefs; kwargs...)
+    return
 end
 
 @testset "Monomial" begin
@@ -105,6 +125,9 @@ end
 @testset "Fixed" begin
     include("fixed.jl")
 end
+@testset "Orthonormal" begin
+    include("orthonormal.jl")
+end
 @testset "Hermite" begin
     include("hermite.jl")
 end