Add 9j symbols.

Project.toml

@@ -1,25 +1,26 @@
 name = "WignerSymbols"
 uuid = "9f57e263-0b3d-5e2e-b1be-24f2bb48858b"
 authors = ["Jutho Haegeman"]
-version = "2.0.0"
+version = "2.1.0"
 
 [deps]
-RationalRoots = "308eb6b3-cc68-5ff3-9e97-c3c4da4fa681"
+Combinatorics = "861a8166-3701-5b0c-9a16-15d98fcdc6aa"
 HalfIntegers = "f0d1745a-41c9-11e9-1dd9-e5d34d218721"
-Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae"
 LRUCache = "8ac3fa9e-de4c-5943-b1dc-09c6b5f20637"
+Primes = "27ebfcd6-29c5-5fa9-bf4b-fb8fc14df3ae"
+RationalRoots = "308eb6b3-cc68-5ff3-9e97-c3c4da4fa681"
 
 [compat]
-RationalRoots = "0.1 - 0.9"
 HalfIntegers = "1"
-Primes = "0.4 - 0.9"
 LRUCache = "1.3"
+Primes = "0.4 - 0.9"
+RationalRoots = "0.1 - 0.9"
 julia = "1.5"
 
 [extras]
-Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
 test = ["Test", "LinearAlgebra", "Random"]

README.md

@@ -89,9 +89,6 @@ Also uses ideas from
 
 [2] [J. Rasch and A. C. H. Yu, SIAM Journal on Scientific Compututing 25 (2003), 1416–1428](https://doi.org/10.1137/S1064827503422932)
 
-for caching the computed 3j and 6j symbols.
+for caching the computed 3j and 6j symbols. Wigner 9-j symbol implementation is based on
 
-## Todo
-*   Wigner 9-j symbols, as explained in [1] and based on
-
-    [3] [L. Wei, New formula for 9-j symbols and their direct calculation, Computers in Physics, 12 (1998), 632–634.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.481.5946&rep=rep1&type=pdf)
+[3] [L. Wei, New formula for 9-j symbols and their direct calculation, Computers in Physics, 12 (1998), 632–634.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.481.5946&rep=rep1&type=pdf)

src/WignerSymbols.jl

@@ -1,11 +1,12 @@
 module WignerSymbols
-export δ, Δ, clebschgordan, wigner3j, wigner6j, racahV, racahW, HalfInteger
+export δ, Δ, clebschgordan, wigner3j, wigner6j, wigner9j, racahV, racahW, HalfInteger
 
 using HalfIntegers
 using RationalRoots
 using LRUCache
 const RRBig = RationalRoot{BigInt}
 import RationalRoots: _convert
+using Combinatorics
 
 include("growinglist.jl")
 include("bigint.jl") # additional GMP BigInt functionality not wrapped in Base.GMP.MPZ
@@ -14,16 +15,21 @@ convert(BigInt, primefactorial(401)) # trigger compilation and generate some fix
 
 const Key3j = Tuple{UInt,UInt,UInt,Int,Int}
 const Key6j = NTuple{6,UInt}
+const Key9j = NTuple{9,HalfInteger}
 
 const Wigner3j = LRU{Key3j,Tuple{Rational{BigInt},Rational{BigInt}}}(; maxsize = 10^6)
 const Wigner6j = LRU{Key6j,Tuple{Rational{BigInt},Rational{BigInt}}}(; maxsize = 10^6)
+const Wigner9j = LRU{Key9j,Tuple{Rational{BigInt},Rational{BigInt}}}(; maxsize = 10^6)
 
 function set_buffer3j_size!(; maxsize)
     resize!(Wigner3j; maxsize = maxsize)
 end
 function set_buffer6j_size!(; maxsize)
     resize!(Wigner6j; maxsize = maxsize)
 end
+function set_buffer9j_size!(; maxsize)
+    resize!(Wigner9j; maxsize = maxsize)
+end
 
 # check integerness and correctness of (j,m) angular momentum
 ϵ(j, m) = (abs(m) <= j && ishalfinteger(j) && isinteger(j-m) && isinteger(j+m))
@@ -242,6 +248,75 @@ function racahW(T::Type{<:Real}, j₁, j₂, J, j₃, J₁₂, J₂₃)
     end
 end
 
+"""
+    wigner9j(T::Type{<:Real} = RationalRoot{BigInt}, j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉) -> ::T
+
+Compute the value of the Wigner-9j symbol
+⎧ j₁ j₂ j₃ ⎫
+⎨	j₄ j₅ j₆ ⎬
+⎩	j₇ j₈ j₉ ⎭
+as a type `T` real number. By default `T = RationalRoot{BigInt}`.
+
+Returns `zero(T)` if any of the triangle conditions TODO 
+are not satisfied, but throws a `DomainError` if
+the `jᵢ`s are not integer or halfinteger.
+"""
+wigner9j(j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉) = wigner9j(RRBig, j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉)
+function wigner9j(T::Type{<:Real}, j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉) 
+    # check validity of `jᵢ`s
+    for jᵢ in (j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉)
+        (ishalfinteger(jᵢ) && jᵢ >= zero(jᵢ)) || throw(DomainError("invalid jᵢ", jᵢ))
+    end
+    return _wigner9j(T, HalfInteger.((j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉))...)
+end
+
+_perms9j = [(i, j) for i in Combinatorics.permutations([1, 2, 3]), 
+                       j in Combinatorics.permutations([1, 2, 3])]
+
+function _wigner9j(T::Type{<:Real}, j₁::HalfInteger, j₂::HalfInteger, j₃::HalfInteger, 
+                                    j₄::HalfInteger, j₅::HalfInteger, j₆::HalfInteger,
+                                    j₇::HalfInteger, j₈::HalfInteger, j₉::HalfInteger)
+
+    # check triangle conditions 
+    if !(δ(j₁, j₂, j₃) && δ(j₄, j₅, j₆) && δ(j₇, j₈, j₉) && δ(j₁, j₄, j₇) && δ(j₂, j₅, j₈) && δ(j₃, j₆, j₉))
+        return zero(T)
+    end
+
+    # dictionary lookup, check all 72 permutations 
+    k = [j₁ j₂ j₃; j₄ j₅ j₆; j₇ j₈ j₉]
+    for p in _perms9j
+        kk = tuple(reshape(k[p...], 9))
+        kkT = tuple(reshape(transpose(k[p...]), 9))
+        if haskey(Wigner9j, kk)
+            r, s = Wigner9j[kk]
+            return _convert(T, s) * convert(T, signedroot(r))
+        elseif haskey(Wigner9j, kkT)
+            r, s = Wigner9j[kkT]
+            return _convert(T, s) * convert(T, signedroot(r)) 
+        end
+    end
+
+    # order irrelevant: product remains the same
+    n₁, d₁ = Δ²(j₁, j₂, j₃)
+    n₂, d₂ = Δ²(j₄, j₅, j₆)
+    n₃, d₃ = Δ²(j₇, j₈, j₉)
+    n₄, d₄ = Δ²(j₁, j₄, j₇)
+    n₅, d₅ = Δ²(j₂, j₅, j₈)
+    n₆, d₆ = Δ²(j₃, j₆, j₉) 
+
+    snum, rnum = splitsquare(n₁ * n₂ * n₃ * n₄ * n₅ * n₆)
+    sden, rden = splitsquare(d₁ * d₂ * d₃ * d₄ * d₅ * d₆)
+    snum, sden = divgcd!(snum, sden)
+    rnum, rden = divgcd!(rnum, rden) 
+    s = Base.unsafe_rational(convert(BigInt, snum), convert(BigInt, sden))
+    r = Base.unsafe_rational(convert(BigInt, rnum), convert(BigInt, rden))
+    s *= compute9jseries(j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉)
+
+    Wigner9j[(j₁, j₂, j₃, j₄, j₅, j₆, j₇, j₈, j₉)] = (r, s)
+    return _convert(T, s) * convert(T, signedroot(r))
+end
+
+
 # COMPUTATIONAL ROUTINES
 #------------------------
 # squared triangle coefficient
@@ -312,7 +387,6 @@ function compute3jseries(β₁, β₂, β₃, α₁, α₂)
     end
     den = commondenominator!(nums, dens)
     totalnum = sumlist!(nums)
-    totalden = convert(BigInt, den)
     for n = 1:length(den.powers)
         p = bigprime(n)
         while den.powers[n] > 0
@@ -365,13 +439,80 @@ function compute6jseries(β₁, β₂, β₃, α₁, α₂, α₃, α₄)
     return Base.unsafe_rational(totalnum, totalden)
 end
 
+# compute the sum appearing in the 9j symbol
+function compute9jseries(a, b, c, d, e, f, g, h, j)
+    I1 = max(abs(h - d), abs(b - f), abs(a - j))
+    I2 = min(h + d, b + f, a + j)
+    krange = I1:I2
+
+    sum(krange) do k
+        p = iseven(2k) ? big(2k + 1) : -big(2k + 1)
+
+        b₁ = let (m₁, m₂, m₃, m₄, m₅, m₆) = (a, b, c, f, j, k)     
+            α₁ = convert(BigInt, m₁ + m₅ - m₆)
+            α₂ = convert(BigInt, m₁ - m₅ + m₆)
+            α₃ = convert(BigInt, -m₁ + m₅ + m₆)
+            β₁ = convert(BigInt, m₁ + m₅ + m₆)
+            β₂ = convert(BigInt, m₂ + m₄ + m₆)
+            β₃ = convert(BigInt, m₃ + m₄ + m₅)
+            β₀ = convert(BigInt, m₁ + m₂ + m₃)
+            wei9jbracket(α₁, α₂, α₃, β₁, β₂, β₃, β₀)
+        end
+
+        b₂ = let (m₁, m₂, m₃, m₄, m₅, m₆) = (f, d, e, h, b, k)
+            α₁ = convert(BigInt, m₁ + m₅ - m₆)
+            α₂ = convert(BigInt, m₁ - m₅ + m₆)
+            α₃ = convert(BigInt, -m₁ + m₅ + m₆)
+            β₁ = convert(BigInt, m₁ + m₅ + m₆)
+            β₂ = convert(BigInt, m₂ + m₄ + m₆)
+            β₃ = convert(BigInt, m₃ + m₄ + m₅)
+            β₀ = convert(BigInt, m₁ + m₂ + m₃)
+            wei9jbracket(α₁, α₂, α₃, β₁, β₂, β₃, β₀)
+        end
+
+        b₃ = let (m₁, m₂, m₃, m₄, m₅, m₆) = (h, j, g, a, d, k) 
+            α₁ = convert(BigInt, m₁ + m₅ - m₆)
+            α₂ = convert(BigInt, m₁ - m₅ + m₆)
+            α₃ = convert(BigInt, -m₁ + m₅ + m₆)
+            β₁ = convert(BigInt, m₁ + m₅ + m₆)
+            β₂ = convert(BigInt, m₂ + m₄ + m₆)
+            β₃ = convert(BigInt, m₃ + m₄ + m₅)
+            β₀ = convert(BigInt, m₁ + m₂ + m₃)
+            wei9jbracket(α₁, α₂, α₃, β₁, β₂, β₃, β₀)
+        end
+end
+
+export wei9jbracket
+# Wei square bracket terms appearing the 9j series
+function wei9jbracket(α₁::BigInt, α₂::BigInt, α₃::BigInt, β₁::BigInt, β₂::BigInt, β₃::BigInt, β₀::BigInt) 
+    p = max(β₁, β₂, β₃, β₀)
+    q = min(α₁ + β₂, α₂ + β₃, α₃ + β₀)
+    lrange = p:q
+    T = PrimeFactorization{eltype(eltype(factorialtable))}
+
+    terms = Vector{T}(undef, length(lrange))
+    for (j, l) in enumerate(lrange)
+        b₁ = iseven(l) ? copy(primebinomial(big(l + 1), l - β₁)) : neg!(copy(primebinomial(big(l + 1), l - β₁)))
+        b₂ = primebinomial(α₁, l - β₂)
+        b₃ = primebinomial(α₂, l - β₃)
+        b₄ = primebinomial(α₃, l - β₀)
+
+        terms[j] = mul!(mul!(mul!(b₁, b₂), b₃), b₄)
+    end
+    total = sumlist!(terms)
+    return total
+end
+
 function _precompile_()
     @assert precompile(wigner3j, (Type{Float64}, Int, Int, Int, Int, Int, Int))
     @assert precompile(wigner6j, (Type{Float64}, Int, Int, Int, Int, Int, Int))
+    @assert precompile(wigner9j, (Type{Float64}, Int, Int, Int, Int, Int, Int, Int, Int, Int))
     @assert precompile(wigner3j, (Type{BigFloat}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}))
     @assert precompile(wigner6j, (Type{BigFloat}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}))
+    @assert precompile(wigner9j, (Type{BigFloat}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}, Rational{Int}))
     @assert precompile(wigner3j, (Type{RRBig}, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt))
     @assert precompile(wigner6j, (Type{RRBig}, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt))
+    @assert precompile(wigner9j, (Type{RRBig}, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt, HalfInt))
 end
 _precompile_()
 

src/primefactorization.jl

@@ -384,3 +384,31 @@ function sumlist!(list::Vector{<:PrimeFactorization}, ind = 1:length(list))
     end
     return MPZ.mul!(s, _convert!(i, g))
 end
+
+#=
+# A cached binomial implementation
+bcache = LRU{Tuple{BigInt, BigInt}, PrimeFactorization}(; maxsize=10^6)
+function primebinomial(n::BigInt, k::BigInt)
+    T = PrimeFactorization{eltype(eltype(factorialtable))}
+    if k == 0
+        return one(T)
+    end # guard
+    if haskey(bcache, (n, k))
+        return bcache[(n, k)]
+    else
+        den = primefactor(k)
+        num = mul!(copy(primefactor(n + 1 - k)), primebinomial(n, k - 1)) 
+        res = divexact!(num, den)
+        bcache[(n, k)] = res
+        return res
+    end
+end
+=#
+
+function primebinomial(n::BigInt, k::BigInt)
+    num = copy(primefactorial(n))
+    den = mul!(copy(primefactorial(k)), primefactorial(n - k))
+    return divexact!(num, den)
+end
+
+

test/runtests.jl

@@ -13,6 +13,7 @@ Random.seed!(1234)
 smalljlist = 0:1//2:10
 largejlist = 0:1//2:1000
 
+
 @threads for i = 1:N
     @testset "triangle coefficient, thread $i" begin
         for k = i:N:length(smalljlist)
@@ -207,3 +208,23 @@ end
         end
     end
 end
+
+@threads for i = 1:N
+    @testset "wigner9j: relation to sum over 6j products, thread $i" begin
+        for k = 1:10_000
+            @testset let (j1, j2, j3, j4, j5, j6, j7, j8, j9) = rand(smalljlist, 9)
+                @test wigner9j(j1, j2, j3, 
+                           j4, j5, j6, 
+                           j7, j8, j9) ≈ sum(largejlist) do x # lazy choice for range of this sum, but good enough
+                                            (iseven(2x) ? (2x + 1) : -(2x + 1)) *
+                                            wigner6j(j1, j4, j7,
+                                                     j8, j9, x ) *
+                                            wigner6j(j2, j5, j8,
+                                                     j4, x , j6) *
+                                            wigner6j(j3, j6, j9,
+                                                     x , j1, j2)
+                               end
+            end
+        end
+    end
+end