mirror of
https://github.com/tgorordo/WignerSymbols.jl.git
synced 2026-06-13 02:02:14 -07:00
add wigner9j symbols. update README, .gitignore Manifest
This commit is contained in:
parent
473e618cf0
commit
152d52595e
GPG key ID:
0CBED22BB0D94490
6 changed files with 206 additions and 12 deletions
|
|
@ -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: permutations
|
||||
|
||||
include("growinglist.jl")
|
||||
include("bigint.jl") # additional GMP BigInt functionality not wrapped in Base.GMP.MPZ
|
||||
|
|
@ -14,9 +15,11 @@ 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)
|
||||
|
|
@ -24,6 +27,9 @@ 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 +248,74 @@ 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
|
||||
|
||||
const _perms9j = [(i, j) for i in permutations([1, 2, 3]),
|
||||
j in 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 +386,6 @@ 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 +438,83 @@ 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
|
||||
|
||||
p * b₁ * b₂ * b₃
|
||||
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_()
|
||||
|
||||
|
|
|
|||
|
|
@ -384,3 +384,32 @@ 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 = copy(primefactorial(k))
|
||||
den = mul!(den, primefactorial(n - k))
|
||||
res = divexact!(num, den)
|
||||
return res
|
||||
end
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue