mirror of
https://github.com/tgorordo/WignerSymbols.jl.git
synced 2026-06-05 15:42:15 -07:00
add MPZ for 0.6 compatibility; update REQUIRE, add testsets
This commit is contained in:
Jutho Haegeman
parent
cb09b427c9
commit
258246f854
5 changed files with 249 additions and 114 deletions
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@ -1,9 +1,9 @@
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# WignerSymbols
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[](https://travis-ci.org/jutho/WignerSymbols.jl)
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[](https://travis-ci.org/Jutho/WignerSymbols.jl)
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[](LICENSE.md)
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[](https://coveralls.io/github/Jutho/WignerSymbols.jl?branch=master)
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[](http://codecov.io/github/Jutho/WignerSymbols.jl?branch=master)
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[](http://codecov.io/github/Jutho/WignerSymbols.jl?branch=master)
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Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.
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2
REQUIRE
2
REQUIRE
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@ -1,2 +1,2 @@
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julia 0.7-
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julia 0.6
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Primes
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116
src/mpz.jl
Normal file
116
src/mpz.jl
Normal file
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@ -0,0 +1,116 @@
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# taken from Julia 0.7-Dev base library
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module MPZ
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# wrapping of libgmp functions
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# - "output parameters" are labeled x, y, z, and are returned when appropriate
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# - constant input parameters are labeled a, b, c
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# - a method modifying its input has a "!" appendend to its name, according to Julia's conventions
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# - some convenient methods are added (in addition to the pure MPZ ones), e.g. `add(a, b) = add!(BigInt(), a, b)`
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# and `add!(x, a) = add!(x, x, a)`.
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using Base.GMP: BigInt, Limb
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const mpz_t = Ref{BigInt}
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const bitcnt_t = Culong
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gmpz(op::Symbol) = (Symbol(:__gmpz_, op), :libgmp)
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init!(x::BigInt) = (ccall((:__gmpz_init, :libgmp), Void, (mpz_t,), x); x)
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init2!(x::BigInt, a) = (ccall((:__gmpz_init2, :libgmp), Void, (mpz_t, bitcnt_t), x, a); x)
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realloc2!(x, a) = (ccall((:__gmpz_realloc2, :libgmp), Void, (mpz_t, bitcnt_t), x, a); x)
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realloc2(a) = realloc2!(BigInt(), a)
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sizeinbase(a::BigInt, b) = Int(ccall((:__gmpz_sizeinbase, :libgmp), Csize_t, (mpz_t, Cint), a, b))
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for op in (:add, :sub, :mul, :fdiv_q, :tdiv_q, :fdiv_r, :tdiv_r, :gcd, :lcm, :and, :ior, :xor)
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op! = Symbol(op, :!)
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@eval begin
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$op!(x::BigInt, a::BigInt, b::BigInt) = (ccall($(gmpz(op)), Void, (mpz_t, mpz_t, mpz_t), x, a, b); x)
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$op(a::BigInt, b::BigInt) = $op!(BigInt(), a, b)
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$op!(x::BigInt, b::BigInt) = $op!(x, x, b)
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end
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end
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invert!(x::BigInt, a::BigInt, b::BigInt) =
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ccall((:__gmpz_invert, :libgmp), Cint, (mpz_t, mpz_t, mpz_t), x, a, b)
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invert(a::BigInt, b::BigInt) = invert!(BigInt(), a, b)
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invert!(x::BigInt, b::BigInt) = invert!(x, x, b)
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for op in (:add_ui, :sub_ui, :mul_ui, :mul_2exp, :fdiv_q_2exp, :pow_ui, :bin_ui)
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op! = Symbol(op, :!)
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@eval begin
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$op!(x::BigInt, a::BigInt, b) = (ccall($(gmpz(op)), Void, (mpz_t, mpz_t, Culong), x, a, b); x)
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$op(a::BigInt, b) = $op!(BigInt(), a, b)
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$op!(x::BigInt, b) = $op!(x, x, b)
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end
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end
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ui_sub!(x::BigInt, a, b::BigInt) = (ccall((:__gmpz_ui_sub, :libgmp), Void, (mpz_t, Culong, mpz_t), x, a, b); x)
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ui_sub(a, b::BigInt) = ui_sub!(BigInt(), a, b)
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for op in (:scan1, :scan0)
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@eval $op(a::BigInt, b) = Int(ccall($(gmpz(op)), Culong, (mpz_t, Culong), a, b))
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end
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mul_si!(x::BigInt, a::BigInt, b) = (ccall((:__gmpz_mul_si, :libgmp), Void, (mpz_t, mpz_t, Clong), x, a, b); x)
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mul_si(a::BigInt, b) = mul_si!(BigInt(), a, b)
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mul_si!(x::BigInt, b) = mul_si!(x, x, b)
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for op in (:neg, :com, :sqrt, :set)
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op! = Symbol(op, :!)
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@eval begin
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$op!(x::BigInt, a::BigInt) = (ccall($(gmpz(op)), Void, (mpz_t, mpz_t), x, a); x)
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$op(a::BigInt) = $op!(BigInt(), a)
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end
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op == :set && continue # MPZ.set!(x) would make no sense
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@eval $op!(x::BigInt) = $op!(x, x)
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end
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for (op, T) in ((:fac_ui, Culong), (:set_ui, Culong), (:set_si, Clong), (:set_d, Cdouble))
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op! = Symbol(op, :!)
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@eval begin
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$op!(x::BigInt, a) = (ccall($(gmpz(op)), Void, (mpz_t, $T), x, a); x)
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$op(a) = $op!(BigInt(), a)
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end
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end
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popcount(a::BigInt) = Int(ccall((:__gmpz_popcount, :libgmp), Culong, (mpz_t,), a))
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mpn_popcount(d::Ptr{Limb}, s::Integer) = Int(ccall((:__gmpn_popcount, :libgmp), Culong, (Ptr{Limb}, Csize_t), d, s))
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mpn_popcount(a::BigInt) = mpn_popcount(a.d, abs(a.size))
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function tdiv_qr!(x::BigInt, y::BigInt, a::BigInt, b::BigInt)
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ccall((:__gmpz_tdiv_qr, :libgmp), Void, (mpz_t, mpz_t, mpz_t, mpz_t), x, y, a, b)
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x, y
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end
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tdiv_qr(a::BigInt, b::BigInt) = tdiv_qr!(BigInt(), BigInt(), a, b)
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powm!(x::BigInt, a::BigInt, b::BigInt, c::BigInt) =
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(ccall((:__gmpz_powm, :libgmp), Void, (mpz_t, mpz_t, mpz_t, mpz_t), x, a, b, c); x)
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powm(a::BigInt, b::BigInt, c::BigInt) = powm!(BigInt(), a, b, c)
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powm!(x::BigInt, b::BigInt, c::BigInt) = powm!(x, x, b, c)
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function gcdext!(x::BigInt, y::BigInt, z::BigInt, a::BigInt, b::BigInt)
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ccall((:__gmpz_gcdext, :libgmp), Void, (mpz_t, mpz_t, mpz_t, mpz_t, mpz_t), x, y, z, a, b)
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x, y, z
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end
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gcdext(a::BigInt, b::BigInt) = gcdext!(BigInt(), BigInt(), BigInt(), a, b)
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cmp(a::BigInt, b::BigInt) = Int(ccall((:__gmpz_cmp, :libgmp), Cint, (mpz_t, mpz_t), a, b))
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cmp_si(a::BigInt, b) = Int(ccall((:__gmpz_cmp_si, :libgmp), Cint, (mpz_t, Clong), a, b))
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cmp_ui(a::BigInt, b) = Int(ccall((:__gmpz_cmp_ui, :libgmp), Cint, (mpz_t, Culong), a, b))
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cmp_d(a::BigInt, b) = Int(ccall((:__gmpz_cmp_d, :libgmp), Cint, (mpz_t, Cdouble), a, b))
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mpn_cmp(a::Ptr{Limb}, b::Ptr{Limb}, c) = ccall((:__gmpn_cmp, :libgmp), Cint, (Ptr{Limb}, Ptr{Limb}, Clong), a, b, c)
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mpn_cmp(a::BigInt, b::BigInt, c) = mpn_cmp(a.d, b.d, c)
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get_str!(x, a, b::BigInt) = (ccall((:__gmpz_get_str,:libgmp), Ptr{Cchar}, (Ptr{Cchar}, Cint, mpz_t), x, a, b); x)
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set_str!(x::BigInt, a, b) = Int(ccall((:__gmpz_set_str, :libgmp), Cint, (mpz_t, Ptr{UInt8}, Cint), x, a, b))
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get_d(a::BigInt) = ccall((:__gmpz_get_d, :libgmp), Cdouble, (mpz_t,), a)
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limbs_write!(x::BigInt, a) = ccall((:__gmpz_limbs_write, :libgmp), Ptr{Limb}, (mpz_t, Clong), x, a)
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limbs_finish!(x::BigInt, a) = ccall((:__gmpz_limbs_finish, :libgmp), Void, (mpz_t, Clong), x, a)
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import!(x::BigInt, a, b, c, d, e, f) = ccall((:__gmpz_import, :libgmp), Void,
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(mpz_t, Csize_t, Cint, Csize_t, Cint, Csize_t, Ptr{Void}), x, a, b, c, d, e, f)
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end # module MPZ
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@ -1,6 +1,13 @@
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using Primes.isprime
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import Base.divgcd
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if VERSION <= v"0.7.0-DEV.262"
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include("mpz.jl")
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using .MPZ
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else
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using Base.GMP.MPZ
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end
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const primetable = [2,3,5]
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const factortable = [UInt8[], UInt8[1], UInt8[0,1], UInt8[2], UInt8[0,0,1]]
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@ -115,7 +122,7 @@ function Base.convert(::Type{BigInt}, a::PrimeFactorization)
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A = big(a.sign)
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for (n, e) in enumerate(a.powers)
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if !iszero(e)
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Base.GMP.MPZ.mul!(A, bigprime(n, e))
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MPZ.mul!(A, bigprime(n, e))
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end
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end
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return A
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@ -236,10 +243,10 @@ function sumlist!(list::Vector{<:PrimeFactorization}, ind = 1:length(list))
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# do sum
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s = big(0)
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for k in ind
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Base.GMP.MPZ.add!(s, convert(BigInt, list[k]))
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MPZ.add!(s, convert(BigInt, list[k]))
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end
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end
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return Base.GMP.MPZ.mul!(s, convert(BigInt, g))
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return MPZ.mul!(s, convert(BigInt, g))
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end
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# Mutating vector methods that also grow and shrink as required
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228
test/runtests.jl
228
test/runtests.jl
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if VERSION < v"0.7.0-DEV.2005"
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const Test = Base.Test
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end
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using Test
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using WignerSymbols
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using Base.Test
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smalljlist = 0:1//2:10
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largejlist = 0:1//2:1000
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# test triangle coefficient
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for j1 in smalljlist, j2 in smalljlist
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for j3 = abs(j1-j2):(j1+j2)
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@test Δ(j1,j2,j3) ≈ sqrt(factorial(float(j1+j2-j3))*factorial(float(j1-j2+j3))*
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factorial(float(j2+j3-j1))/factorial(float(j1+j2+j3+1)))
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@testset "triangle coefficient" begin
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for j1 in smalljlist, j2 in smalljlist
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for j3 = abs(j1-j2):(j1+j2)
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@test Δ(j1,j2,j3) ≈ sqrt(factorial(float(j1+j2-j3))*factorial(float(j1-j2+j3))*
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factorial(float(j2+j3-j1))/factorial(float(j1+j2+j3+1)))
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end
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end
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end
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# test 3j:
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#--------
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# test cg orthogonality
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for j1 in smalljlist, j2 in smalljlist
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d1 = 2*j1+1
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d2 = 2*j2+1
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M = zeros(d1*d2, d1*d2)
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ind1 = 1
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for m1 in -j1:j1, m2 in -j2:j2
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ind2 = 1
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for j3 in abs(j1-j2):(j1+j2)
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for m3 in -j3:j3
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M[ind1,ind2] = clebschgordan(j1,m1,j2,m2,j3,m3)
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ind2 += 1
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end
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end
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ind1 += 1
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end
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@test M'*M ≈ one(M)
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end
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# test recurrence relations: Phys Rev E 57, 7274 (1998)
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for k = 1:10
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j2 = convert(BigFloat,rand(0:1//2:1000))
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j3 = convert(BigFloat,rand(0:1//2:1000))
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m2 = convert(BigFloat,rand(-j2:j2))
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m3 = convert(BigFloat,rand(-j3:j3))
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for j in max(abs(j2-j3),abs(m2+m3))+1:(j2+j3)-1
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X = j*sqrt(((j+1)^2-(j2-j3)^2)*((j2+j3+1)^2-(j+1)^2)*((j+1)^2-(m2+m3)^2))
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Y = (2*j+1)*((m2+m3)*(j2*(j2+1)-j3*(j3+1)) - (m2-m3)*j*(j+1))
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Z = (j+1)*sqrt((j^2-(j2-j3)^2)*((j2+j3+1)^2-j^2)*(j^2-(m2+m3)^2))
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tol = 10*max(abs(X),abs(Y),abs(Z))*eps(BigFloat)
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@test (X*wigner3j(BigFloat,j+1,j2,j3,-m2-m3,m2,m3) + Z*wigner3j(BigFloat,j-1,j2,j3,-m2-m3,m2,m3))≈(-Y*wigner3j(BigFloat,j,j2,j3,-m2-m3,m2,m3)) atol=tol
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end
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end
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# test 6j
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#----------
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# test orthogonality
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for j1 in smalljlist, j2 in smalljlist, j4 in smalljlist
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for j5 in max(abs(j1-j2-j4),abs(j1-j2+j4),abs(j1+j2-j4)):(j1+j2+j4)
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j6range = max(abs(j2-j4),abs(j1-j5)):min((j2+j4),(j1+j5))
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j3range = max(abs(j1-j2),abs(j4-j5)):min((j1+j2),(j4+j5))
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@test length(j6range) == length(j3range)
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M = zeros(length(j3range),length(j6range))
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for (k2,j6) in enumerate(j6range)
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for (k1,j3) in enumerate(j3range)
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M[k1,k2] = sqrt(2*j3+1)*sqrt(2*j6+1)*wigner6j(j1,j2,j3,j4,j5,j6)
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@testset "clebschgordan: test orthogonality" begin
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for j1 in smalljlist, j2 in smalljlist
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d1 = 2*j1+1
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d2 = 2*j2+1
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M = zeros(d1*d2, d1*d2)
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ind1 = 1
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for m1 in -j1:j1, m2 in -j2:j2
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ind2 = 1
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for j3 in abs(j1-j2):(j1+j2)
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for m3 in -j3:j3
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M[ind1,ind2] = clebschgordan(j1,m1,j2,m2,j3,m3)
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ind2 += 1
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end
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end
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ind1 += 1
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end
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@test M'*M ≈ one(M)
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end
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end
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# test special case
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for j1 in smalljlist, j2 in smalljlist
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j6 = 0
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j4 = j2
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j5 = j1
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for j3 in abs(j1-j2):(j1+j2)
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@test wigner6j(j1,j2,j3,j4,j5,j6) ≈ (-1)^(j1+j2+j3)/sqrt((2*j1+1)*(2*j2+1))
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end
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end
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# test recurrence relations: Phys Rev E 57, 7274 (1998)
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for k = 1:10
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j2 = convert(BigFloat,rand(largejlist))
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j3 = convert(BigFloat,rand(largejlist))
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l1 = convert(BigFloat,rand(largejlist))
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l2 = convert(BigFloat,rand(abs(l1-j3):(l1+j3)))
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l3 = convert(BigFloat,rand(abs(l1-j2):min(l1+j2)))
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@testset "wigner3j: test recurrence relations" begin
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for k = 1:10
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j2 = convert(BigFloat,rand(0:1//2:1000))
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j3 = convert(BigFloat,rand(0:1//2:1000))
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m2 = convert(BigFloat,rand(-j2:j2))
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m3 = convert(BigFloat,rand(-j3:j3))
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for j in intersect(abs(j2-j3):(j2+j3), abs(l2-l3):(l2+l3))
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X = j*sqrt(((j+1)^2-(j2-j3)^2)*((j2+j3+1)^2-(j+1)^2)*((j+1)^2-(l2-l3)^2)*((l2+l3+1)^2 - (j+1)^2))
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Y = (2*j+1)*( j*(j+1)*( -j*(j+1) + j2*(j2+1) + j3*(j3+1) - 2*l1*(l1+1)) +
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l2*(l2+1)*( j*(j+1) + j2*(j2+1) - j3*(j3+1) ) +
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l3*(l3+1)*( j*(j+1) - j2*(j2+1) + j3*(j3+1) ) )
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Z = (j+1)*sqrt((j^2-(j2-j3)^2)*((j2+j3+1)^2-j^2)*(j^2-(l2-l3)^2)*((l2+l3+1)^2 - j^2))
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tol = 10*max(abs(X),abs(Y),abs(Z))*eps(BigFloat)
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@test (X*wigner6j(BigFloat,j+1,j2,j3,l1,l2,l3) + Z*wigner6j(BigFloat,j-1,j2,j3,l1,l2,l3))≈(-Y*wigner6j(BigFloat,j,j2,j3,l1,l2,l3)) atol=tol
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end
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end
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# test recoupling, i.e. relation between 3j and 6j symbols (but use Clebsch-Gordan and RacahW)
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#---------------------------------------------------------------------------------------------
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smallerjlist = 0:1//2:5
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for j1 in smallerjlist, j2 in smallerjlist, j3 in smallerjlist
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m1range = -j1:j1
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m2range = -j2:j2
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m3range = -j3:j3
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V1 = Array{Float64}(length(m1range),length(m2range),length(m3range))
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V2 = Array{Float64}(length(m1range),length(m2range),length(m3range))
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for J in max(abs(j1-j2-j3),abs(j1-j2+j3),abs(j1+j2-j3)):(j1+j2+j3)
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J12range = max(abs(j1-j2),abs(J-j3)):min((j1+j2),(J+j3))
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J23range = max(abs(j2-j3),abs(j1-J)):min((j2+j3),(j1+J))
|
||||
for J12 in J12range, J23 in J23range
|
||||
M = J # only test for J, should be independent
|
||||
fill!(V1,0)
|
||||
fill!(V2,0)
|
||||
for (k1,m1) in enumerate(m1range)
|
||||
for (k2,m2) in enumerate(m2range)
|
||||
abs(m1+m2)<=J12 || continue
|
||||
for (k3,m3) in enumerate(m3range)
|
||||
abs(m2+m3)<=J23 || continue
|
||||
m1+m2+m3==M || continue
|
||||
V1[k1,k2,k3] = clebschgordan(j1,m1,j2,m2,J12)*clebschgordan(J12,m1+m2,j3,m3,J)
|
||||
V2[k1,k2,k3] = clebschgordan(j2,m2,j3,m3,J23)*clebschgordan(j1,m1,J23,m2+m3,J)
|
||||
end
|
||||
end
|
||||
end
|
||||
@test racahW(j1,j2,J,j3,J12,J23) ≈ vecdot(V2,V1)/sqrt((2*J12+1)*(2*J23+1)) atol=10*eps(Float64)
|
||||
for j in max(abs(j2-j3),abs(m2+m3))+1:(j2+j3)-1
|
||||
X = j*sqrt(((j+1)^2-(j2-j3)^2)*((j2+j3+1)^2-(j+1)^2)*((j+1)^2-(m2+m3)^2))
|
||||
Y = (2*j+1)*((m2+m3)*(j2*(j2+1)-j3*(j3+1)) - (m2-m3)*j*(j+1))
|
||||
Z = (j+1)*sqrt((j^2-(j2-j3)^2)*((j2+j3+1)^2-j^2)*(j^2-(m2+m3)^2))
|
||||
tol = 10*max(abs(X),abs(Y),abs(Z))*eps(BigFloat)
|
||||
@test (X*wigner3j(BigFloat,j+1,j2,j3,-m2-m3,m2,m3) + Z*wigner3j(BigFloat,j-1,j2,j3,-m2-m3,m2,m3))≈(-Y*wigner3j(BigFloat,j,j2,j3,-m2-m3,m2,m3)) atol=tol
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
# test 6j
|
||||
#----------
|
||||
@testset "wigner6j: test orthogonality" begin
|
||||
for j1 in smalljlist, j2 in smalljlist, j4 in smalljlist
|
||||
for j5 in max(abs(j1-j2-j4),abs(j1-j2+j4),abs(j1+j2-j4)):(j1+j2+j4)
|
||||
j6range = max(abs(j2-j4),abs(j1-j5)):min((j2+j4),(j1+j5))
|
||||
j3range = max(abs(j1-j2),abs(j4-j5)):min((j1+j2),(j4+j5))
|
||||
@test length(j6range) == length(j3range)
|
||||
M = zeros(length(j3range),length(j6range))
|
||||
for (k2,j6) in enumerate(j6range)
|
||||
for (k1,j3) in enumerate(j3range)
|
||||
M[k1,k2] = sqrt(2*j3+1)*sqrt(2*j6+1)*wigner6j(j1,j2,j3,j4,j5,j6)
|
||||
end
|
||||
end
|
||||
@test M'*M ≈ one(M)
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@testset "wigner6j: test special cases" begin
|
||||
for j1 in smalljlist, j2 in smalljlist
|
||||
j6 = 0
|
||||
j4 = j2
|
||||
j5 = j1
|
||||
for j3 in abs(j1-j2):(j1+j2)
|
||||
@test wigner6j(j1,j2,j3,j4,j5,j6) ≈ (-1)^(j1+j2+j3)/sqrt((2*j1+1)*(2*j2+1))
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@testset "wigner6j: test recurrence relation" begin
|
||||
for k = 1:10
|
||||
j2 = convert(BigFloat,rand(largejlist))
|
||||
j3 = convert(BigFloat,rand(largejlist))
|
||||
l1 = convert(BigFloat,rand(largejlist))
|
||||
l2 = convert(BigFloat,rand(abs(l1-j3):(l1+j3)))
|
||||
l3 = convert(BigFloat,rand(abs(l1-j2):min(l1+j2)))
|
||||
|
||||
for j in intersect(abs(j2-j3):(j2+j3), abs(l2-l3):(l2+l3))
|
||||
X = j*sqrt(((j+1)^2-(j2-j3)^2)*((j2+j3+1)^2-(j+1)^2)*((j+1)^2-(l2-l3)^2)*((l2+l3+1)^2 - (j+1)^2))
|
||||
Y = (2*j+1)*( j*(j+1)*( -j*(j+1) + j2*(j2+1) + j3*(j3+1) - 2*l1*(l1+1)) +
|
||||
l2*(l2+1)*( j*(j+1) + j2*(j2+1) - j3*(j3+1) ) +
|
||||
l3*(l3+1)*( j*(j+1) - j2*(j2+1) + j3*(j3+1) ) )
|
||||
Z = (j+1)*sqrt((j^2-(j2-j3)^2)*((j2+j3+1)^2-j^2)*(j^2-(l2-l3)^2)*((l2+l3+1)^2 - j^2))
|
||||
tol = 10*max(abs(X),abs(Y),abs(Z))*eps(BigFloat)
|
||||
@test (X*wigner6j(BigFloat,j+1,j2,j3,l1,l2,l3) + Z*wigner6j(BigFloat,j-1,j2,j3,l1,l2,l3))≈(-Y*wigner6j(BigFloat,j,j2,j3,l1,l2,l3)) atol=tol
|
||||
end
|
||||
end
|
||||
end
|
||||
|
||||
@testset "test recoupling relation between 3j/clebschgordan and 6j/racahW symbols" begin
|
||||
smallerjlist = 0:1//2:5
|
||||
for j1 in smallerjlist, j2 in smallerjlist, j3 in smallerjlist
|
||||
m1range = -j1:j1
|
||||
m2range = -j2:j2
|
||||
m3range = -j3:j3
|
||||
V1 = Array{Float64}(length(m1range),length(m2range),length(m3range))
|
||||
V2 = Array{Float64}(length(m1range),length(m2range),length(m3range))
|
||||
for J in max(abs(j1-j2-j3),abs(j1-j2+j3),abs(j1+j2-j3)):(j1+j2+j3)
|
||||
J12range = max(abs(j1-j2),abs(J-j3)):min((j1+j2),(J+j3))
|
||||
J23range = max(abs(j2-j3),abs(j1-J)):min((j2+j3),(j1+J))
|
||||
for J12 in J12range, J23 in J23range
|
||||
M = J # only test for J, should be independent
|
||||
fill!(V1,0)
|
||||
fill!(V2,0)
|
||||
for (k1,m1) in enumerate(m1range)
|
||||
for (k2,m2) in enumerate(m2range)
|
||||
abs(m1+m2)<=J12 || continue
|
||||
for (k3,m3) in enumerate(m3range)
|
||||
abs(m2+m3)<=J23 || continue
|
||||
m1+m2+m3==M || continue
|
||||
V1[k1,k2,k3] = clebschgordan(j1,m1,j2,m2,J12)*clebschgordan(J12,m1+m2,j3,m3,J)
|
||||
V2[k1,k2,k3] = clebschgordan(j2,m2,j3,m3,J23)*clebschgordan(j1,m1,J23,m2+m3,J)
|
||||
end
|
||||
end
|
||||
end
|
||||
@test racahW(j1,j2,J,j3,J12,J23) ≈ vecdot(V2,V1)/sqrt((2*J12+1)*(2*J23+1)) atol=10*eps(Float64)
|
||||
end
|
||||
end
|
||||
end
|
||||
end
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue