README and benchmarks

LICENSE.md

@@ -1,17 +1,17 @@
 The WignerSymbols.jl package is licensed under the MIT "Expat" License:
 
-> Copyright (c) 2017: Jutho.
+> Copyright (c) 2017: Jutho Haegeman.
-> 
+>
 > Permission is hereby granted, free of charge, to any person obtaining a copy
 > of this software and associated documentation files (the "Software"), to deal
 > in the Software without restriction, including without limitation the rights
 > to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 > copies of the Software, and to permit persons to whom the Software is
 > furnished to do so, subject to the following conditions:
-> 
+>
 > The above copyright notice and this permission notice shall be included in all
 > copies or substantial portions of the Software.
-> 
+>
 > THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 > IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 > FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
@@ -19,4 +19,4 @@ The WignerSymbols.jl package is licensed under the MIT "Expat" License:
 > LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 > OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 > SOFTWARE.
-> 
+>

README.md

@@ -1,7 +1,53 @@
 # WignerSymbols
 
-[![Build Status](https://travis-ci.org/jutho/WignerSymbols.jl.svg?branch=master)](https://travis-ci.org/jutho/WignerSymbols.jl)
+[![Build Status](https://travis-ci.org/Jutho/WignerSymbols.jl.svg?branch=master)](https://travis-ci.org/jutho/WignerSymbols.jl)
+[![License](http://img.shields.io/badge/license-MIT-brightgreen.svg?style=flat)](LICENSE.md)
+[![Coverage Status](https://coveralls.io/repos/Jutho/WignerSymbols.jl/badge.svg?branch=master&service=github)](https://coveralls.io/github/Jutho/WignerSymbols.jl?branch=master)
+[![codecov.io](http://codecov.io/github/jutho/WignerSymbols.jl/coverage.svg?branch=master)](http://codecov.io/github/Jutho/WignerSymbols.jl?branch=master)
 
-[![Coverage Status](https://coveralls.io/repos/jutho/WignerSymbols.jl/badge.svg?branch=master&service=github)](https://coveralls.io/github/jutho/WignerSymbols.jl?branch=master)
+Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.
 
-[![codecov.io](http://codecov.io/github/jutho/WignerSymbols.jl/coverage.svg?branch=master)](http://codecov.io/github/jutho/WignerSymbols.jl?branch=master)
+## Requirements
+This requires a recent master edition of Julia (i.e. v0.7.0-DEV), because it depends on some changes in `Base.GMP`. In particular, it uses the mutating functions for reducing allocation overhead while working with `BigInts` (namely [JuliaLang/julia#21654](https://github.com/JuliaLang/julia/pull/21654)). It also depends on `Primes.jl` for generating prime numbers.
+
+## Installation
+Until it is registered, install via `Pkg.clone("https://github.com/Jutho/WignerSymbols.jl.git")`.
+
+## Available functions
+While the following function signatures are probably self-explanatory, you can query help for them in the Julia REPL to get further details.
+*   `wigner3j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₁) -> ::T`
+*   `wigner6j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, j₄, j₅, j₆) -> ::T`
+*   `clebschgordan(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = m₁+m₂) -> ::T`
+*   `racahV(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) -> ::T`
+*   `racahW(T::Type{<:AbstractFloat} = Float64, j₁, j₂, J, j₃, J₁₂, J₂₃) -> ::T`
+*   `δ(j₁, j₂, j₃) -> ::Bool`
+*   `Δ(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃) -> ::T`
+
+## Implementation
+Largely based on reading the paper (but not the code):
+
+[1] [H. T. Johansson and C. Forssén, SIAM Journal on Scientific Compututing 38 (2016) 376-384](https://doi.org/10.1137/15M1021908) ([arXiv:1504.08329](https://arxiv.org/abs/1504.08329))
+
+with some additional modifications to further improve efficiency for large `j` (angular momenta quantum numbers).
+
+In particular, 3j and 6j symbols are computed exactly, in the format `√(r) * s` where `r` and `s` are exactly computed as `Rational{BigInt}`,
+using an intermediate representation based on prime number factorization. As a consequence thereof, all of the above functions can be called
+requesting `BigFloat` precision for the result. There is currently no convenient syntax for obtaining `r` and `s` directly (see TODO).
+
+Most intermediate calculations (prime factorizations of numbers and their factorials, conversion between prime powers and `BigInt`s) are cached to improve the efficiency, but this can result in large use of memory when querying Wigner symbols for large values of `j`.
+
+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.
+
+# Benchmark:
+
+## 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.](ftp://gravity.physics.uwa.edu.au/pub/Clebsch-Gordan/Papers/Wei9j.pdf)
+
+*   Convenient syntax to get the exact results in the `√(r) * s` format, but in such a way that it can be used by
+    the Julia type system and can be converted afterwards.

benchmark/benchmarks.jl

@@ -0,0 +1,106 @@
+# some random benchmarks; for j=50000 you need more than 8GB of memory
+using WignerSymbols
+
+@time wigner3j(BigFloat, 15, 30, 40, 2, 2, -4)
+@time wigner3j(BigFloat, 200, 200, 200, -10, 60, -50)
+@time wigner3j(BigFloat, 50000, 50000, 50000, 1000, -6000, 5000)
+
+j = 8
+@time wigner6j(BigFloat, j, j, j, j, j, j)
+j = 200
+@time wigner6j(BigFloat, j, j, j, j, j, j)
+j = 600
+@time wigner6j(BigFloat, j, j, j, j, j, j)
+j = 10000
+@time wigner6j(BigFloat, j, j, j, j, j, j)
+j = 50000
+@time wigner6j(BigFloat, j, j, j, j, j, j)
+
+function compute3jmax(jmax)
+    for j₁ = 0:1//2:jmax
+        for j₂ = 0:1//2:j₁
+            for j₃ = abs(j₁-j₂):j₂
+                if isinteger(j₁)
+                    m₁ = 0
+                    M₂ = min(j₂, j₃-m₁)
+                    for m₂ = M₂:(-1):0
+                        wigner3j(j₁,j₂,j₃,0, m₂)
+                    end
+                    for m₁ = 1:j₁
+                        M₂ = min(j₂, j₃-m₁)
+                        for m₂ = -M₂:M₂
+                            wigner3j(j₁, j₂, j₃, m₁, m₂)
+                        end
+                    end
+                else
+                    for m₁ = 1//2:j₁
+                        M₂ = min(j₂, j₃-m₁)
+                        for m₂ = -M₂:M₂
+                            wigner3j(j₁, j₂, j₃, m₁, m₂)
+                        end
+                    end
+                end
+            end
+        end
+    end
+    return nothing
+end
+
+@time compute3jmax(10)
+
+function computeinteger3jmax(jmax)
+    for j₁ = 0:jmax
+        for j₂ = 0:j₁
+            for j₃ = abs(j₁-j₂):j₂
+                m₁ = 0
+                M₂ = min(j₂, j₃-m₁)
+                for m₂ = M₂:(-1):0
+                    wigner3j(j₁,j₂,j₃,0, m₂)
+                end
+                for m₁ = 1:j₁
+                    M₂ = min(j₂, j₃-m₁)
+                    for m₂ = -M₂:M₂
+                        wigner3j(j₁, j₂, j₃, m₁, m₂)
+                    end
+                end
+            end
+        end
+    end
+    return nothing
+end
+
+function compute6jmax(jmax)
+    for j₁ = 0:1//2:jmax
+        for j₂ = 0:1//2:j₁
+            for j₃ = abs(j₁-j₂):j₂
+                for j₄ = 0:1//2:jmax
+                    for j₅ = abs(j₃-j₄):min(jmax,j₃+j₄)
+                        for j₆ = min(abs(j₂-j₄),abs(j₁-j₅)):min(j₂+j₄,j₁+j₅,jmax)
+                            wigner6j(j₁, j₂, j₃, j₄, j₅, j₆)
+                        end
+                    end
+                end
+            end
+        end
+    end
+    return nothing
+end
+@time compute6jmax(10)
+
+
+function computeinteger6jmax(jmax)
+    for j₁ = 0:jmax
+        for j₂ = 0:j₁
+            for j₃ = abs(j₁-j₂):j₂
+                for j₄ = 0:jmax
+                    for j₅ = abs(j₃-j₄):min(jmax,j₃+j₄)
+                        for j₆ = min(abs(j₂-j₄),abs(j₁-j₅)):min(j₂+j₄,j₁+j₅,jmax)
+                            wigner6j(j₁, j₂, j₃, j₄, j₅, j₆)
+                        end
+                    end
+                end
+            end
+        end
+    end
+    return nothing
+end