README and benchmarks

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.