tgorordo/WignerSymbols.jl
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@ -5,10 +5,8 @@
[![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)
Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.
## Requirements
Latest version is compatible with Julia v0.7 only, but older versions can be installed on Julia v0.6.
Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan
coefficients and Racah's symbols.
## Installation
Install with the new package manager via `]add WignerSymbols` or
@ -18,7 +16,8 @@ Pkg.add("WignerSymbols")
```
## 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.
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, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) -> ::T`
@ -27,20 +26,31 @@ While the following function signatures are probably self-explanatory, you can q
* `δ(j₁, j₂, j₃) -> ::Bool`
* `Δ(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃) -> ::T`
The package also defines the `HalfInteger` type that can be used to represent half-integer values. Construct if as `HalfInteger(a::Real)` or `HalfInteger(numerator::Integer, denominator::Integer)`. Furthermore, the range operator `a:b` can be used to create ranges of `HalfInteger` values (a `HalfIntegerRange`).
~~The package also defines the `HalfInteger` type that can be used to represent half-
integer values. Construct if as `HalfInteger(a::Real)` or `HalfInteger(numerator::Integer,
denominator::Integer)`. Furthermore, the range operator `a:b` can be used to create ranges
of `HalfInteger` values (a `HalfIntegerRange`).~~
The package now relies on [HalfIntegers.jl](https://github.com/sostock/HalfIntegers.jl) to
support and use arithmetic with half integer numbers.
## 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).
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).
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`.
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
@ -53,5 +63,5 @@ for caching the computed 3j and 6j symbols.
[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)
* 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.
* 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.