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* Call the HalfInteger field twofold Makes it more immediately obvious what the meaning of the value stored in the field is. * Introduce new constructors for HalfInteger The primary inner constructor mirrors the two-argument constructor of the Rational type, where the user provides the numerator and denominator values. There is also a single argument outer constructor that makes HalfInteger behave like a normal numeric type such that HalfInteger(n) == n. * Move HalfInteger tests to a separate file The using statements in halfinteger.jl are there so that it would be possible to run the file separately from the other tests. * Test the single-argument HalfInteger constructor * Organize halfinteger.jl a bit Prioritise the convert methods. * Add multiplication with integer to HalfInteger * Implement parsing and printing for HalfInteger * parse(::HalfInteger, x) method * Overload show to pretty-print HalfInteger * Overload Base.numerator/denominator And add tests for the other supplementary functions and methods as well. * Add HalfIntegerRange type Can be constructed using the range operator :. Currently only supports unit steps in the positive direction. * Address feedback * Rename .twofold -> .numerator * Consistent variable names * Remove unnecessary methods for HalfIntegerRange * Allow constructing HalfIntegerRange with non-integer difference * Add docs and ceil(::HalfInteger)
58 lines
3.6 KiB
Markdown
58 lines
3.6 KiB
Markdown
# WignerSymbols
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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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Compute Wigner's 3j and 6j symbols, and related quantities such as Clebsch-Gordan coefficients and Racah's symbols.
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## Requirements
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Latest version is compatible with Julia v0.7 only, but older versions can be installed on Julia v0.6.
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## Installation
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Install with the new package manager via `]add WignerSymbols` or
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```julia
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using Pkg
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Pkg.add("WignerSymbols")
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```
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## Available functions
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While the following function signatures are probably self-explanatory, you can query help for them in the Julia REPL to get further details.
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* `wigner3j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₂-m₁) -> ::T`
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* `wigner6j(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, j₄, j₅, j₆) -> ::T`
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* `clebschgordan(T::Type{<:AbstractFloat} = Float64, j₁, m₁, j₂, m₂, j₃, m₃ = m₁+m₂) -> ::T`
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* `racahV(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃, m₁, m₂, m₃ = -m₁-m₂) -> ::T`
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* `racahW(T::Type{<:AbstractFloat} = Float64, j₁, j₂, J, j₃, J₁₂, J₂₃) -> ::T`
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* `δ(j₁, j₂, j₃) -> ::Bool`
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* `Δ(T::Type{<:AbstractFloat} = Float64, j₁, j₂, j₃) -> ::T`
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The package also defines the `HalfInteger` type that can be used to represent half-integer values.
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Furthermore, the range operator `a:b` can be used to create ranges of `HalfInteger` values (a `HalfIntegerRange`).
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## Implementation
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Largely based on reading the paper (but not the code):
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[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))
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with some additional modifications to further improve efficiency for large `j` (angular momenta quantum numbers).
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In particular, 3j and 6j symbols are computed exactly, in the format `√(r) * s` where `r` and `s` are exactly computed as `Rational{BigInt}`,
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using an intermediate representation based on prime number factorization. As a consequence thereof, all of the above functions can be called
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requesting `BigFloat` precision for the result. There is currently no convenient syntax for obtaining `r` and `s` directly (see TODO).
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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`.
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Also uses ideas from
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[2] [J. Rasch and A. C. H. Yu, SIAM Journal on Scientific Compututing 25 (2003), 1416–1428](https://doi.org/10.1137/S1064827503422932)
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for caching the computed 3j and 6j symbols.
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## Todo
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* Wigner 9-j symbols, as explained in [1] and based on
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[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)
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* Convenient syntax to get the exact results in the `√(r) * s` format, but in such a way that it can be used by
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the Julia type system and can be converted afterwards.
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