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A Julia package for computing Wigner symbols and related quantities. Forked to add 9j symbols - see branch.
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Morten Piibeleht 8ebb2c791b Partially revamp the HalfInteger type (#4) 
* 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)
2019-01-10 21:50:46 +01:00
benchmark README and benchmarks 2017-08-11 15:32:10 +02:00
src Partially revamp the HalfInteger type (#4) 2019-01-10 21:50:46 +01:00
test Partially revamp the HalfInteger type (#4) 2019-01-10 21:50:46 +01:00
.gitignore WignerSymbols.jl generated files. 2017-08-08 11:23:18 +02:00
.travis.yml update require and travis.yml 2018-10-24 09:51:00 +02:00
LICENSE.md README and benchmarks 2017-08-11 15:32:10 +02:00
README.md Partially revamp the HalfInteger type (#4) 2019-01-10 21:50:46 +01:00
REQUIRE update require and travis.yml 2018-10-24 09:51:00 +02:00

README.md

WignerSymbols

Build Status License Coverage Status codecov.io

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.

Installation

Install with the new package manager via ]add WignerSymbols or

using Pkg
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.

  • 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
  • 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

The package also defines the HalfInteger type that can be used to represent half-integer values. Furthermore, the range operator a:b can be used to create ranges of HalfInteger values (a HalfIntegerRange).

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 (arXiv: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 BigInts) 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

for caching the computed 3j and 6j symbols.

Todo