dchang10 / jacobielliptic.jl

Elliptic integrals and Jacobi elliptic functions that are GPU friendly

JacobiElliptic

JacobiElliptic is an implementation of Toshio Fukushima's & Billie C. Carlson's for calculating Elliptic Integrals and Jacobi Elliptic Functions. The defaul algorithms are set to the Carlson algorithms.

Features

• Type stable and preserving
• Metal.jl and CUDA.jl compatible
• Automatic Differentiable. Compatible with Zygote.jl, ForwardDiff.jl and Enzyme.jl

Documentation

Repo Status

Incomplete Elliptic Integrals

Function	Definition
F(φ, m)	$F(\phi|m)$: Incomplete elliptic integral of the first kind
E(φ, m)	$E(\phi|m)$: Incomplete elliptic integral of the second kind
Pi(n, φ, m)	$\Pi(n;\,\phi|\, m)$: Incomplete elliptic integral of the third kind
J(n, φ, m)	$J (n;\, \phi |\,m)=\frac{\Pi(n;\,\phi|\, m) - F(\phi,m)}{n}$: Associated incomplete elliptic integral of the third kind

Complete Elliptic Integrals

Function	Definition
K(m)	$K(m)$: Complete elliptic integral of the first kind
E(m)	$E(m)$: Complete elliptic integral of the second kind
Pi(n, m)	$\Pi(n|\, m)$: Complete elliptic integral of the third kind
J(n, m)	$J (n|\,m)=\frac{\Pi(n|\, m) - K(m)}{n}$: Associated incomplete elliptic integral of the third kind

Jacobi Elliptic Functions

Function	Definition
sn(u, m)	$\text{sn}(u | m) = \sin(\text{am}(u | m))$
cn(u, m)	$\text{cn}(u | m) = \cos(\text{am}(u | m))$
asn(u, m)	$\text{asn}(u | m) = \text{sn}^{-1}(u | m)$
acn(u, m)	$\text{acn}(u | m) = \text{cn}^{-1}(u | m)$

jacobielliptic.jl's Issues

NaNs from ForwardDiff

For the specific value of phi=1.57079632 and m=0 in JacobiElliptic.E(phi, m), ForwardDiff outputs NaNs in the partial derivatives.
Here's a minimum working example:

using JacobiElliptic                  
using ForwardDiff                     
                                             
phi = 1.57079632                      
m = 0.0                               
                                      
func(x) = JacobiElliptic.E(x[1], x[2])
                                      
x = [phi, m]                          
ForwardDiff.gradient(func, x)         

EllipticFunctions.ellipticE() reports this gradient to be [1.0, 0.0]
Is this fixable?

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Extend Elliptic Pi for n > 1

The Carlson Elliptic Pi does not currently work for n > 1

Extend Elliptic E for negative m

The Incomplete elliptic integral of the second kind needs to be extended to work for values of m < 0

Jacobi J does not exist

When I try to call Jacobi J I get a symbol not found error. MWE

using JacobiElliptic

J(2, 0.5, 0.5)

ERROR: UndefVarError: `J` not defined
Stacktrace:
 [1] J(::Int64, ::Float64, ::Vararg{Float64})
   @ JacobiElliptic ~/.julia/dev/JacobiElliptic/src/JacobiElliptic.jl:44
 [2] top-level scope
   @ REPL[6]:1

From looking around, I see that CarlsonAlg doesn't define a J function.