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Explicit derivatives for complex analytic functions #727

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Explicit derivatives for complex analytic functions #727

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114cfe9
Added explicit derivatives for analytic functions
clguillot Dec 4, 2024
b55fc60
Manual handling of inv
clguillot Dec 5, 2024
e4aa3cf
More subtypes handled ?
clguillot Dec 5, 2024
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61 changes: 60 additions & 1 deletion src/dual.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -238,13 +238,30 @@
val = $Mf(x)
deriv = $(DiffRules.diffrule(M, f, :x))
end)
return quote
real_diff_exp = quote
@inline function $M.$f(d::$FD.Dual{T}) where T
x = $FD.value(d)
$work
return $FD.dual_definition_retval(Val{T}(), val, deriv, $FD.partials(d))
end
end
if (M, f) in ((:Base, :abs), (:Base, :abs2), (:Base, :inv))
return real_diff_exp
else
complex_diff_expr = quote
@inline function $M.$f(d::Complex{<:$FD.Dual{T}}) where{T}
x = complex($FD.value(real(d)), $FD.value(imag(d)))
$work
re_deriv, im_deriv = reim(deriv)
re_partials = $FD.partials(real(d))
im_partials = $FD.partials(imag(d))
re_retval = $FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
im_retval = $FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
return complex(re_retval, im_retval)
end
end
return Expr(:block, real_diff_exp, complex_diff_expr)
end
end

function binary_dual_definition(M, f)
Expand Down Expand Up @@ -709,6 +726,22 @@
Dual{Tz}(muladd(x, y, value(z)), partials(z)) # z_body
)

# inv(Complex) #
#--------#

function Base.inv(d::Complex{<:Dual{T}}) where{T}
FD = ForwardDiff
x = complex(FD.value(real(d)), FD.value(imag(d)))
val = inv(x)
deriv = - val * val
re_deriv, im_deriv = reim(deriv)
re_partials = FD.partials(real(d))
im_partials = FD.partials(imag(d))
re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
return complex(re_retval, im_retval)

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end

# sin/cos #
#--------#

Expand All @@ -727,6 +760,32 @@
return (Dual{T}(sd, cd * partials(d)), Dual{T}(cd, -sd * partials(d)))
end

function Base.sin(d::Complex{<:Dual{T}}) where{T}
FD = ForwardDiff
x = complex(FD.value(real(d)), FD.value(imag(d)))
val = sin(x)
deriv = cos(x)
re_deriv, im_deriv = reim(deriv)
re_partials = FD.partials(real(d))
im_partials = FD.partials(imag(d))
re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
return complex(re_retval, im_retval)

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end

function Base.cos(d::Complex{<:Dual{T}}) where{T}
FD = ForwardDiff
x = complex(FD.value(real(d)), FD.value(imag(d)))
val = cos(x)
deriv = -sin(x)
re_deriv, im_deriv = reim(deriv)
re_partials = FD.partials(real(d))
im_partials = FD.partials(imag(d))
re_retval = FD.dual_definition_retval(Val{T}(), real(val), re_deriv, re_partials, -im_deriv, im_partials)
im_retval = FD.dual_definition_retval(Val{T}(), imag(val), im_deriv, re_partials, re_deriv, im_partials)
return complex(re_retval, im_retval)

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end

# sincospi #
#----------#

Expand Down
7 changes: 7 additions & 0 deletions test/DerivativeTest.jl
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Expand Up @@ -113,4 +113,11 @@ end
@test ForwardDiff.derivative(x -> (1+im)*x, 0) == (1+im)
end

@testset "analytic functions" begin
dexp(x) = ForwardDiff.derivative(y -> exp(complex(0, y)), x)
@test ForwardDiff.derivative(dexp, 0.0) ≈ -1
@test ForwardDiff.derivative(x -> exp(1im*x), 0.7) ≈ im * cis(0.7)
@test ForwardDiff.derivative(x -> sqrt(im + (1+im) * x), 1.23) ≈ (1+im) / (2 * sqrt(im + (1+im)*1.23))
end

end # module
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