Change SVD to use Composite

[nickrobinson251]

• part of Update Cholesky and SVD to use Composite #151
• Very WIP
• Currently presumes we have Make getproperty on unset Composite properties return Zero().  ChainRulesCore.jl#107

[willtebbutt]

Very excited for replacing zero(large_array) with Zero().

[nickrobinson251]

when returning Zero test fails with comparing to finite differences (which gives an array of approx zeros)

julia> extern(Zero()) ≈ zeros(2, 2)
ERROR: MethodError: no method matching isapprox(::Bool, ::Array{Float64,2})
Closest candidates are:
  isapprox(::Missing, ::Any; kwargs...) at missing.jl:90
  isapprox(::Any, ::Missing; kwargs...) at missing.jl:91
  isapprox(::Number, ::Number; atol, rtol, nans) at floatfuncs.jl:274
  ...
Stacktrace:
 [1] top-level scope at REPL[15]:1

julia> extern(Zero()) == zeros(2, 2)
false

[nickrobinson251]

i guess i can just write the tests to account for this, i.e. compare element-wise, but the output on failure is less good:

# now; where in practice we have `Zero` instead of `zeros(2, 2)`
julia> @test all(isapprox.(zeros(2, 2), ones(2, 2)))
Test Failed at REPL[34]:1
  Expression: all(isapprox.(zeros(2, 2), ones(2, 2)))
ERROR: There was an error during testing

# before; better message but didn't work with `Zero` instead of `zeros(2, 2)`
julia> @test isapprox(zeros(2, 2), ones(2, 2))
Test Failed at REPL[35]:1
  Expression: isapprox(zeros(2, 2), ones(2, 2))
   Evaluated: isapprox([0.0 0.0; 0.0 0.0], [1.0 1.0; 1.0 1.0])
ERROR: There was an error during testing

[nickrobinson251]

The current version of this PR is what it would look like if we extended various linear algebra function to handle Zero

(We do already extend conj)

This maybe leaves some optimizations still up for grabs, but it's a lot cleaner than naively checking A isa Zero many places.

[nickrobinson251]

What's up with Compostie and accumulate? What's up with accumulate at all? (xref JuliaDiff/ChainRulesCore.jl#62)

is accumulate[!](x::C, y::C) where c<:Composite = C(; elementwise_add(backing(x), backing(y))...) or ??

[willtebbutt]

I can't see anything to disagree with here.

[oxinabox]

Other than the stuff commented this LGTM

[nickrobinson251]

https://github.com/JuliaDiff/ChainRulesCore.jl/pull/114/files