Add documentation for array symbolic registration

docs/src/manual/functions.md

@@ -66,9 +66,77 @@ is the partial derivative of the Julia `min(x,y)` function with respect to `x`.
     is required in the derivative expression is defined. Note that you only need to define
     the partial derivatives which are actually computed.
 
+## Registration of Array Functions
+
+Similar to scalar functions, array functions can be registered to define new primitives for
+functions which either take in or return arrays. This is done by using the `@register_array_symbolic`
+macro. It acts similarly to the scalar function registration but requires a calculation of the
+input and output sizes. For example, let's assume we wanted to have a function that computes the
+solution to `Ax = b`, i.e. a linear solve, using an SVD factorization. In Julia, the code for this
+would be `svdsolve(A,b) = svd(A)\b`. We would create this function as follows:
+
+```@example array_registration
+using LinearAlgebra, Symbolics
+
+svdsolve(A, b) = svd(A)\b
+@register_array_symbolic svdsolve(A::AbstractMatrix, b::AbstractVector) begin
+    size = size(b)
+    eltype = promote_type(eltype(A), eltype(b))
+end
+```
+
+Now using the function `svdsolve` with symbolic array variables will be kept lazy:
+
+```@example array_registration
+@variables A[1:3, 1:3] b[1:3]
+svdsolve(A,b)
+```
+
+Note that at this time array derivatives cannot be defined.
+
 ## Registration API
 
 ```@docs
 @register_symbolic
 @register_array_symbolic
 ```
+
+## Direct Registration API (Advanced, Experimental)
+
+!!! warn
+
+    This is a lower level API which is not as stable as the macro APIs.
+
+In some circumstances you may need to use the direct API in order to define
+registration on functions or types without using the macro. This is done
+by directly defining dispatches on symbolic objects.
+
+A good exmample of this is DataInterpolations.jl's interpolations object.
+On an interpolation by a symbolic variable, we generate the symbolic
+function (the `term`) for the interpolation function. This looks like:
+
+```julia
+using DataInterpolations, Symbolics, SymbolicUtils
+(interp::AbstractInterpolation)(t::Num) = SymbolicUtils.term(interp, unwrap(t))
+```
+
+In order for this to work, it is required that we define the `symtype` for the
+symbolic type inference. This is done via:
+
+```julia
+SymbolicUtils.promote_symtype(t::AbstractInterpolation, args...) = Real
+```
+
+Additionally a symbolic name is required:
+
+```julia
+Base.nameof(interp::AbstractInterpolation) = :Interpolation
+```
+
+The derivative is defined similarly to the macro case:
+
+```julia
+function Symbolics.derivative(interp::AbstractInterpolation, args::NTuple{1, Any}, ::Val{1})
+    Symbolics.unwrap(derivative(interp, Symbolics.wrap(args[1])))
+end
+```