broadcasting .+ versus "algebraic" +

[StefanKarpinski]

https://groups.google.com/d/topic/julia-dev/CXUBlOPov3k/discussion
https://groups.google.com/d/topic/julia-dev/E47eV6B8w78/discussion

I'm opening this issue even though I now actually think we don't need separate .+ and + operators. The reason why is here:

Basically, matrix multiplication distributes correctly over broadcasting addition if it is even meaningful for it to do so. There are, however, cases where "distributing" doesn't make sense:

• matrix left-multiplication can't sensibly distribute over addition of a matrix and a row vector
• matrix right-multiplication can't sensibly distribute over addition of a matrix and a column vector.

From this I tend to conclude that it's reasonable to just have + broadcast singleton dimensions instead of introducing a new .+ operator for broadcasting addition.

cc: @toivoh

[JeffBezanson]

Ok.

[ViralBShah]

Also see #1019

[nolta]

Broadcasting + is a little too magical for my tastes. Personally, i'd rather A+u throws an error.

[StefanKarpinski]

Would it be acceptable to have .+ be broadcasting and have + require compatible shapes?

[nolta]

Sure. I thought we had come up with a nice solution in the first email thread (e.g., https://groups.google.com/d/msg/julia-dev/CXUBlOPov3k/cyUZjzQIJW0J).

[toivoh]

@StefanKarpinski: How about the case of adding a matrix and a scalar:

M*v + s*v    = (M + s*eye)*v   = (M + s)*v
M.*v .+ s.*v = (M + s*ones).*v = (M .+ s).*v

where M, v, and s is a matrix, vector, and scalar (Number or Array{T,0}), respectively.
There, matrix multiplication does not distribute correctly over broadcasting addition.

[toivoh]

Would it be acceptable to have .+ be broadcasting and have + require compatible shapes?

I'm for this. I think that this was the original conclusion, with + being pretty conservative about shapes, probably not even adding trailing singleton dimensions.

Also, there was the prospect to have

matrix + scalar = matrix + one(matrix)*scalar  # for square matrices

i.e. to multiply the scalar with an appropriate identity matrix before adding. Starting out with a conservative +, that could of course be added afterwards.

[JeffBezanson]

Broadcasting .+ now implemented.