Julia: cumsum, accumulate should not have default dim=1

[StefanKarpinski]

This is inconsistent with how non-cumulative functions work in Julia, e.g.:

julia> A = rand(3,4)
3×4 Array{Float64,2}:
 0.239183  0.420538  0.760069  0.349576
 0.768802  0.924563  0.934626  0.446843
 0.767848  0.761547  0.518525  0.758667

julia> sum(A)
7.650786585136748

julia> cumsum(A)
3×4 Array{Float64,2}:
 0.239183  0.420538  0.760069  0.349576
 1.00798   1.3451    1.6947    0.796419
 1.77583   2.10665   2.21322   1.55509

Correspondingly, cumsum(A) should, rather than asymmetrically defaulting to the first dimension, either be an error, or return an accumulation of the elements of A such that cumsum(A)[end] == sum(A) and such that cumsum(v')' == cumsum(v) for any vector v. One possibility is performing a cumulative sum in column-major order. Another is that each output value is the sum of values to the values to the left or above that slot in the intput. I.e.:

julia> [sum(A[1:i,1:j]) for i=1:size(A,1), j=1:size(A,2)]
3×4 Array{Float64,2}:
 0.239183  0.65972  1.41979  1.76937
 1.00798   2.35308  4.04778  4.8442
 1.77583   3.88248  6.0957   7.65079

This answer has the above properties, can be computed efficiently, and is useful.

[andreasnoack]

I agree that cumsum shouldn't default to dim=1 but I think it is better to throw an error if no dimension is specified and ndims>1. I don't like that storage order becomes significant for anything but performance and floating point errors. Short term, there would also be a deprecation issue.

[StefanKarpinski]

I agree that it's not desirable for storage order to become semantically visible. That's why I proposed the [sum(A[1:i,1:j]) for i=1:size(A,1), j=1:size(A,2)] definition as well. Wanting cumsum(v')' == cumsum(v) may tie into #4774 depending on how that pans out.

[StefanKarpinski]

See also https://github.com/JuliaLang/julia/pull/23542, https://github.com/JuliaLang/julia/issues/20041.

[StefanKarpinski]

The minimal plan of action here is:

1. Make not supplying a dimension an error unless all dimensions but the first are singleton.
2. Make accumulate on a row vector work as accumulate(f, r.').'.

In the future, this could be generalized to the n-dimensional prefix summation behavior I proposed above, which is generally useful and ties the desired behavior for vectors and row vectors together.