On Tue, Aug 15, 2023 at 2:44 PM <john.dawson@camlingroup.com> wrote:
> From my point of view, such function is a bit of a corner-case to be added to numpy. And it doesn’t justify it’s naming anymore. It is not one operation anymore. It is a cumsum and prepending 0. And it is very difficult to argue why prepending 0 to cumsum is a part of cumsum.

That is backwards. Consider the array [x0, x1, x2].

The sum of the first 0 elements is 0.
The sum of the first 1 elements is x0.
The sum of the first 2 elements is x0+x1.
The sum of the first 3 elements is x0+x1+x2.

Hence, the array of partial sums is [0, x0, x0+x1, x0+x1+x2].

Thus, the operation [x0, x1, x2] -> [0, x0, x0+x1, x0+x1+x2] is a natural and primitive one.

You are describing ndarray.sum() behavior here inside an array as intermediate results; sum is an aggregator that produces single item from a list of items. Then you can argue about missing items behavior and the values you have provided are exactly the values the accumulator would get. However, cumsum, cumprod, diff etc. are "array functions". In other words they provide fast vectorized access to otherwise laborious for loops. You have to consider the equivalent for loops working on the array *data*, not the ideal math framework over the number field. You don't start with the array element that is before the first element for an array function hence no elements -> 0 is only applicable to sum but not to the array function. Or at least that would be my argument.

If you have no element meaning 0 elements the cumulative sum is not 0, it is the empty array. Because there is no array to cumulatively "sum" (remember we are working on the array to generate another array, not aggregating). You can argue what empty set translates to under summation etc. but I don't think it applies here. But that's my opinion. I'm not sure why folks wanted to have this at all. It is the same as asking whether this code

for k in range(0):
    ...some code ...

should at least spin once (fortran-ish behavior). I don't know why it should. But then again, it becomes a bikeshedding with some conflicting idealistic mathy axioms thrown at each other. 

NumPy cumsum returns empty array for empty array (I think all software does this including matlab). ndarray.sum() however returns scalar 0 (and I think most software does this too), because that's pretty much a no-op over the initialization value and aggregated, in the example above

for k in range(0):
    x += 1
return x # returns 0

I think all these point to the missing convenient functionality that extends arrays. In matlab "[0 arr 10]" nicely extends the array to a new one but in NumPy you need to punch quite some code and some courage to remember whether it is hstack or vstack or concat or block as the correct naming which decreases the "code morale". So if people want to quickly extend arrays they either have to change the code for their needs or create larger arrays which is pretty much #6044. So I think this is a feature request of "prepend", "append" in a convenient fashion not to ufuncs but to ndarray. Because concatenation is just pain in NumPy and ubiquitous operation all around. Hence probably we should get a decision on that instead of discussing each case separately.