`cumsum` computes the sum of the first k summands for every k from 1. Judging
by my experience, it is more often useful to compute the sum of the first k
summands for every k from 0, as `cumsum`'s behaviour leads to fencepost-like
problems.
https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error
For example, `cumsum` is not the inverse of `diff`. I propose adding a function
to NumPy to compute cumulative sums beginning with 0, that is, an inverse of
`diff`. It might be called `cumsum0`. The following code is probably not the
best way to implement it, but it illustrates the desired behaviour.
```
def cumsum0(a, axis=None, dtype=None, out=None):
"""
Return the cumulative sum of the elements along a given axis,
beginning with 0.
cumsum0 does the same as cumsum except that cumsum computes the sum
of the first k summands for every k from 1 and cumsum, from 0.
Parameters
----------
a : array_like
Input array.
axis : int, optional
Axis along which the cumulative sum is computed. The default
(None) is to compute the cumulative sum over the flattened
array.
dtype : dtype, optional
Type of the returned array and of the accumulator in which the
elements are summed. If `dtype` is not specified, it defaults to
the dtype of `a`, unless `a` has an integer dtype with a
precision less than that of the default platform integer. In
that case, the default platform integer is used.
out : ndarray, optional
Alternative output array in which to place the result. It must
have the same shape and buffer length as the expected output but
the type will be cast if necessary. See
:ref:`ufuncs-output-type` for more details.
Returns
-------
cumsum0_along_axis : ndarray.
A new array holding the result is returned unless `out` is
specified, in which case a reference to `out` is returned. If
`axis` is not None the result has the same shape as `a` except
along `axis`, where the dimension is smaller by 1.
See Also
--------
cumsum : Cumulatively sum array elements, beginning with the first.
sum : Sum array elements.
trapz : Integration of array values using the composite trapezoidal rule.
diff : Calculate the n-th discrete difference along given axis.
Notes
-----
Arithmetic is modular when using integer types, and no error is
raised on overflow.
``cumsum0(a)[-1]`` may not be equal to ``sum(a)`` for floating-point
values since ``sum`` may use a pairwise summation routine, reducing
the roundoff-error. See `sum` for more information.
Examples
--------
>>> a = np.array([[1, 2, 3], [4, 5, 6]])
>>> a
array([[1, 2, 3],
[4, 5, 6]])
>>> np.cumsum0(a)
array([ 0, 1, 3, 6, 10, 15, 21])
>>> np.cumsum0(a, dtype=float) # specifies type of output value(s)
array([ 0., 1., 3., 6., 10., 15., 21.])
>>> np.cumsum0(a, axis=0) # sum over rows for each of the 3 columns
array([[0, 0, 0],
[1, 2, 3],
[5, 7, 9]])
>>> np.cumsum0(a, axis=1) # sum over columns for each of the 2 rows
array([[ 0, 1, 3, 6],
[ 0, 4, 9, 15]])
``cumsum(b)[-1]`` may not be equal to ``sum(b)``
>>> b = np.array([1, 2e-9, 3e-9] * 1000000)
>>> np.cumsum0(b)[-1]
1000000.0050045159
>>> b.sum()
1000000.0050000029
"""
empty = a.take([], axis=axis)
zero = empty.sum(axis, dtype=dtype, keepdims=True)
later_cumsum = a.cumsum(axis, dtype=dtype)
return concatenate([zero, later_cumsum], axis=axis, dtype=dtype, out=out)
```
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