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# numpy.cumulative_sum

numpy.cumulative_sum(x, /, *, axis=None, dtype=None, out=None, include_initial=False)

  Return the cumulative sum of the elements along a given axis.

  This function is an Array API compatible alternative to `numpy.cumsum`.

  Parameters:
    x array_like

      Input array.

    axis int, optional

      Axis along which the cumulative sum is computed. The default (None) is only allowed for one-dimensional arrays. For arrays with more than one dimension `axis` is required.

    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 `x`, unless `x` 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 Output type determination for more details.

    include_initial bool, optional

      Boolean indicating whether to include the initial value (zeros) as the first value in the output. With `include_initial=True` the shape of the output is different than the shape of the input. Default: `False`.

  Returns:
    cumulative_sum_along_axis ndarray

      A new array holding the result is returned unless `out` is specified, in which case a reference to `out` is returned. The result has the same shape as `x` if `include_initial=False`.

  See also

  `sum`

    Sum array elements.

  `trapezoid`

    Integration of array values using 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.

  `cumulative_sum(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.cumulative_sum(a)
      array([ 1,  3,  6, 10, 15, 21])
      >>> np.cumulative_sum(a, dtype=float)  # specifies type of output value(s)
      array([  1.,   3.,   6.,  10.,  15.,  21.])

      >>> b = np.array([[1, 2, 3], [4, 5, 6]])
      >>> np.cumulative_sum(b,axis=0)  # sum over rows for each of the 3 columns
      array([[1, 2, 3],
             [5, 7, 9]])
      >>> np.cumulative_sum(b,axis=1)  # sum over columns for each of the 2 rows
      array([[ 1,  3,  6],
             [ 4,  9, 15]])

  `cumulative_sum(c)[-1]` may not be equal to `sum(c)`

      >>> c = np.array([1, 2e-9, 3e-9] * 1000000)
      >>> np.cumulative_sum(c)[-1]
      1000000.0050045159
      >>> c.sum()
      1000000.0050000029