DOC: Fix a typo in description and add an example of numpy.tensordot

[John-Venti]

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Updated comment on 7.2.2023:

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1. I modified the description,

   When axes is integer_like, the sequence for evaluation will be: first
   the -Nth axis in a and 0th axis in b, and the -1th axis in a and
   Nth axis in b last.

   to

   When axes is integer_like, the sequence of axes for evaluation
   will be: from the -Nth axis to the -1th axis in a,
   and from the 0th axis to (N-1)th axis in b.
   For example, axes = 2 is the equal to
   axes = [[-2, -1], [0, 1]].
   When N-1 is smaller than 0, or when -N is larger than -1,
   the element of 'a' and 'b' are defined as the 'axes'.

   This is because the source code of np.tensordot() is

   except Exception:
   axes_a = list(range(-axes, 0))
   axes_b = list(range(0, axes))
   else:
   axes_a, axes_b = axes

   Therefore, (N-1)th is more proper than Nth. for example, when N=10, list(range(0,10)) and list(range(-10,0)) are the following, respectively:

    [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] 
    and
    [-10, -9, -8, -7, -6, -5, -4, -3, -2, -1]

Because tensordot() in other libraries ,like Pytorch, doesn't have the usage of integer_like as axes input. This method is only exist in Numpy, therefore enough instructions are needed. And I think this method is created for the author of numpy.tensordot() hopes that users can write more concise codes.
The results are different for axes = [[-2, -1], [0, 1]] and axes = [[-1, -2], [0, 1]]. Therefore, I think it's important to emphasise that, and I guess the author of this part also hope to express that.

2. I add an example to enhance the description of integer_like:

     "traditional" examples:
    >>> # An example on integer_like
    >>> a_0 = np.array([[1, 2], [3, 4]])
    >>> b_0 = np.array([[5, 6], [7, 8]])
    >>> c_0 = np.tensordot(a_0, b_0, axes=0)
    >>> c_0.shape
    (2, 2, 2, 2)
    >>> c_0
    array([[[[ 5,  6],
             [ 7,  8]],
            [[10, 12],
             [14, 16]]],
           [[[15, 18],
             [21, 24]],
            [[20, 24],
             [28, 32]]]])

---

Reference

I think this is an important issue, for many people can't understand the description of the tutorial of np.tensordot(), especially the using of the integer_like parameters,
for example:

1. https://medium.com/analytics-vidhya/tensordot-explained-6673cfa5697f shows that
   
2. https://stackoverflow.com/questions/51989572/how-does-numpy-tensordot-function-works-step-by-step shows that
   
3. https://stackoverflow.com/questions/66214690/numpy-tensordot-axes-2 shows that
   
4. ....

Many people have to do many experiments to understand what happens, for integer_like is a specific usage of tensordot() in numpy. Spending time to understand how to use it does take much time and kill people's interests to use it. I also take 2 days to understand it and find the conflict. For efficiency reasons, many people prefer to replace this function with some other functions, which I think will make the function lose it's meaning.

Therefore, I hope to fix the problem as soon as possible.

Thank you for your review and help.

[John-Venti]

@charris I fixed the variable name conflict in my last PR. I think this version is stable.
could you check it again?
you can find the problem description at numpy/numpy-tutorials#174 or #23586

[John-Venti]

@charris
I rectified some conflicts I submitted before, for I didn't notice original codes behind the example I added use the same variable name.
I believe this version is stable and correct now.
could you check that again?
I think it is an important modification for the tutorial of np.tensordot(). Because the description of the argument exists some typos which will mislead many new users, and I find many people shows they can't understand the tutorial.

[mattip]

I think this whole section should be removed. It is less clear than the explanation above (in the Parameters section).

[John-Venti]

I also think this part is a little redundant.
However, I think this part also has certain significance.

1. I think it should be claimed what would happen when N is less than 0, because there is no limitation or error warning when N < 0.
2. the orders of a and b axes are important and it's not clear in the parameters section.

Therefore, I add some details to this part to make it clearer and more meaningful.

[mattip]

Let's see what other reviewers think

[John-Venti]

about reason 2.
please let me explain more.
the results are different for axes = [[-2, -1], [0, 1]] and axes = [[-1, -2], [0, 1]]. Therefore, I think it's important to emphasise that, and I guess the author of this part also hope to express that.
Because tensordot() in other libraries ,like Pytorch, doesn't have the usage of integer_like as axes input. This method is only exist in Numpy, therefore enough instructions are needed. And I think this method is created for the author of numpy.tensordot() hopes that users can write more concise codes.

[John-Venti]

Agree with @mattip, this part was a little unclear and redundant before.
Therefore, I add some details to help express more clearly.

[John-Venti]

about reason 2.
please let me explain more.
the results are different for axes = [[-2, -1], [0, 1]] and axes = [[-1, -2], [0, 1]]. Therefore, I think it's important to emphasise that, and I guess the author of this part also hope to express that.
Because tensordot() in other libraries ,like Pytorch, doesn't have the usage of integer_like as axes input. This method is only exist in Numpy, therefore enough instructions are needed. And I think this method is created for the author of numpy.tensordot() hopes that users can write more concise codes.

[linus-md]

I have a few minor suggestions.

[John-Venti]

I have a few minor suggestions.

Thanks for your suggestions! I modified them.