Multimethods for mathematical functions

[joaosferreira]

This pull request further adds multimethods that target mathematical functions. I'll name the multimethods added according to the NumPy docs sectioning:

Trigonometric functions

• degrees
• radians
• unwrap

Rounding

• around (and its aliase round_)
• fix

Sums, products, differences

• cumprod
• cumsum
• nancumprod
• nancumsum
• ediff1d
• cross
• trapz

Other special functions

• i0
• sinc

Arithmetic operations

• float_power

Handling complex numbers

• angle
• real
• imag
• conjugate

Miscellaneous

• convolve
• clip
• nan_to_num
• real_if_close
• interp

Some thoughts about things to address moving forward with the PR:

1. There are no default implementations in this initial commit. I intend to fix the failing tests first before I start pushing some. The failing tests are all in the XND backend and are almost all the same.
2. I noticed that the multimethod for diff has a different signature from the NumPy's function (the latter has more arguments). I'm wondering if these were added after the unumpy's multimethod was written.
3. I think some older multimethods for ufuncs don't have tests so I might add those as well.

[hameerabbasi]

What's going on here? Concatenating an empty array does nothing.

[joaosferreira]

You are right, I could do the other way around like:

if to_end is not None:
    diffs = concatenate([diffs, ravel(to_end)])

Seems better this way!

[hameerabbasi]

Append to a list, check for trivial case and return early.

[joaosferreira]

There's a type error arising from the call to where in nancumprod and nancumsum defaults. Should I ignore them?

[hameerabbasi]

Yes, just add type: ignore

[joaosferreira]

The default for interp was based on NumPy's implementation and uses the equation of slopes to compute the y-coordinates element-wise.

I'm still not sure how to add the left and right cases. Also, it needs to take into consideration complex numbers.

[peterbell10]

Instead of clipping x, couldn't you do something like this:

# Where we return currently
result = (y0 * x1_dist + y1 * x0_dist) / x_dist 

left = fp[0] if left is None else left
right = fp[-1] if right is None else right
result = where(x < xp[0], left, result)
return where(x > xp[-1], right, result)

You will need to fixup idxs as well to make sure there are no out of bounds accesses when interpolating.

[joaosferreira]

I've pushed the changes to _interp_default that add the left and right cases. It also works for complex numbers.

[joaosferreira]

NumPy's implementation for unwrap uses item assignment which is not supported in backends with read-only arrays. Maybe there's a way to implement the default without using item assignment?

[hameerabbasi]

I'd recommend you first do the calculation as if slice1 was only used for reading, not assignment, then, when it's being used for assignment: https://github.com/numpy/numpy/blob/v1.19.0/numpy/lib/function_base.py#L1540

I'd recommend you replace that line with a concatenate in the appropriate axis with a sub-array in the appropriate axis. A 1D version is below, I'll leave it as an exercise to come up with N-D (hint: tuple with slice(None) and 0, and axis in concatenate).

up_slice1 = p[slice1] + ph_correct.cumsum(axis)
up = concatenate([p[0], up_slice1])

[peterbell10]

You can use slice(0, 1) to grab the 0 value without also collapsing the dimension. That will avoid the need to use expand_dims later.

[joaosferreira]

Good idea, that way we avoid an unnecessary multimethod call.

[peterbell10]

I think this is the same as trunc.

The truncated value of the scalar x is the nearest integer i which is closer to zero than x is.

[joaosferreira]

They seem to be the same, yes. Refactor the default in terms of trunc then? Something like:

def _fix_default(x, out=None):
    x = trunc(x)

    if out is None:
        return x
    else:
        copyto(out, x)

[hameerabbasi]

Why not just return trunc(x, out=out)?

[joaosferreira]

You are right! Forgot about the out kwarg in ufuncs. 😅

[hameerabbasi]

We've merged the fix we talked about in the meeting in Quansight-Labs/uarray#246. Pushing an empty commit with git commit --allow-empty (or any changes) should fix CI if there are no other breaking changes.

[joaosferreira]

I think this can be merged now.

[hameerabbasi]

Thanks, @joaosferreira!