issues

• [x] Closes #2217
• [x] Tests added
• [ ] Passes isort . && black . && mypy . && flake8
• [ ] User visible changes (including notable bug fixes) are documented in whats-new.rst
• [ ] New functions/methods are listed in api.rst

Reading #2217, I've implemented fast algorithms for union and intersection of arrays with numerical tolerance. This works fine in the "normal" case, when each array has all its values different (outside the tolerance) and the first and second arrays have some values in common within the tolerance. Conversely, the behavior is not well defined when one array has some values that are within the tolerance (or are equal) of each other. In this case, the behavior of union and intersection is not well defined anyway. The "bad" cases could be checked to raise an Exception (duplicate values within the tolerance), but this is not implemented yet.

I've also implemented a function to test index equality within the tolerance.

At last, the logic of xarray.align has been changed to deal with the tolerance. It was not possible to avoid some changes in align.

I'd appreciate some tests and code review, because there are certainly some rough corners I've not though about.

What is your issue?

Xarray is permissive regarding the type of the attributes. If using a wrong type, the error reveals the valid types: For serialization to netCDF files, its value must be of one of the following types: str, Number, ndarray, number, list, tuple

Using a non iterable type used to raise an Exception when reading the saved netcdf, but this is now solved with #7085

The pending question is whether it is valid to save netcdf attributes with type other than a string or not. The following lines are working (in a notebook):

```python xr.DataArray([1, 2, 3], attrs={'units': 1}, name='x').to_netcdf("tmp.nc") !ncdump tmp.nc

xr.DataArray([1, 2, 3], attrs={'units': np.nan}, name='x').to_netcdf("tmp.nc") !ncdump tmp.nc

xr.DataArray([1, 2, 3], attrs={'units': ['xarray', 'is', 'very', 'permissive', ]}, name='x').to_netcdf("tmp.nc") !ncdump tmp.nc ``` On the other hand, the following line raises an error:

```python xr.DataArray([1, 2, 3], attrs={'units': None}, name='x').to_netcdf("tmp.nc")

```

• [x] Closes #4465
• [x] Tests added
• [ ] Passes isort . && black . && mypy . && flake8
• [ ] User visible changes (including notable bug fixes) are documented in whats-new.rst
• [ ] New functions/methods are listed in api.rst

Is your feature request related to a problem? Please describe.

When a coordinate in different dataset / netcdf files has slightly different values, combine_by_coords considers the coordinate are different and attempts a concatenation of the coordinates.

Concretely, I produce netcdf with (lat, lon, time) coordinates, annually. Apparently the lat is not the same in all the files (difference is 1e-14), which I suspect is due to different pyproj version used to produce the lon,lat grid. Reprocessing all the annual netcdf is not an option. When using open_mfdataset on these netcdf, the lat coordinate is concatenated which leads to a MemoryError in my case.

Describe the solution you'd like Two options: - add a coord_tolerance argument to xr.combine_by_coords and use np.allclose to compare the coordinates. In line 69 combine.py the comparison uses strict equality "if not all(index.equals(indexes[0]) for index in indexes[1:]):". This does not break the compatibility because coord_tolerance=0 should be the default.

• add an argument to explicity list the coordinates to NOT concatenate. I tried to play with the coords argument to solve my problem, but was not succesfull.

Describe alternatives you've considered

• I certainly could find a workaround for this specific case, but I often had issue with the magic in combine_by_coords, and imho adding more control by the user would be useful in general.

Additional context