Some more Python 2 cleanup
The kwarg option norm=inverse leads to scaling of the transforms inverse (opposite) to that of the default option norm=None; i.e. the forward transform is normalized with 1/n whereas the backward one with 1. The fft routines and their tests have been modified to reflect the changes; all tests have been passed successfully. Closes #16126 * modified _unitary, fft & ifft and their tests; tests passed * modified rfft & irfft and their tests; tests passed * modified hfft & ihfft and all tests; all modifications are now done * correction in ihfft; all 79 fft tests passed successfully * undo unnecessary docstrings changes made in my previous commits * use norm=forward kwarg value name * Update numpy/fft/__init__.py Co-authored-by: Leo Fang <leofang@bnl.gov> * add code review suggestions Co-authored-by: Leo Fang <leofang@bnl.gov> * add default norm=None alias norm=backward * streamline private normalization functions * modify hermitian FFTs * add review suggestions Co-authored-by: Leo Fang <leofang@bnl.gov> * add review suggestions v2 (dict as module const) * make review suggestions v3 * Apply suggestions from code review Co-authored-by: Leo Fang <leofang@bnl.gov>
* Cleanup unused imports (F401) of mostly standard Python modules, or some internal but unlikely referenced modules * Where internal imports are potentially used, mark with noqa * Avoid redefinition of imports (F811)
Inheriting from object was necessary for Python 2 compatibility to use new-style classes. In Python 3, this is unnecessary as there are no old-style classes. Dropping the object is more idiomatic Python.
As numpy is Python 3 only, these import statements are now unnecessary and don't alter runtime behavior.
Relaxed test_fft_with_order for float32. Infinity norm round-off error of FFT is shown in G.U. Ramos, "Roundoff Error Analyss of the Fast Fourier Transform," Mathematics of Computation, vol. 25, no. 116, Oct. 1971, p. 757 to be bounded by sqrt(N)*K*eps.