Hi Ilhan,

Overall I think something like this would be great. However, I wonder if you considered having a specialized container with a structure tag instead of trying to discover the structure. If it's a container, it can neatly wrap various lapack storage schemes and dispatch to an appropriate lapack functionality. Possibly even sparse storage schemes. And it seems a bit more robust than trying to discover the structure (e.g. what about off-band elements of \sim 1e-16 etc).

The next question is of course if this should live in scipy/numpy .linalg or as a separate repo, at least for some time (maybe in the scipy organization?). So that it can iterate faster, among other things. (I'd be interested in contributing FWIW)

Cheers,

Evgeni

On Wed, Jun 30, 2021 at 1:22 AM Ilhan Polat ilhanpolat@gmail.com wrote:

Dear all,

I'm writing some helper Cythpm functions for scipy.linalg which is kinda performant and usable. And there is still quite some wiggle room for more.

In many linalg routines there is a lot of performance benefit if the structure can be discovered in a cheap and reliable way at the outset. For example if symmetric then eig can delegate to eigh or if triangular then triangular solvers can be used in linalg.solve and lstsq so forth

Here is the Cythonized version for Jupyter notebook to paste to discover the lower/upper bandwidth of square array A that competes well with A != 0 just to use some low level function (note the latter returns an array hence more cost is involved) There is a higher level supervisor function that checks C-contiguousness otherwise specializes to different versions of it

Initial cell

%load_ext Cython %load_ext line_profiler import cython import line_profiler

Then another cell

%%cython # cython: language_level=3 # cython: linetrace=True # cython: binding = True # distutils: define_macros=CYTHON_TRACE=1 # distutils: define_macros=CYTHON_TRACE_NOGIL=1

cimport cython cimport numpy as cnp import numpy as np import line_profiler ctypedef fused np_numeric_t: cnp.int8_t cnp.int16_t cnp.int32_t cnp.int64_t cnp.uint8_t cnp.uint16_t cnp.uint32_t cnp.uint64_t cnp.float32_t cnp.float64_t cnp.complex64_t cnp.complex128_t cnp.int_t cnp.long_t cnp.longlong_t cnp.uint_t cnp.ulong_t cnp.ulonglong_t cnp.intp_t cnp.uintp_t cnp.float_t cnp.double_t cnp.longdouble_t

@cython.linetrace(True) @cython.initializedcheck(False) @cython.boundscheck(False) @cython.wraparound(False) cpdef inline (int, int) band_check_internal(np_numeric_t[:, ::1]A): cdef Py_ssize_t n = A.shape[0], lower_band = 0, upper_band = 0, r, c cdef np_numeric_t zero = 0

`for r in xrange(n): # Only bother if outside the existing band: for c in xrange(r-lower_band): if A[r, c] != zero: lower_band = r - c break for c in xrange(n - 1, r + upper_band, -1): if A[r, c] != zero: upper_band = c - r break return lower_band, upper_band`

Final cell for use-case ---------------

# Make arbitrary lower-banded array n = 50 # array size k = 3 # k'th subdiagonal R = np.zeros([n, n], dtype=np.float32) R[[x for x in range(n)], [x for x in range(n)]] = 1 R[[x for x in range(n-1)], [x for x in range(1,n)]] = 1 R[[x for x in range(1,n)], [x for x in range(n-1)]] = 1 R[[x for x in range(k,n)], [x for x in range(n-k)]] = 2

Some very haphazardly put together metrics

%timeit band_check_internal(R) 2.59 µs ± 84.7 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

%timeit np.linalg.solve(R, zzz) 824 µs ± 6.24 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

%timeit R != 0. 1.65 µs ± 43.1 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

So the worst case cost is negligible in general (note that the given code is slower as it uses the fused type however if I go with tempita standalone version is faster)

Two questions:

This is missing np.half/float16 functionality since any arithmetic with float16 is might not be reliable including nonzero check. IS it safe to view it as np.uint16 and use that specialization? I'm not sure about the sign bit hence the question. I can leave this out since almost all linalg suite rejects this datatype due to well-known lack of supprt.

Should this be in NumPy or SciPy linalg? It is quite relevant to be on SciPy but then again this stuff is purely about array structures. But if the opinion is for NumPy then I would need a volunteer because NumPy codebase flies way above my head.

All feedback welcome

Best ilhan

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