Add the shirokov inverse

[hugohadfield]

Does what it says on the tin, see #369

[hugohadfield]

Thinking about numerical accuracy etc. I knocked up a quick comparison:

from clifford import Cl
import numpy as np

def measure_error(layout):
    rand_mv = layout.randomMV()
    unity = layout.scalar.value
    error_shirokov = np.linalg.norm(unity - (rand_mv*rand_mv.shirokov_inverse()).value)
    error_hitzer = np.linalg.norm(unity - (rand_mv*rand_mv.hitzer_inverse()).value)
    return error_shirokov, error_hitzer

for p in range(1, 6):
    layout, blades = Cl(p=p)
    error_shirokov, error_hitzer = zip(*[measure_error(layout) for i in range(200)])
    print('P: ', p)
    print('Shirokov', 'Hitzer')
    print(np.mean(error_shirokov), np.mean(error_hitzer))
    print(np.std(error_shirokov), np.std(error_hitzer))
    print()

Which produces

P:  1
Shirokov Hitzer
3.649271807406798e-16 3.649271807406798e-16
1.4566808659104976e-15 1.4566808659104976e-15

P:  2
Shirokov Hitzer
6.508691648288092e-16 6.508691648288092e-16
2.487589974051038e-15 2.487589974051038e-15

P:  3
Shirokov Hitzer
2.8485322231863833e-15 3.848218632954542e-16
1.813845517640471e-14 4.0117595566385547e-16

P:  4
Shirokov Hitzer
7.526109166859676e-16 4.448032513664149e-16
6.899802749548069e-16 3.8918760978280135e-16

P:  5
Shirokov Hitzer
1.7353364908081022e-14 1.7874116654496292e-14
7.064224091878557e-14 9.111388037030311e-14

[eric-wieser]

Flake8 is complaining because:

[eric-wieser]

Well the new tests were all fast, but that's mainly because they all failed due to accuracy issues.

[hugohadfield]

Well the new tests were all fast, but that's mainly because they all failed due to accuracy issues.

Yes indeed... any smart ideas for attempting to improve the accuracy? I had a brief look but made no headway

[eric-wieser]

Well, the simple answer is "reduce the expected accuracy of the high-dimensional test". The failing ones are all very high dimensional:

FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[0-8-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[1-7-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[2-6-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[3-5-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[4-4-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[5-3-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[6-2-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[7-1-1]
FAILED clifford/test/test_multivector_inverse.py::TestClosedForm::test_shirokov_inverse[8-0-1]

[eric-wieser]

Wait actually I think your tests is just bad. The higher dimensional your space, the more multivectors there are which are simply not invertible (and the closer you get to those, the more inaccurate your results are). The failures are all in the algebras with r=0 because in those algebras the spaces of non-invertible multivectors are much higher in dimension

[hugohadfield]

Yeah I think that is a good point, perhaps this just needs to be some reasonable range about 0 then

[eric-wieser]

Suggested change

	* The default multivector inverse method is now the arbitrary signature algorithm described
	in Theorem 4, page 16 of Dmitry Shirokov's ICCA 2020 paper :cite:`shirokov2020inverse`.
	* A new multivector inverse method is available, :meth:`clifford.MultiVector.shirokov_inverse`,
	which is the arbitrary signature algorithm described
	in Theorem 4, page 16 of Dmitry Shirokov's ICCA 2020 paper :cite:`shirokov2020inverse`.

[eric-wieser]

https://github.com/pygae/clifford/pull/373/files#r549352578 still applies. At a guess:

Suggested change

	(p, total_dims - p)
	pytest.param(p, total_dims - p, marks=[pytest.mark.slow] if total_dims >= 6 else [])

[eric-wieser]

Test failures are real at the moment; perhaps adjust the precision based on the dimension? Since it's all multiplication, I'd perhaps expect the number of digits of error to scale with the dimension of the algebra.

[hugohadfield]

Build failure is unrelated... @eric-wieser shall we merge this thing finally?