For many polynomial operations (e.g. add, multiply, negate) the operation takes place on the coefficients of the operands and then a new Polynomial is created. For the creation of the new Polynomial we can skip the validation of input arguments, as they are either passed from the operands (which have already been validated) or constructed with internal methods (which have validated output).

Benchmark results:

p + q: Mean +- std dev: [main0] 31.5 us +- 2.1 us -> [prx] 17.4 us +- 0.6 us: 1.81x faster
p * q: Mean +- std dev: [main0] 32.4 us +- 1.3 us -> [prx] 18.3 us +- 1.0 us: 1.77x faster
2 * p: Mean +- std dev: [main0] 24.3 us +- 1.3 us -> [prx] 11.7 us +- 0.4 us: 2.08x faster
-p: Mean +- std dev: [main0] 14.6 us +- 0.6 us -> [prx] 2.50 us +- 0.11 us: 5.84x faster

Geometric mean: 2.50x faster

with script:

import pyperf
from numpy.polynomial import Polynomial

setup = """
from numpy.polynomial import Polynomial

p=Polynomial([.5,4,5])
q=Polynomial([1,0,5])
"""

runner = pyperf.Runner()
runner.timeit(name=f"p + q", stmt="_ = p + q", setup=setup)
runner.timeit(name=f"p * q", stmt="_ = p * q", setup=setup)
runner.timeit(name=f"2 * p", stmt="_ = 2 * q", setup=setup)
runner.timeit(name=f"-p", stmt="_ = - p", setup=setup)

Notes:

• The as_series and ABCPolyBase.__init__ have an additional flag _validate_input. The default is True (so the PR is backwards compatible), but for internal operations we set it to False
• Testing was performed by with MAINT: Optimize performance of polynomial operations #24499 included in both main and the PR. Without MAINT: Optimize performance of polynomial operations #24499 there is still a significant performance gain
• We can do even more optimization if we refactor the polyadd and/or polyutils._add to have a _validate_input flag as well. But this requires more refactoring and the polyadd is public, so this has not yet been done.
• The modifications have been made for __add__, __neg__, __mul__ and __rmul__. If this PR is considered a merge candidate, we can modify other operations as well.