Bound on absolute tolerance 'xtol' in 'optimize/zeros.py' is too strict or not properly documented

[miccoli]

A number of root finding functions do not accept xtol = 0 although the docs clearly state that 'The parameter must be nonnegative.'

I had no time to thoroughly inspect the code, but I suspect that imposing xtol > 0 (instead of xtol >= 0) is overzealous and I would classify this issue as a bug.

Alternatively if erroring when xtol == 0 is strictly intended behaviour, the docs should read "The parameter must be positive."

In both cases I can provide a PR.

Reproducing code example:

import scipy.optimize
f = lambda x: x - 1

scipy.optimize.root_scalar(f, method='bisect', xtol=0, bracket=[0, 2])
scipy.optimize.root_scalar(f, method='ridder', xtol=0, bracket=[0, 2])
scipy.optimize.root_scalar(f, method='brentq', xtol=0, bracket=[0, 2])
scipy.optimize.root_scalar(f, method='brenth', xtol=0, bracket=[0, 2])
scipy.optimize.root_scalar(f, method='toms748', xtol=0, bracket=[0, 2])

Error message:

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File ".../lib/python3.8/site-packages/scipy/optimize/_root_scalar.py", line 249, in root_scalar
    r, sol = methodc(f, a, b, args=args, **kwargs)
  File ".../lib/python3.8/site-packages/scipy/optimize/zeros.py", line 546, in bisect
    raise ValueError("xtol too small (%g <= 0)" % xtol)
ValueError: xtol too small (0 <= 0)

Scipy/Numpy/Python version information:

>>> import sys, scipy, numpy; print(scipy.__version__, numpy.__version__, sys.version_info)
1.6.3 1.18.5 sys.version_info(major=3, minor=8, micro=7, releaselevel='final', serial=0)

Code reference

Culprit for method = bisect

scipy/scipy/optimize/zeros.py

Lines 487 to 490 in 4ec4ab8

	xtol : number, optional
	The computed root ``x0`` will satisfy ``np.allclose(x, x0,
	atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
	parameter must be nonnegative.

scipy/scipy/optimize/zeros.py

Lines 545 to 546 in 4ec4ab8

	if xtol <= 0:
	raise ValueError("xtol too small (%g <= 0)" % xtol)

etc.