Show that init_session() runs init_printing()
polyroots: always refine all roots

Previously if the given root was already converged, it wasn't refined. But this
seems to increase the number of iterations needed to converge considerably.
polyroots: Raise error if convergence not achieved
polyroots: chop small numbers in "cleanup" mode

Previously only small real or imaginary parts were chopped. But if abs(x) is
close to zero, we want to chop this case as well.
polyroots: use `extraprec` argument

Previously this argument was not used at all by a mistake and instead the
default value 10 was assigned directly.
polyroots: fix tolerance calculation

We now set the tolerance `tol` directly equal to `ctx.eps`. This is the right
way to do it, since if more working precision is needed, the user needs to
assign the `extraprec` argument.
Refactor nroots() to make it more robust

The following changes were made:

* The polynomial coefficients are always converted to integers. Previously, a
  floating point number was used instead, which loses accuracy, and then for
  higher order polynomials (e.g. Legendre of order 64) one has to use large
  working precision just to represent polynomial coefficients accurately
  enough (otherwise one gets inaccurate roots, for example complex roots
  solution instead of real ones). Now this problem is completely avoided.

* We set extra precision in polyroots() based on the polynomial order.
  "10*order" seem to work well. This formula can be adjusted. This takes care
  of the problem that higher order polynomials suffer from numerical
  cancellation.

* We remove the `error` argument. Now the error is always either lower than `n`
  digits, or an exception is raised.

* The roots are now ordered by mpmath, which returns the correct ordering (real
  roots first, then complex ones).
Add tests for nroots()

We test roots of Legendre polynomial of order 64, as well as some smaller
polynomials. Besides accuracy, these tests also ensure the proper ordering of
the roots.
polyroots: use NoConvergence instead of Exception

A test written as well.
Add a note to tests about the roots order
ODE:Corrected classify_ode of separable_reduced Issue #6950

There were few equations like sin(x)*cos(f(x)) + cos(x)*sin(f(x))*f(x).diff(x) which separable_reduced classified but was not able to solve because it was not fitting in its general form. So, this patch modifies classify_ode of separable_reduced so as not to classify those equations which are out of its scope.
Port changes from mpmath
Update tests: nroots does not take the error arg

Instead it raises an exception if the root is not converged.
Put back the original ordering

This is required by tests like the following:

    f = 7*z**4 - 19*z**3 + 20*z**2 + 17*z + 20

    assert intervals(f) == []

    real_part, complex_part = intervals(f, all=True, sqf=True)

    assert real_part == []
    assert all(re(a) < re(r) < re(b) and im(
        a) < im(r) < im(b) for (a, b), r in zip(complex_part, nroots(f)))

For now, we do not want to modify the symbolic algorithms in intervals(), so we
returned to the original ordering. We documented this in the relevant parts of
the code and tests.
Make the coefficients working for other fields

This just replicates the logic that was used there before for such fields.
Skip the init_printing() "doctest" in the tutorial
Merge pull request #7375 from asmeurer/init_printing

Show that init_session() runs init_printing()
Merge pull request #7393 from certik/nroots

Make nroots() robust
Merge pull request #7371 from KDMOE/6950_separable_reduced_wrong_dsolve_output

ODE: Corrected classify_ode of separable_reduced fixes #6950
Add Kundan Kumar to AUTHORS/aboutus. Welcome to SymPy!