More control on the numerical ODE solver for integrated curves and geodesics

[egourgoulhon]

This ticket allows to pass extra parameters to the method IntegratedCurve.solve() in order to control the numerical ODE solver. In particular, this allows to fix an issue with the odeint solver by increasing its precision via some tolerance parameters (see comment:1 for details).

Besides, this ticket makes the odeint solver (introduced in #25936) the default one, instead of rk4_maxima, since the former is much faster than the latter, as already noticed in #25936. The method name 'ode_int' has been changed to 'odeint', in order to match with desolve_odeint() and the ​SciPy name (for backward compatibility, method='ode_int' is still accepted though).

Moreover, this ticket performs some improvement of the documentation of integrated curves.

[egourgoulhon]

In Sage 8.9, when integrating a null geodesic in Schwarzschild spacetime with an impact parameter close to critical (all details are in ​this notebook), the solution obtained with solve(method='ode_int') (green in the figure below) differs significantly from those obtained by means of other solvers, for instance the rkf45 solver (red in the figure below):

The geodesic arises from the upper right ((x,y) approx. (10, 5)), circles twice around the black hole (the grey area) and then departs away to the lower left (rkf45 "exact" solution) or to the lower right (ode_int bad solution). This issue is due to the default values of 1.49012e-8 for the parameters rtol and atol of ​scipy.integrate.odeint being not sufficient in the present case. Now, with the Sage 8.9 implementation of IntegratedCurve.solve(), there is no way to change the values of rtol and atol. Thanks to the new argument **control_param introduced here this becomes possible. Morevoer, the default values of rtol and atol are changed to 1.e-10. Accordingly, the curves obtained from the odeint and rkf45 become almost undistinguishable:

See the ​notebook for a detailed study of the agreement between odeint and the other solvers when using the code of this ticket branch.

[egourgoulhon]

New commits:

​7b44ef4	Add control parameters to IntegratedCurve.solve()
​e31f6cb	Expose methods of scipy.integrate.ode in IntegratedCurve.solve()
​0cc2077	Change default method in IntegratedCurve.solve() from rk4_maxima to odeint
​bd1392b	Improve documentation of integrated curves
​511e26d	More improvements in the documentation of integrated curves

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2e918f4

Correct some doctests after the change of default method in IntegratedCurve.solve()

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​aca08a8

Merge branch public/manifolds/geod_integrator_options into Sage 9.0.beta5

[egourgoulhon]

The above commit fixes a merge conflict with Sage 9.0.beta5, due to #28669.

[gh-FlorentinJ]

Seems perfect to me. Many thanks Éric.

(The failed plugins are only there because of the merge with #28669, no issue here)

[egourgoulhon]

Replying to gh-FlorentinJ:

Seems perfect to me. Many thanks Éric.

Thank you for the review!

(The failed plugins are only there because of the merge with #28669, no issue here)

I think it's rather an issue with this patchbot: there are many errors and all reports say that they happen "on 0 non-empty lines". The "0" is highly suspicious here...

[gh-FlorentinJ]

The diff for the failed plugin only contain 2 lines:

--- 9.0.beta5
+++ 9.0.beta5 + #28707

Isn't that the reason it failed ?