Scalar Field Restrictions

[gh-DeRhamSource]

How is a scalar field implemented which is piecewisely defined with different expressions in one particular chart?

Take for instance a scalar field f on the real line with standard "top" chart x, defined via f(x)=0 for x<-1, f(x)=x+1 for -1<=x<0, f(x)=1-x for 0<=x<1 and f(x)=0 for x>=1. Currently, this is solved by using

f = M.scalar_field( unit_step(x + 1)*unit_step(1 - x)*(1 - abs(x)) )

(see ​https://trac.sagemath.org/ticket/28519#comment:46).

This solution is quite unhandy and becomes even more so for more complicated scalar fields.

This ticket is part of the metaticket #28519.

This ticket includes:

• display method modified in such a way that all distinct expressions are shown (a small slowdown in computation time)
• set_restriction method added smilar to tensor fields

[gh-DeRhamSource]

Eric, in the ticket mentioned above, I suggested to display all known expressions (on the greatest domain) and compute new expressions only if a particular chart is stated. It seems, there is no big difference to the previous code then, and the additional computation time might be negligible. What do you say? Is this a good compromise for you? Moreover, this approach seems to be the easiest one.

Or one implements an additional flag display_all for which the user is aware of additional computation time when using it?

However, do you have an example for me with much computation time to test my current algorithm how long it takes? Then, I can compare computation times. I have a feeling that the current algorithm is not that much slower than before because it starts from the top charts -- what happens anyway and, I guess, it only makes a significant difference if the scalar field is picewisely defined.

[gh-DeRhamSource]

New commits:

​0c645ca

'set_restriction' added + 'display' modified + naming modified

[gh-DeRhamSource]

I compared computation times.

Before:

...$ ./sage -t src/sage/manifolds/scalarfield.py 
too few successful tests, not using stored timings
Running doctests with ID 2019-10-04-19-03-38-e810720b.
Git branch: scalar_field_restrictions
Using --optional=build,dochtml,memlimit,python2,sage
Doctesting 1 file.
sage -t src/sage/manifolds/scalarfield.py
    [715 tests, 31.77 s]
----------------------------------------------------------------------
All tests passed!
----------------------------------------------------------------------
Total time for all tests: 32.0 seconds
    cpu time: 33.2 seconds
    cumulative wall time: 31.8 seconds

With this ticket:

...$ ./sage -t src/sage/manifolds/scalarfield.py
too few successful tests, not using stored timings
Running doctests with ID 2019-10-04-20-45-12-df0ec794.
Git branch: scalar_field_restrictions
Using --optional=build,dochtml,memlimit,python2,sage
Doctesting 1 file.
sage -t src/sage/manifolds/scalarfield.py
    [735 tests, 34.59 s]
----------------------------------------------------------------------
All tests passed!
----------------------------------------------------------------------
Total time for all tests: 34.8 seconds
    cpu time: 35.9 seconds
    cumulative wall time: 34.6 seconds

What do you say?

[git]

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

​b85f6ec

Typos fixed

[git]

Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​1d68a1f

'set_restriction' added + some restriction modifications