Meta-ticket: Piecewise functions, polyhedral complexes, piecewise functions of several variables, periodic piecewise functions

[mkoeppe]

This is a meta-ticket about piecewise functions.

Fractional part, modulo, ...:

• #9935: Make a symbolic mod function
• #21232: Fractional part function {x}
• #21274: frac(x) immediate simplifications
• #21745: Standardize modulo operator % on real numbers (step 1)
• #21747: Standardize modulo operator % on real numbers (step 2)
• #21748: change the mod method of rationals

Piecewise functions of 1 variable:

• #1773: piecewise functions and integration / arithmetic do not play well together
• #8603: Partial sums are off for Fourier series of piecewise functions
• #11225: improve piecewise plotting
• #6895: Detect discontinuities when plotting the ceil function
• #10597: Riemann sums, added trapezoid option
• #21215: Periodic piecewise functions
• #21412: Convert piecewise functions to SymPy?
• #21618: plot in piecewise regression (?)
• #21709: Categories for piecewise functions

RealSets, underlying the current implementation of piecewise:

• #16063: Unify the classes RealSet and UnionOfIntervals
• #21249: RealSets fail their TestSuite: _test_category, _test_pickling
• #21277: sage_input for RealSet and piecewise
• #21243: RealSet: add methods closure, interior, boundary, is_open, is_closed
• #31941: Make empty RealSet false
• #31927: RealSet: Replace is_included_in by is_subset / issubset
• #31930: Replace UnionOfIntervals by RealSet
• #31914: RealSet.contains: Handle symbolic argument
• #31877: Refine category of RealSet
• #32181: RealSet: Faster operations by scan-line (merging) techniques
• #31740: Meta-ticket: Families, posets, complexes of manifold subsets
• #21242: Allow initializing RealSet from RealInterval or RBF elements (dup: #17791)

As a generalization, this meta-ticket will implement piecewise functions of several real variables. The pieces are polyhedral complexes (not necessarily triangulations) or more generally polyhedral dissections (i.e., not necessarily face-to-face). In my application, the functions will be piecewise linear, possibly discontinuous; but the implementation should be general enough for splines.

• #31748 PolyhedralComplex
• #26512 Piecewise linear functions of several variables

Discussion welcome, in particular in regard to how this code should interact with the existing classes for abstract complexes (CW complexes) and PointConfiguration.

• #20878: PointConfiguration: Add plot method
• #20879: Expose PPL's Not-Necessarily-Closed Polyhedra using Sage Polyhedra -- could play the same role as half-open intervals do in the context of the new piecewise functions.
• #22710: Meta-ticket: polymake
• #24943: Interface to polymake's polyhedral complexes
• #21151: Update TOPCOM to 0.17.7, disable outdated libsoplex support
• #26887: Implement CombinatorialPolyhedron
• #27063: Transition of combinatorial computations of Polyhedron to Combinatorial Type
• #31842: CombinatorialPolyhedralComplex
• #18197: Add CGAL package

General piecewise functions:

• #23923: Interface cases function with SymPy's piecewise
• #20191: implement SymPy Piecewise equivalent
• #21412: Convert piecewise functions to SymPy

[mkoeppe]

Setting new milestone based on a cursory review of ticket status, priority, and last modification date.

[caruso]

I'm interested in implementing piecewise affine functions in one variable with specific methods that are easier to implement in this setting (e.g. compose, is_bijective, inverse, is_convex, convex_hull, laplace_transform, etc.) Mostly, I'm interested in the case where the coefficients of the affine parts are in QQ but I might also want to consider the case where they are in a fixed number field.

I'm wondering whether I should derive from the class sage.functions.piecewise.PiecewiseFunction (which is mostly concerned with symbolic expressions) or if it is okay if I rewrite my own class from scratch.

[mkoeppe]

I have code in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/fast_piecewise.py, based on the old Sage implementation of piecewise, which may be a starting point. It is not tied to symbolics and uses the parent-element framework

[mkoeppe]

Also ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/move_semigroup.py

[caruso]

Replying to mkoeppe:

I have code in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/fast_piecewise.py, based on the old Sage implementation of piecewise, which may be a starting point. It is not tied to symbolics and uses the parent-element framework

That's great! Do you plan to propose this code for integration in sage at some point or not?

[yzh]

Replying to caruso:

Replying to mkoeppe:

I have code in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/fast_piecewise.py, based on the old Sage implementation of piecewise, which may be a starting point. It is not tied to symbolics and uses the parent-element framework

That's great! Do you plan to propose this code for integration in sage at some point or not?

I'm familiar with the piecewise linear function code in the above github folder, and I'd love to have that integrated into sage. This has been on my wishlist for a few years, but the integration has never happened, partially because I don't have a clear view on where this new class should be derived from and what methods are useful to general users.

Besides the class hierarchy, I think that the main modification concerns changing the use of close or open or half-open intervals defined in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/intervals.py to RealSet.

[mkoeppe]

Replying to yzh:

Besides the class hierarchy, I think that the main modification concerns changing the use of close or open or half-open intervals defined in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/intervals.py to RealSet.

Yes, that's an important task. Related: #32181

[yzh]

Right, I notice that #32181 is related, and I added it into the description of this meta-ticket in comment:43. Replying to mkoeppe:

Replying to yzh:

Besides the class hierarchy, I think that the main modification concerns changing the use of close or open or half-open intervals defined in ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/intervals.py to RealSet.

Yes, that's an important task. Related: #32181

[mkoeppe]

Regarding the class hierarchy, class PiecewiseLinearFunction_1d (ModuleElement) ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/fast_piecewise.py#L17 for the element is already good, I think. The corresponding parent is ​https://github.com/mkoeppe/cutgeneratingfunctionology/blob/master/cutgeneratingfunctionology/igp/fast_piecewise.py#L1221

There are many details so that it works correctly with all operations provided by the category.

If this is added to Sage, it would be best to put it somewhere that's not sage.functions because this package concerns symbolic functions.

If the scope remains piecewise linear functions (or even piecewise polynomial functions where the pieces are polyhedra such as box splines or simplex splines), then somewhere in sage.geometry would make sense because of its close ties to ​https://doc.sagemath.org/html/en/reference/discrete_geometry/sage/geometry/polyhedral_complex.html

Another connection: both indicator_function and convex_indicator_function (proposed to add in #32201) of polyhedra are, of course, piecewise linear functions.