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
• #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
• #10777: Implement CombinatorialPolyhedron
• #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.