|
Size: 14160
Comment: PolyhedralComplex
|
← Revision 119 as of 2024-08-19 06:03:06 ⇥
Size: 0
Comment: Moved to https://github.com/sagemath/sage/releases/tag/9.4 including attachment
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 1: | Line 1: |
| = Sage 9.4 Release Tour = current development cycle (2021) <<TableOfContents>> == Symbolics == === Extended interface with SymPy === The [[https://www.sympy.org/en/index.html|SymPy]] package has been updated to version 1.8. !SageMath has a bidirectional interface with !SymPy. Symbolic expressions in Sage provide a `_sympy_` method, which converts to !SymPy; also, Sage attaches `_sage_` methods to various !SymPy classes, which provide the opposite conversion. In Sage 9.4, several conversions have been added. Now there is a bidirectional interface as well for matrices and vectors. [[https://trac.sagemath.org/ticket/31942|#31942]] {{{ sage: M = matrix([[sin(x), cos(x)], [-cos(x), sin(x)]]); M [ sin(x) cos(x)] [-cos(x) sin(x)] sage: sM = M._sympy_(); sM Matrix([ [ sin(x), cos(x)], [-cos(x), sin(x)]]) sage: sM.subs(x, pi/4) # computation in SymPy Matrix([ [ sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2]]) }}} Work is underway to make !SymPy's symbolic linear algebra methods available in Sage via this route. Callable symbolic expressions, such as those created using the Sage preparser's `f(...) = ...` syntax, now convert to a !SymPy `Lambda`. [[https://trac.sagemath.org/ticket/32130|#32130]] {{{ sage: f(x, y) = x^2 + y^2; f (x, y) |--> x^2 + y^2 sage: f._sympy_() Lambda((x, y), x**2 + y**2) }}} Sage has added a formal set membership function `element_of` for use in symbolic expressions; it converts to a !SymPy's `Contains` expression. [[https://trac.sagemath.org/ticket/24171|#24171]] Moreover, all sets and algebraic structures (`Parent`s) of !SageMath are now accessible to !SymPy by way of a wrapper class `SageSet`, which implements the [[https://docs.sympy.org/latest/modules/sets.html#set|SymPy Set API]]. [[https://trac.sagemath.org/ticket/31938|#31938]] {{{ sage: F = Family([2, 3, 5, 7]); F Family (2, 3, 5, 7) sage: sF = F._sympy_(); sF SageSet(Family (2, 3, 5, 7)) # this is how the wrapper prints sage: sF._sage_() is F True # bidirectional sage: bool(sF) True sage: len(sF) 4 sage: sF.is_finite_set # SymPy property True }}} Finite or infinite, we can wrap it: {{{ sage: W = WeylGroup(["A",1,1]) sage: sW = W._sympy_(); sW SageSet(Weyl Group of type ['A', 1, 1] (as a matrix group acting on the root space)) sage: sW.is_finite_set False sage: sW.is_iterable True sage: sB3 = WeylGroup(["B", 3])._sympy_(); sB3 SageSet(Weyl Group of type ['B', 3] (as a matrix group acting on the ambient space)) sage: len(sB3) 48 }}} Some parents or constructions have a more specific conversion to !SymPy. [[https://trac.sagemath.org/ticket/31931|#31931]], [[https://trac.sagemath.org/ticket/32015|#32015]] {{{ sage: ZZ3 = cartesian_product([ZZ, ZZ, ZZ]) sage: sZZ3 = ZZ3._sympy_(); sZZ3 ProductSet(Integers, Integers, Integers) sage: (1, 2, 3) in sZZ3 sage: NN = NonNegativeIntegers() sage: NN._sympy_() Naturals0 sage: (RealSet(1, 2).union(RealSet.closed(3, 4)))._sympy_() Union(Interval.open(1, 2), Interval(3, 4)) sage: X = Set(QQ).difference(Set(ZZ)); X Set-theoretic difference of Set of elements of Rational Field and Set of elements of Integer Ring sage: X._sympy_() Complement(Rationals, Integers) sage: X = Set(ZZ).difference(Set(QQ)); X Set-theoretic difference of Set of elements of Integer Ring and Set of elements of Rational Field sage: X._sympy_() EmptySet }}} See [[https://trac.sagemath.org/ticket/31926|Meta-ticket #31926: Connect Sage sets to SymPy sets]] === ConditionSet === Sage 9.4 introduces a class `ConditionSet` for subsets of a parent (or another set) consisting of elements that satisfy the logical "and" of finitely many predicates. The name `ConditionSet` is borrowed from !SymPy. In fact, if the given predicates (condition) are symbolic, a `ConditionSet` can be converted to a !SymPy `ConditionSet`; the `_sympy_` method falls back to creating a `SageSet` wrapper otherwise. === symbolic_expression(lambda x, y: ...) === Sage 9.4 has added a new way to create callable symbolic expressions. [[https://trac.sagemath.org/ticket/32103|#32103]] The global function `symbolic_expression` now accepts a callable such as those created by `lambda` expressions. The result is a callable symbolic expression, in which the formal arguments of the callable are the symbolic arguments. Example: {{{ symbolic_expression(lambda x,y: x^2+y^2) == (SR.var("x")^2 + SR.var("y")^2).function(SR.var("x"), SR.var("y")) }}} This provides a convenient syntax in particular in connection to `ConditionSet`. Instead of {{{ sage: predicate(x, y, z) = sqrt(x^2 + y^2 + z^2) < 12 # preparser syntax, creates globals sage: ConditionSet(ZZ^3, predicate) }}} one is now able to write {{{ sage: ConditionSet(ZZ^3, symbolic_expression(lambda x, y, z: ....: sqrt(x^2 + y^2 + z^2) < 12)) }}} == Convex geometry == === ABC for convex sets === Sage 9.4 has added an abstract base class `ConvexSet_base` (as well as abstract subclasses `ConvexSet_closed`, `ConvexSet_compact`, `ConvexSet_relatively_open`, `ConvexSet_open`) for convex subsets of finite-dimensional real vector spaces. The abstract methods and default implementations of methods provide a unifying API to the existing classes `Polyhedron_base`, `ConvexRationalPolyhedralCone`, `LatticePolytope`, and `PolyhedronFace`. [[https://trac.sagemath.org/ticket/31919|#31919]], [[https://trac.sagemath.org/ticket/31959|#31959]], [[https://trac.sagemath.org/ticket/31990|#31990]] As part of the API, there are new methods for point-set topology such as `is_open`, `relative_interior`, and `closure`. For example, taking the `relative_interior` of a polyhedron constructs an instance of `RelativeInterior`, a simple object that provides a `__contains__` method and all other methods of the `ConvexSet_base` API. [[https://trac.sagemath.org/ticket/31916|#31916]] {{{ sage: P = Polyhedron(vertices=[(1,0), (-1,0)]) sage: ri_P = P.relative_interior(); ri_P Relative interior of a 1-dimensional polyhedron in ZZ^2 defined as the convex hull of 2 vertices sage: (0, 0) in ri_P True sage: (1, 0) in ri_P False }}} `ConvexSet_base` is a subclass of the new abstract base class `Set_base`. [[https://trac.sagemath.org/ticket/32013|#32013]] This makes various methods that were previously only defined for sets constructed using the `Set` constructor available for polyhedra and other convex sets. As an example, we can now do: {{{ sage: polytopes.cube().union(polytopes.tetrahedron()) Set-theoretic union of Set of elements of A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices and Set of elements of A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 4 vertices }}} === Polyhedral geometry === Sage 9.4 defines a new subclass of `GenericCellComplex` for (geometric) polyhedral complexes. [[https://trac.sagemath.org/ticket/31748|#31748]] {{{ sage: pc = PolyhedralComplex([ ....: Polyhedron(vertices=[(1/3, 1/3), (0, 0), (1/7, 2/7)]), ....: Polyhedron(vertices=[(1/7, 2/7), (0, 0), (0, 1/4)])]) sage: [p.Vrepresentation() for p in pc.cells_sorted()] [(A vertex at (0, 0), A vertex at (0, 1/4), A vertex at (1/7, 2/7)), (A vertex at (0, 0), A vertex at (1/3, 1/3), A vertex at (1/7, 2/7)), (A vertex at (0, 0), A vertex at (0, 1/4)), ... (A vertex at (1/7, 2/7),), (A vertex at (1/3, 1/3),)] sage: pc.plot() Graphics object consisting of 10 graphics primitives sage: pc.is_pure() True sage: pc.is_full_dimensional() True sage: pc.is_compact() True sage: pc.boundary_subcomplex() Polyhedral complex with 4 maximal cells sage: pc.is_convex() True sage: pc.union_as_polyhedron().Hrepresentation() (An inequality (1, -4) x + 1 >= 0, An inequality (-1, 1) x + 0 >= 0, An inequality (1, 0) x + 0 >= 0) sage: pc.face_poset() Finite poset containing 11 elements sage: pc.is_connected() True sage: pc.connected_component() == pc True }}} == Manifolds == === Defining submanifolds and manifold subsets by pullbacks from Sage sets === Given a continuous map `Φ` from a topological or differentiable manifold `N` and a subset `S` of the codomain of `Φ`, we define the pullback (preimage) of `S` as the subset of `N` of points `p` with `Φ(p)` in `S`. [[https://trac.sagemath.org/ticket/31688|#31688]] Generically, such pullbacks are represented by instances of the new class `ManifoldSubsetPullback`. But because `Φ` is continuous, topological closures and interiors pull back accordingly. Hence, in some cases we are able to give the pullback additional structure, such as creating a submanifold rather than merely a manifold subset. In addition to the case when `Φ` is a continuous map between manifolds, there are two situations that connect Sage manifolds to sets defined by other components of Sage: * If `Φ: N -> R` is a real scalar field, then any `RealSet` `S` (i.e., a finite union of intervals) can be pulled back. * If `Φ` is a chart – viewed as a continuous function from the chart's domain to `R^n` – then any subset of `R^n` can be pulled back to define a manifold subset. This can be a polyhedron, a lattice, a linear subspace, a finite set, or really any object with a `__contains__` method. For example, defining a "chart polyhedron" by pulling back a polyhedron: {{{ sage: M = Manifold(2, 'R^2', structure='topological') sage: c_cart.<x,y> = M.chart() # Cartesian coordinates on R^2 sage: P = Polyhedron(vertices=[[0, 0], [1, 2], [2, 1]]); P A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices sage: McP = c_cart.preimage(P); McP Subset x_y_inv_P of the 2-dimensional topological manifold R^2 sage: M((1, 2)) in McP True sage: M((2, 0)) in McP False }}} Pulling back the interior of a polytope under a chart: {{{ sage: int_P = P.interior(); int_P Relative interior of a 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices sage: McInt_P = c_cart.preimage(int_P, name='McInt_P'); McInt_P Open subset McInt_P of the 2-dimensional topological manifold R^2 sage: M((0, 0)) in McInt_P False sage: M((1, 1)) in McInt_P True }}} In a similar direction, the new method `Polyhedron.affine_hull_manifold` makes the affine hull of a polyhedron available as a Riemannian submanifold embedded into the ambient Euclidean space. [[https://trac.sagemath.org/ticket/31659|#31659]] === Families and posets of manifold subsets === [[https://trac.sagemath.org/ticket/31740|Meta-ticket #31740]] == Configuration changes == === Support for system gcc/g++/gfortran 11 added === Sage can now be built using GCC 11. [[https://trac.sagemath.org/ticket/31786|#31786]] This enables building Sage using the default compiler on Fedora 34, and on macOS with homebrew using the default compilers. (Previously, in Sage 9.3, specific older versions of the compilers had to be installed.) === Support for system Python 3.6 dropped === It was already deprecated in Sage 9.3. [[https://trac.sagemath.org/ticket/30551|#30551]] It is still possible to build the Sage distribution on systems with old Python versions, but Sage will build its own copy of Python 3.9.x in this case. === Support for optional packages on systems with gcc 4.x dropped === Sage is phasing out its support for building from source using very old compilers from the gcc 4.x series. As of Sage 9.4, on systems such as `ubuntu-trusty` (Ubuntu 14.04), `debian-jessie` (8), `linuxmint-17`, and `centos-7` that only provide gcc from the outdated 4.x series, it is still supported to build Sage from source with the system compilers. However, building optional and experimental packages is no longer supported, and we have removed these configurations from our CI. [[https://trac.sagemath.org/ticket/31526|#31526]] Users in scientific computing environments using these platforms should urge their system administrators to upgrade to a newer distribution, or at least to a newer toolchain. === For developers: ./configure --prefix=SAGE_LOCAL --with-sage-venv=SAGE_VENV === Sage 9.4 makes it possible to configure the build to make a distinction between: * the installation tree for non-Python packages (`SAGE_LOCAL`, which defaults to `SAGE_ROOT/local` and can be set using the `configure` option `--prefix`), and * the installation tree (virtual environment) for Python packages (`SAGE_VENV`). By default, `SAGE_VENV` is just the same as `SAGE_LOCAL`, but it can be set to an arbitrary directory using the `configure` option `--with-sage-venv`. Package installation records are kept within each tree, and thus separately. This allows developers to switch between different system Python versions without having to rebuild the whole set of non-Python packages. See [[https://trac.sagemath.org/ticket/29013|#29013]] for details. == Package upgrades == * https://repology.org/projects/?inrepo=sagemath_develop * many upgrades were enabled by dropping support for Python 3.6 == Availability of Sage 9.4 and installation help == The first beta of the 9.4 series, 9.4.beta0, was tagged on 2021-05-26. * See [[https://groups.google.com/forum/#!forum/sage-devel|sage-devel]] for development discussions and [[https://groups.google.com/forum/#!forum/sage-release|sage-release]] for announcements of beta versions and release candidates. == More details == * [[https://trac.sagemath.org/query?status=needs_info&status=needs_review&status=needs_work&status=new&summary=~Meta&col=id&col=summary&col=status&col=type&col=priority&col=milestone&col=component&order=priority|Open Meta-Tickets]] * [[https://trac.sagemath.org/query?milestone=sage-9.4&groupdesc=1&group=status&max=1500&col=id&col=summary&col=author&col=reviewer&col=time&col=changetime&col=component&col=keywords&order=component|Trac tickets with milestone 9.4]] |
