Releases: sagemath/sage
9.6
Release Tour
Sage 9.6 was released on May 15, 2022 (changelog)
- adds support for GCC 12; major package upgrades
- major improvements to user interface and graphics; new developer tools
- 497 tickets (PRs) merged
A total of 83 people were involved as authors or reviewers of code contributions to Sage 9.6.
Here is an overview of some of the main changes in this version.
User interface, plotting and graphics
JupyterLab 3.3
JupyterLab, the latest web-based interactive development environment for notebooks, code, and data, is slated to replace the now-classic Jupyter notebook interface. The version of JupyterLab in the Sage distribution has been upgraded to the major new version 3.3. #32069, #33607
After ./sage -i jupyterlab_widgets, you can run it using
./sage -n jupyterlabAlso two new interface variants are provided:
./sage -n nbclassicand
./sage -n retrolabLaTeX displays in JupyterLab
Users of Sage in JupyterLab got used to expressions displayed at center in the LaTeX display mode. For compatibility with displays in classic Jupyter, we decided to change the behavior so that now expressions are displayed aligned left by default.
If you belong to the minority preferring centered displays, you can set your preference by
dm = get_display_manager()
dm.preferences.align_latex = 'center' # or 'left'in the ~/.sage/init.sage script.
Interactive graph editing with phitigra
With the new optional package phitigra (use ./sage -i phitigra to install), graphs can be edited by interactively placing vertices, edges, etc. This works both in the classic Jupyter notebook and in JupyterLab. It can also be used to make animations (see the demo notebook at https://github.com/jfraymond/phitigra for examples). Done in #30540 and #33639.
Hyperbolic plots
- Added the ability to choose the hyperbolic model for hyperbolic plots. #22081
Graphics with TikZ
The TikzPicture module which was developed in the slabbe package for more than 5 years is now in Sage. This was done in ticket #20343. The module is within the new file sage/misc/latex_standalone.py and its documentation in the reference manual is available here: https://doc.sagemath.org/html/en/reference/misc/sage/misc/latex_standalone.html. Below are some usage examples.
First example shows that it takes any tikz picture string as input:
sage: from sage.misc.latex_standalone import TikzPicture
sage: s = '\\begin{tikzpicture}\n\\draw[->,green,very thick](0,0) -- (1,1);\\end{tikzpicture}'
sage: t = TikzPicture(s)
sage: t # in Jupyter, rich representation will show the image instead
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\draw[->,green,very thick](0,0) -- (1,1);\end{tikzpicture}
\end{document}
sage: path_to_file = t.pdf() # and opens the image in a viewer
Of course, conversion to pdf format necessitates pdflatex or lualatex. If lualatex is available it uses it in preference to pdflatex because it handles better the very big pictures in terms of memory limits.
One can provide a local filename to save to, or convert the image to other formats (using pdftocairo or imagemagick external packages):
sage: path_to_file = t.pdf('file.pdf') # when providing a filename, it just saves
# the file locally, does not open in a viewer
sage: path_to_file = t.png() # conversion to png
sage: path_to_file = t.svg() # to svg
sage: path_to_file = t.tex() # print the tex source to a fileAnother example with graphs where additional usepackage are necessary to compile the image correctly:
sage: from sage.misc.latex_standalone import TikzPicture
sage: g = graphs.PetersenGraph()
sage: t = TikzPicture(latex(g), standalone_config=["border=4mm"], usepackage=['tkz-graph'])
sage: t # in Jupyter, rich representation will show the image instead
\documentclass[tikz]{standalone}
\standaloneconfig{border=4mm}
\usepackage{tkz-graph}
\begin{document}
\begin{tikzpicture}
\definecolor{cv0}{rgb}{0.0,0.0,0.0}
\definecolor{cfv0}{rgb}{1.0,1.0,1.0}
\definecolor{clv0}{rgb}{0.0,0.0,0.0}
\definecolor{cv1}{rgb}{0.0,0.0,0.0}
---
65 lines not printed (3695 characters in total).
Use print to see the full content.
---
\Edge[lw=0.1cm,style={color=cv6v8,},](v6)(v8)
\Edge[lw=0.1cm,style={color=cv6v9,},](v6)(v9)
\Edge[lw=0.1cm,style={color=cv7v9,},](v7)(v9)
%
\end{tikzpicture}
\end{document}
sage: _ = t.pdf() # or t.png() or t.svg()
sage: from sage.misc.latex_standalone import TikzPicture
sage: V = [[1,0,1],[1,0,0],[1,1,0],[0,0,-1],[0,1,0],[-1,0,0],[0,1,1],[0,0,1],[0,-1,0]]
sage: P = Polyhedron(vertices=V).polar()
sage: s = P.projection().tikz([674,108,-731],112)
sage: t = TikzPicture(s)
sage: _ = t.pdf() # or t.png() or t.svg()
The module also contains a class Standalone, from which the class TikzPicture inherits:
sage: from sage.misc.latex_standalone import Standalone
sage: s = Standalone('Hello World', usepackage=['amsmath'], standalone_config=['beamer=true','border=1mm'])
sage: s # in Jupyter, rich representation will show the image instead
\documentclass{standalone}
\standaloneconfig{beamer=true}
\standaloneconfig{border=1mm}
\usepackage{amsmath}
\begin{document}
Hello World
\end{document}
sage: _ = s.pdf() # or s.png() or s.svg()
Another example using Standalone with a tableau:
sage: P = Permutations(10)
sage: p = P.random_element()
sage: p
[3, 10, 1, 9, 5, 6, 7, 2, 8, 4]
sage: t = p.to_tableau_by_shape([3,3,3,1])
sage: t
[[2, 8, 4], [5, 6, 7], [10, 1, 9], [3]]
sage: s = Standalone(latex(t), standalone_config=["border=1mm"])
sage: s
\documentclass{standalone}
\standaloneconfig{border=1mm}
\begin{document}
{\def\lr#1{\multicolumn{1}{|@{\hspace{.6ex}}c@{\hspace{.6ex}}|}{\raisebox{-.3ex}{$#1$}} }
\raisebox{-.6ex}{$\begin{array}[b]{*{3}c}\cline{1-3}
\lr{2}&\lr{8}&\lr{4}\\\cline{1-3}
\lr{5}&\lr{6}&\lr{7}\\\cline{1-3}
\lr{10}&\lr{1}&\lr{9}\\\cline{1-3}
\lr{3}\\\cline{1-1}
\end{array}$}
}
\end{document}
sage: _ = s.pdf() # or s.png() or s.svg()
In a next step, a method tikz() will be added to graphs, polytopes, posets, etc. to return an object of type TikzPicture see #33002.
Complex plots
The complex plotting package phase_mag_plot has been incorporated into Sage. Now complex_plot allows contouring, tiling, and matplotlib-compatible colormaps. This was added in ticket #33416.
To use a colormap, one can pass in a string as in
sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight')
Contouring or tiling are enabled through keyword options. To look smooth, it's typically necessary to plot the function on additional points through the use of plot_points. This looks like
sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight', plot_points=500, contoured=True)
sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight', plot_points=500, tiled=True)
Linear algebra
NumPy integration
The new classes Matrix_numpy_integer_dense and Vector_numpy_integer_dense implement matrices and vectors with 64-bit integer entries backed by numpy arrays. #32465.
As a first application, several methods of GenericGraph that return matrices, such as adjacency_matrix, now accept keyword arguments that can select the matrix implementation. #33377, #33387, #33388, #33389
sage: graphs.PathGraph(5).adjacency_matrix(sparse=False, implementation='numpy')
[0 1 0 0 0]
[1 0 1 0 0]
[0 1 0 1 0]
[0 0 1 0 1]
[0 0 0 1 0]
sage: type(_)
<class 'sage.matrix.matrix_numpy_integer_dense.Matrix_numpy_integer_dense'>CombinatorialFreeModule improvements
Performing sums and similar constructions for CombinatorialFreeModule have been made faster. #33267
#...
9.5
Release Tour
SageMath version 9.5 was released on Jan 30, 2022 (changelog)
- adds support for Apple M1, Linux distributions using glibc ≥ 2.34, and system Python 3.10
- new code from 2021 Google Summer of Code projects in algebra and number theory
- 663 tickets (PRs) merged, 94 contributors
Symbolics
Changes to symbolic expressions
symbolic_expressionis now able to create vectors and matrices of symbolic expressions for more general inputs. #16761
For example, if the input is a list or tuple of lists/tuples/vectors:
sage: M = symbolic_expression([[1, x, x^2], (x, x^2, x^3), vector([x^2, x^3, x^4])]); M
[ 1 x x^2]
[ x x^2 x^3]
[x^2 x^3 x^4]
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Symbolic Ring
- Symbolic expressions can no longer be called with positional arguments. #14270
This was deprecated since Sage 4.0, although a bug prevented the deprecation warning from being issued in Sage versions 8.4 to 9.3. #32319
Instead of (x+1)(2), write either (x+1)(x=2), or (x+1).subs(x=2), or ((x+1).function(x))(2).
Interface to Mathics, a free implementation of the Wolfram language
Sage now has an optional package providing Mathics, a free (open-source) general-purpose computer algebra system featuring Mathematica-compatible syntax and functions, and an interface to it.#31778
Linear Algebra
- The Cholesky decomposition for sparse RDF/CDF matrices now uses a specialized fast algorithm when cvxopt is available. #13674
- The
is_hermitian()method for sparse RDF/CDF matrices now uses a small tolerance by default to mitigate numerical issues. This brings it to parity with its dense counterpart. #33031
Manifolds
The full list of changes is available in this changelog.
De Rham cohomology and characteristic classes
The de Rham cohomology has been made an algebra (#32270).
The method characteristic_class for vector bundles is now outdated and replaced by the method characteristic_cohomology_class. This change reflects the difference between characteristic classes, seen as natural transformations, and characteristic cohomology classes in a more rigorous way (see #29581). The previous usability and syntax remains intact. Among other things, the following has been changed:
- The performance of computing characteristic forms has been improved significantly by using a Faddeev-LeVerrier-like algorithm.
- The characteristic forms of Pontryagin/Chern/Euler classes w.r.t. to a given connection are cached in order to speed up computations of all characteristic forms w.r.t. the same connection.
Furthermore, new features have been added. For example, characteristic cohomology classes now form an algebra:
sage: M = Manifold(4, 'M')
sage: E = M.vector_bundle(2, 'E', field='complex')
sage: R = E.characteristic_cohomology_class_ring(); R
Algebra of characteristic cohomology classes of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M
sage: R.gens()
[Characteristic cohomology class (c_1)(E) of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M,
Characteristic cohomology class (c_2)(E) of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M]
sage: c_1, c_2 = R.gens()
Therefore, elements can be added and multiplied:
sage: c_1 + c_2
Characteristic cohomology class (c_1 + c_2)(E) of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M
sage: c_1 * c_1
Characteristic cohomology class (c_1^2)(E) of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M
Additive, multiplicative and Pfaffian cohomology classes are now related to the generators of the characteristic cohomology class ring via additive/multiplicative sequences:
sage: ch = E.characteristic_cohomology_class('ChernChar'); ch
Characteristic cohomology class ch(E) of the Differentiable complex vector bundle E -> M of rank 2 over the base space 4-dimensional differentiable manifold M
sage: ch == 2 + c_1 + c_1^2 / 2 - c_2 # additive sequence of exp(x)
True
As for the tangent bundle of a manifold, as long as an orientation and a metric is provided, the characteristic form of the Euler class (and therefore all Pfaffian classes) w.r.t. the Levi-Civita connection is now computed automatically (previously, a compatible curvature form matrix had to be provided by the user):
sage: M.<x,y> = manifolds.Sphere(2, coordinates='stereographic')
sage: g = M.metric()
sage: nab = g.connection()
sage: nab.set_immutable()
sage: TM = M.tangent_bundle()
sage: e = TM.characteristic_cohomology_class('Euler'); e
Characteristic cohomology class e(TS^2) of the Tangent bundle TS^2 over the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
sage: e_form = e.get_form(nab)
sage: e_form
Mixed differential form e(TS^2, nabla_g) on the 2-sphere S^2 of radius 1 smoothly embedded in the Euclidean space E^3
sage: e_form.display_expansion()
e(TS^2, nabla_g) = 2/(pi + pi*x^4 + pi*y^4 + 2*pi*x^2 + 2*(pi + pi*x^2)*y^2) dx∧dy
Automatic coordinate change in curve plots
The method plot of curves is now allowed to perform a coordinate change to make the plot in terms of the coordinates specified via the argument chart, in case the curve is not known in these coordinates (#32578). For example, a cardioid is defined in terms of polar coordinates:
sage: E.<r, phi> = [[EuclideanSpace]](coordinates='polar')
sage: c = E.curve((1 + cos(phi), phi), (phi, 0, 2*pi))
sage: c.display()
(0, 2*pi) → E^2
phi ↦ (r, phi) = (cos(phi) + 1, phi)
and its plot in terms of Cartesian coordinates can now be obtained simply by
sage: c.plot(chart=E.cartesian_coordinates(), aspect_ratio=1)
The above command has triggered the computation of the curve's expression in terms of Cartesian coordinates:
sage: c.display()
(0, 2*pi) → E^2
phi ↦ (r, phi) = (cos(phi) + 1, phi)
phi ↦ (x, y) = (cos(phi)^2 + cos(phi), (cos(phi) + 1)*sin(phi))
Internal code improvements and bug fixes
Various improvements have been performed in the internal code, some of them in view of SageMath modularization:
- faster generation of non-redundant indices (#32318)
- unnecessary uses of symbolic functions removed from
sage.tensor.modules(#32415) - doctests involving
SRmarked optional insage.tensor.modules(#32712) sage.tensor.modulesmade independent fromsage.manifolds(#32708).
Some bugs have been fixed: #31781, #32457, #32355, #32929.
Number theory
Logarithms
- Logarithms modulo composite integers are now dramatically faster in some important cases (such as prime-power moduli). #32375
- Logarithms in binary finite fields now use index calculus instead of generic-group algorithms when appropriate (via PARI's fflog()). #32842
Binary quadratic forms
- Binary quadratic forms' .solve_integer() method now uses PARI's qfbsolve() instead of a brute-force search, which is often exponentially faster. #32782
Prime counting
- Prime counting and related functions, in particular prime_pi, are now implemented using external libraries, primecount and primesieve. This improved performance and fixed a long-standing bug #24960
Elliptic curves and isogenies
- Composite isogenies of elliptic curves have been added as an experimental feature. This includes support for compact smooth-degree isogenies. #32744
- Isomorphisms of elliptic curves now expose the same interface as other isogenies. [#32502](https://trac.sagemath...
9.4
Release Tour
SageMath 9.4 was released on Aug 22, 2021 (changelog), 442 tickets (PRs) merged, 73 contributors.
- adds support for GCC 11, removes support for Python 3.6
- major advances in symbolics, convex and differential geometry, knot theory, coding theory
Symbolics
Extended interface with SymPy
The 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. #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. #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. #24171
Moreover, all sets and algebraic structures (Parents) of SageMath are now accessible to SymPy by way of a wrapper class SageSet, which implements the SymPy Set API. #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
TrueFinite 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)
48Some parents or constructions have a more specific conversion to SymPy. #31931, #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_()
EmptySetSee 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. #32089
sage: in_small_oblong(x, y) = x^2 + 3 * y^2 <= 42
sage: SmallOblongUniverse = ConditionSet(QQ^2, in_small_oblong)
sage: SmallOblongUniverse
{ (x, y) ∈ Vector space of dimension 2 over Rational Field : x^2 + 3*y^2 <= 42 }
sage: parity_check(x, y) = abs(sin(pi/2*(x + y))) < 1/1000
sage: EvenUniverse = ConditionSet(ZZ^2, parity_check); EvenUniverse
{ (x, y) ∈ Ambient free module of rank 2 over the principal ideal
domain Integer Ring : abs(sin(1/2*pi*x + 1/2*pi*y)) < (1/1000) }
sage: SmallOblongUniverse & EvenUniverse
{ (x, y) ∈ Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 1] : x^2 + 3*y^2 <= 42, abs(sin(1/2*pi*x + 1/2*pi*y)) < (1/1000) }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. #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. #31919, #31959, #31990, #31993
Several methods previously only available for Polyhedron_base instances, are now available for all convex sets. The method an_affine_basis returns a sequence of points that span by affine linear combinations the affine hull, i.e., the smallest affine subspace in which the convex set lies. The method affine_hull returns the latter as a polyhedron. The method affine_hull_projection (renamed from affine_hull in Sage 9.1) computes an affine linear transformation of the convex set to a new ambient vector space, in which the image is full-dimensional. The generalized method also provides additional data: the right inverse (section map) of the projection. #27366, #31963, #31993
As part of the ConvexSet_base 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. #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
FalseConvexSet_base is a subclass of the new abstract base class Set_base. #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()) ...9.3
Release Tour
SageMath 9.3 was released on May 9, 2021 (changelog).
- adds support for macOS 11 "Big Sur"
- major package upgrades
- 679 tickets (PRs) merged, 112 contributors
Linear and multilinear algebra
Bär--Faddeev--LeVerrier algorithm for the Pfaffian of skew-symmetric matrices
According to https://arxiv.org/abs/2008.04247, the Pfaffian of skew-symmetric matrices over commutative torsion-free rings can be computed with a Faddeev--LeVerrier-like algorithm. This algorithm is now implemented under the weaker assumption of the base ring's fraction field being a QQ-algebra (#30681). It leads to a significant increase of computational speed in comparison to the definition involving perfect matchings, which has been the only algorithm available in Sage so far.
Using the definition of the Pfaffian:
sage: A = matrix([(0, 0, 1, 0, -1, -2, -1, 0, 2, 1),
(0, 0, 1, -3/2, 0, -1, 1/2, 3, 3/2, -1/2),
(-1, -1, 0, 2, 0, 5/2, 1, 0, -2, 1),
(0, 3/2, -2, 0, 5/2, -1, 2, 0, -1, -3/2),
(1, 0, 0, -5/2, 0, 0, -1, 1/2, 1, -1),
(2, 1, -5/2, 1, 0, 0, 2, 1, 2, 1),
(1, -1/2, -1, -2, 1, -2, 0, 0, -3, -1),
(0, -3, 0, 0, -1/2, -1, 0, 0, 1/2, 1/2),
(-2, -3/2, 2, 1, -1, -2, 3, -1/2, 0, 1),
(-1, 1/2, -1, 3/2, 1, -1, 1, -1/2, -1, 0)])
sage: %%time
....: A.pfaffian(algorithm='definition')
CPU times: user 18.7 ms, sys: 0 ns, total: 18.7 ms
Wall time: 18.6 ms
817/16
With Bär--Faddeev--LeVerrier:
sage: A = matrix([(0, 0, 1, 0, -1, -2, -1, 0, 2, 1),
(0, 0, 1, -3/2, 0, -1, 1/2, 3, 3/2, -1/2),
(-1, -1, 0, 2, 0, 5/2, 1, 0, -2, 1),
(0, 3/2, -2, 0, 5/2, -1, 2, 0, -1, -3/2),
(1, 0, 0, -5/2, 0, 0, -1, 1/2, 1, -1),
(2, 1, -5/2, 1, 0, 0, 2, 1, 2, 1),
(1, -1/2, -1, -2, 1, -2, 0, 0, -3, -1),
(0, -3, 0, 0, -1/2, -1, 0, 0, 1/2, 1/2),
(-2, -3/2, 2, 1, -1, -2, 3, -1/2, 0, 1),
(-1, 1/2, -1, 3/2, 1, -1, 1, -1/2, -1, 0)])
sage: %%time
....: A.pfaffian(algorithm='bfl')
CPU times: user 554 µs, sys: 41 µs, total: 595 µs
Wall time: 599 µs
817/16
Other improvements
- Speedup access items in gf2e dense matrices: 29853
- Speedup conjugation of double dense matrices: 31283
- New Bunch-Kaufman block_ldlt() factorization for possibly indefinite matrices: 10332
- New numerically stable is_positive_semidefinite() method for matrices: 10332
Polyhedral geometry
New features
The Schlegel diagrams are now repaired (they previously broke convexity). Now, one specifies which facet to use to do the projection 30015:
#!python
sage: fcube = polytopes.hypercube(4)
sage: tfcube = fcube.face_truncation(fcube.faces(0)[0])
sage: tfcube.facets()[-1]
A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices
sage: sp = tfcube.schlegel_projection(tfcube.facets()[-1])
sage: sp.plot() # The proper Schlegel diagram is shown
A different values of position changes the projection:
#!python
sage: sp = tfcube.schlegel_projection(tfcube.facets()[4],1/2)
sage: sp.plot()
Graphics3d Object
sage: sp = tfcube.schlegel_projection(tfcube.facets()[4],4)
sage: sp.plot()
Graphics3d Object
New features:
- 30704: Upgrade to Normaliz 3.8.9 and PyNormaliz 2.13
- 30946: Add "minimal=True" option to affine_hull_projection
- 30954: Implement a proper equality check for polyhedron representation objects
Implementation improvements
The zonotope construction got improved:
Before:
sage: from itertools import combinations
sage: cu = polytopes.cube()
sage: sgmt = [p.vector()-q.vector() for p,q in combinations(cu.vertices(),2)]
sage: sgmt2 = set(tuple(x) for x in sgmt)
sage: # %time polytopes.zonotope(sgmt) # killed due to memory overflow
sage: %time polytopes.zonotope(sgmt2)
CPU times: user 2.06 s, sys: 23.9 ms, total: 2.09 s
Wall time: 2.09 s
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 96 vertices
With 31038:
sage: from itertools import combinations
sage: cu = polytopes.cube()
sage: sgmt = [p.vector()-q.vector() for p,q in combinations(cu.vertices(),2)]
sage: sgmt2 = set(tuple(x) for x in sgmt)
sage: %time polytopes.zonotope(sgmt)
CPU times: user 138 ms, sys: 0 ns, total: 138 ms
Wall time: 138 ms
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 96 vertices
sage: %time polytopes.zonotope(sgmt2)
CPU times: user 58 ms, sys: 0 ns, total: 58 ms
Wall time: 57.6 ms
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 96 vertices
Improvements:
- 30040: Improve face iterator for simple/simplicial polytopes
There are also some bug fixes and other improvements. For more details see the release notes for optimization and polyhedral geometry software interactions in Sage.
Graph theory {#graph_theory}
Major improvements in the backends:
- 30777: Deleting edges
- 30665: Edge iterator and copy
- 30776: Subgraph and equality check
- 30753: Obtaining subgraphs.
- 31117, 31154: Breadth First Search
- 31129: Depth first search
- 31197: Use binary matrix data structure for bitsets.
Algebra
Power Series Ring
-
The method
set_default_precis now deprecated since it led to unwanted behavior (see #18416 for details). If another default precision is needed, a new power series ring must be created:sage: R. = [[PowerSeriesRing]](QQ, default_prec=10)
sage: sin(x)
x - 1/6x^3 + 1/120x^5 - 1/5040x^7 + 1/362880x^9 + O(x^10)
sage: R. = [[PowerSeriesRing]](QQ, default_prec=15)
sage: sin(x)
x - 1/6x^3 + 1/120x^5 - 1/5040x^7 + 1/362880x^9 - 1/39916800x^11 + 1/6227020800x^13 + O(x^15)This change does not affect the behavior of its ring elements. Code that relies on this method needs to be updated.
-
Inversion of power series ring elements now provides the correct parent: #8972
Other additions, improvements, and key bug-fixes
-
Implement the symplectic derivation Lie algebra following https://arxiv.org/abs/2006.06064:
sage: lie_algebras.SymplecticDerivation(QQ, 4)
Symplectic derivation Lie algebra of rank 4 over Rational Field -
Implement the *-insertion algorithm from https://arxiv.org/abs/1911.08732:
sage: from sage.combinat.rsk import [[RuleStar]]
sage: p,q = [RuleStar].forward_rule([1,1,2,2,4,4], [1,3,2,4,2,4])
sage: ascii_art(p, q)2 4 1 1 2
4 2 4
3 4
sage: line1,line2...
9.2
Release Tour
SageMath 9.2 was released October 24, 2020 (changelog), 727 tickets (PRs) merged, 134 contributors.
- Python 3 support expanded to 3.6, 3.7, 3.8, 3.9; Python 2 support dropped
- major package upgrades
Python 3 transition completed
SageMath 9.0 was the first version of Sage running on Python 3 by default. SageMath 9.1 continued to support Python 2.
Support for Python 2 removed
Sage 9.2 has removed support for Python 2. The Sage library now makes use of Python language and library features that are only available in Python 3.6 or newer; and large amounts of compatibility code have been removed.
However, note that this is unrelated to the minimal requirements for a source installation of the Sage distribution: Sage 9.2 is still able to build on a system that only provides Python 2.x or Python 3.5 or older. In this case, the SageMath distribution builds its own copy of Python 3.
Support for Python 3.6, 3.8, and 3.9 added
Sage 9.2 has added support for Python 3.8 in #27754 and Python 3.9 in #30184.
Sage 9.2 has also added support for Python 3.6. This allows Sage to use the system Python on some older Linux distributions that are still in widespread use in scientific computing, including centos-8 and fedora-{26,27,28} (although Python 3.7.x packages are also available for these). See #29033 for more details.
Hence, Sage 9.2 conforms to (and exceeds) NumPy Enhancement Proposal 29 regarding Python version support policies.
If no suitable system Python, versions 3.6.x, 3.7.x, 3.8.x, or 3.9.x is found, Sage installs its own copy of Python 3 from source. The version of Python shipped with the Sage distribution has been upgraded from 3.7.3 to 3.8.5.
For developers: Using Python 3.6+ features in sagelib
Meta-ticket #29756 provides a starting point for a discussion of new features of the Python language and standard library to bring them to systematic use in sagelib. All features provided by Python 3.6 can be used immediately; features introduced in Python 3.7 or later will require backporting or a decision to drop the goal of supporting Python 3.6.
More details
- Trac tickets with keyword/component python3 in milestone 9.2
- See Python3-Switch for more details.
Package upgrades
The removal of support for Python 2 has enabled major package upgrades.
Major user-visible package upgrades below...
matplotlib
Dropping Python 2 support allowed us to make a major jump from matplotlib 2.2.5 to 3.3.1. See matplotlib's release notes for 3.0, 3.1, 3.2,3.3. In addition to improved output, this update will likely enable Sage developers to implement new features for plotting and graphics.
rpy2 and R
The rpy2 Python package is the foundation for SageMath's interface to R. Dropping Python 2 support allowed us to make the major upgrade from 2.8.2 to 3.3.5 in #29441; see the release notes for details.
We only did a minor upgrade of R itself in the Sage distribution, to 3.6.3, the latest in the 3.6.x series. Of course, if R 4.0.x is installed in the system, Sage will use it instead of building its own copy.
The SageMath developers are eager to learn from users how they use the SageMath-R interface, and what needs to be added to it to become more powerful. Let us know at sage-devel.
sphinx
Sage uses Sphinx to build its documentation. Sage 9.2 has updated Sphinx from 1.8.5 to 3.1.2; see Sphinx release notes for more information.
SymPy
SymPy has been updated from 1.5 to 1.6.2 in #29730, #30425. See the Release notes.
IPython, Jupyter notebook, JupyterLab
Dropping support for Python 2 allowed us to upgrade IPython from 5.8.0 to 7.13.0 in #28197. See the release notes for the 6.x and 7.x series.
We have also upgraded the Jupyter notebook from 5.7.6 to 6.1.1 in #26919; see the notebook changelog for more information. Besides, the pdf export of Jupyter notebooks has been fixed, so that LaTeX-typeset outputs are now rendered in the pdf file (#23330).
JupyterLab is now fully supported as an optional, alternative interface #30246, including interacts. To use it, install it first, using the command sage -i jupyterlab_widgets. Then you can start it using ./sage -n jupyterlab.
Normaliz
The optional package Normaliz, a tool for computations in affine monoids, vector configurations, lattice polytopes, rational cones, and algebraic polyhedra has been upgraded from 3.7.2 to 3.8.8, and PyNormaliz to version 2.12.
The upgrade adds support for incremental ("dynamic") computations, the computation of automorphism groups and refined triangulations of cones and polyhedra, and limited support for semiopen cones and polyhedra.
To install Normaliz and PyNormaliz, use sage -i pynormaliz.
SageTeX
Updated to version 3.5, improving Python 3 compatibility, also updated to version 3.5 on CTAN.
sws2rst + usage example
In ticket #28838, the command sage -sws2rst was resurrected via a new pip-installable package sage-sws2rst. It can be installed in Sage 9.2 using sage -i sage_sws2rst.
Below is an example of usage. First we download a sage worksheet (.sws) prepared for Sage Days 20 at CIRM (Marseille, 2010):
$ wget http://slabbe.org/Sage/2010-perpignan/CIRM_Tutorial_1.sws
$ ls
CIRM_Tutorial_1.swsWe translate the sws worksheet into a ReStructuredText syntax file (.rst) using sage -sws2rst. This creates also a directory of images:
$ sage -sws2rst CIRM_Tutorial_1.sws
Processing CIRM_Tutorial_1.sws
File at CIRM_Tutorial_1.rst
Image directory at CIRM_Tutorial_1_media
$ ls
CIRM_Tutorial_1_media CIRM_Tutorial_1.rst CIRM_Tutorial_1.swsThen, we can check that it works properly by looking at the generated rst file. Alternatively, we can translate it to a basic html file using rst2html:
$ rst2html.py CIRM_Tutorial_1.rst CIRM_Tutorial_1.html
CIRM_Tutorial_1.rst:176: (WARNING/2) Explicit markup ends without a blank line; unexpected unindent.
CIRM_Tutorial_1.rst:334: (WARNING/2) Inline strong start-string without end-string.As seen above, there are few warnings sometimes because the translation made by sws2rst is not 100% perfect, but most of it is okay:
$ firefox CIRM_Tutorial_1.htmlMoreover, one can use the sage -rst2ipynb script to translate the rst file obtained above to a Jupyter notebook:
$ sage -rst2ipynb ...9.1
Release Tour
SageMath 9.1 was released on May 21, 2020 (changelog).
- last release supporting both Python 2 and 3
- major portability improvements
- 551 tickets (PRs) merged, 106 contributors
Python 3 transition
SageMath 9.0 was the first version of Sage running on Python 3 by default. Sage 9.1 continues to support both Python 2 and Python 3.
In Sage 9.1, we have made some further improvements regarding Python 3 support. In particular, SageMath now supports underscored numbers PEP 515 (py3); the fix was done in Trac #28490:
sage: 1_000_000 + 3_000
1003000
The next release, SageMath 9.2, will remove support for Python 2. See Python3-Switch for more details.
Portability improvements, increased use of system packages
The SageMath distribution continues to vendor versions of required software packages ("SPKGs") that work well together.
In order to reduce compilation times and the size of the Sage installation, a development effort ongoing since the 8.x release series has made it possible to use many system packages provided by the OS distribution (or by the Homebrew or conda-forge distributions) instead of building SageMath's own copies.
This so-called "spkg-configure" mechanism runs at the beginning of a build from source, during the ./configure phase.
(See the sage-devel threads "Brainstorming about Sage dependencies from system packages" (May 2017) and "conditionalise installation of many spkg's?" (Nov 2017) for its origins and Trac #24919 for its initial implementation.)
Sage 9.1 is adding many packages to this mechanism, including openblas, gsl, r, boost, libatomic, cddlib, tachyon, nauty, sqlite, planarity, fplll, brial, flintqs, ppl, libbraiding, cbc, gfan, and python3. As to the latter, SageMath will now make use of a suitable installation of Python 3.7.x in your system by setting up a venv (Python 3 virtual environment).
New in Sage 9.1 is also a database of system packages equivalent to our SPKGs. At the end of a ./configure run, you will see messages like the following:
configure: notice: the following SPKGs did not find equivalent system packages: arb boost boost_cropped bzip2 ... yasm zeromq zlib
checking for the package system in use... debian
configure: hint: installing the following system packages is recommended and may avoid building some of the above [[SPKGs]] from source:
configure: $ sudo apt-get install libflint-arb-dev ... yasm libzmq3-dev libz-dev
configure: After installation, re-run configure using:
configure: $ ./config.status --recheck && ./config.status
We also use the same database to update our installation manual automatically.
Status of Cygwin support
Thanks to the hard work of our Cygwin maintainers, in particular during the 8.x release cycles, building Sage on Windows using Cygwin64 is fully supported. Sage 9.1 reflects this by integrating the instructions for building from source on Cygwin into its documentation.
For developers
For developers who wish to help improve the portability of SageMath, there is a new power tool: A tox configuration that automatically builds and tests Sage within Docker containers running various Linux distributions (ubuntu-trusty through -focal, debian-jessie through -sid, linuxmint-17 through -19.3, fedora-26 through -32, centos-7 and -8, archlinux, slackware-14.2), each in several configurations regarding what system packages are installed. Thus, it is no longer necessary for developers to have access to a machine running fedora-29, say, to verify whether the Sage distribution works there; instead, you just type:
tox -e docker-fedora-29-standard -- build ptest
The Dockerfiles are generated automatically on the fly using the same database of system packages that provides information to users. See the tutorial for developers: Portability testing of the Sage distribution using Docker and the Sage distro-package database from the [Global Virtual SageDays 109](https://researchseminars.org/seminar/SageDays109) and the new section on "testing on multiple platforms" in the Developer's Guide for details.
An entry point for developers who wish to improve the testing infrastructure is the Meta-Ticket #29060: Add Dockerfiles and CI scripts for integration testing. See also the broader Meta-Meta-Ticket #29133.
For packagers
Although we now have continuous integration environments for testing the interaction of the Sage distribution with most major Linux distributions, we are still missing a few. Adding them will enable all Sage developers to check that their changes do not break things on your distribution.
Package updates
We have only made minor updates to standard packages:
- dateutil -- 2.8.1 (from 2.5.3)
- fplll -- 5.3.2 (from 5.2.1)
- fpylll -- 0.5.1.dev (from 0.4.1dev)
- freetype -- 2.10.1
- m4ri -- 20200115
- m4rie -- 20200115
- matplotlib -- 2.2.5 (from 2.2.4)
- ntl -- 11.4.3 (from 11.3.2)
- numpy -- 1.16.6 (from 1.16.1)
- openblas -- 0.3.9
- pkgconfig -- 1.5.1 (from 1.4.0)
- pyzmq -- 19.0.0 (from 18.1)
- sage_brial -- 1.2.5 (this is the python module of the brial package)
- scipy -- 1.2.3 (from 1.2.0)
- sympy -- 1.5 (from 1.4)
- traitlets -- 4.3.3
We expect to make larger package updates in the 9.2 release.
For developers
Preparing and testing package updates has become easier. The new optional field upstream_url in checksums.ini holds an URL to the upstream package archive, see for example build/pkgs/numpy/checksums.ini. Note that, like the tarball field, the upstream_url is a template; the word VERSION is substituted with the actual version. The package can be updated by simply typing ./sage -package update numpy 3.14.59; this will automatically download the archive and update the build/pkgs/ information.
Developers who wish to test a package update from a Trac branch before the archive is available on a Sage mirror can do so by configuring their Sage tree using ./configure \--enable-download-from-upstream-url.
Every Sage developer now has easy access to "their own" set of 20 (40 for GitHub Pro accounts) two-core virtual machines running Linux, macOS, and Windows through GitHub Actions. To automatically test a branch on a multitude of our supported platforms, it suffices to create a fork of the sagemath/sage repository on GitHub, enable GitHub Actions, add the repository as a remote, create a tag and push the tag to the remote. After ... a ... while, your test results will be available --- like the ones at sagemath/sage Actions. We hope that this new testing infrastructure will reduce the FUD in the process of upgrading packages.
Polyhedral geometry
There is now a catalog for common polyhedral cones, e.g.
sage: cones.nonnegative_orthant(5)
5-d cone in 5-d lattice N
New features for polyhedra:
sage: P = polytopes.cube(intervals='zero_one') # obtain others than the standard cube
sage: P = matrix([[0,1,0],[0,1,1],[1,0,0]])*P # linear transformations
sage: it = P.face_generator() # a (fast and efficient) face generator
sage: next(it)
A 3-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: next(it)
A -1-dimensional face of a Polyhedron in ZZ^3
sage: f = next(it)
sage: f.normal_cone() # normal cone for faces
A 1-dimensional polyhedron in ZZ^3 defined as the convex hull of 1 vertex and 1 ray
sage: P.an_affine_basis() # an_affine_basis and a_maximal_chain
[A vertex at (0, 0, 0),
A vertex at (1, 1, 0),
A vertex at (0, 0, 1),
A vertex at (0, 1, 0)]
sage: P = polytopes.hypercube(4)
sage: P.flag_f_vector(0,3) # flag_f_vector is exposed
64
Regarding the optional package normaliz there are some news as well:
sage: P = polytopes.cube(intervals=[[0,1],[0,2],[0,3]], backend='normaliz')
sage: save(P, '/tmp/this_takes_very_long_so_we_save_it') # saving works now
sage: sage: P.h_star_vector() # compute the h_star_vector with normaliz
[1, 20, 15]
There are also some bug fixes and other improvements. ...
9.0
Release Tour
Sage 9.0 was released on Jan 1, 2020 (changelog), 348 tickets (PRs) merged, 116 contributors.
Python 3 transition
Just in time for the new decade, this is the first version of Sage running on Python 3 by default.
See Python3-Switch for more details
Three.js is the new default 3D viewer
Three.js has become the default viewer for 3D plots, in replacement of Jmol. Note that Jmol is still available, via the option _viewer='jmol' _ in the plot functions.
New plotting capabilities
Polyhedral Geometry
Sage uses a new algorithm to obtain the f-vector for polyhedra. This is the only memory efficient implementation for the f-vector at the time of writing and it is as fast or faster as other implementations:
sage: P = polytopes.permutahedron(7)
sage: %time P.incidence_matrix()
CPU times: user 679 ms, sys: 4.01 ms, total: 683 ms
Wall time: 682 ms
5040 x 127 dense matrix over Integer Ring (use the '.str()' method to see the entries)
sage: %time P.f_vector()
CPU times: user 901 ms, sys: 16 ms, total: 917 ms
Wall time: 916 ms
(1, 5040, 15120, 16800, 8400, 1806, 126, 1)There is ongoing work to improve this implementation. It is implemented in the combinatorial polyhedron of P, which is newly exposed:
sage: P.combinatorial_polyhedron()
A 6-dimensional combinatorial polyhedron with 126 facetsSage has the classical construction of the 120-cell of Coxeter from 1969. This construction is much faster than to realize it as generalized permutahedron so that even without the optional package normaliz you won't waste much time:
sage: %time P = polytopes.one_hundred_twenty_cell(backend='normaliz')
CPU times: user 942 ms, sys: 81.8 ms, total: 1.02 s
Wall time: 457 ms
sage: %time P = polytopes.one_hundred_twenty_cell(backend='field')
CPU times: user 15.9 s, sys: 87.2 ms, total: 16 s
Wall time: 16 s
sage: %time P = polytopes.one_hundred_twenty_cell(backend='normaliz', construction='as_permutahedron')
CPU times: user 18.6 s, sys: 137 ms, total: 18.8 s
Wall time: 18 sEhrhart polynomials are computable for lattice polytopes defined with base ring QQ:
sage: P = polytopes.cube()*1/1
sage: P.base_ring()
Rational Field
sage: P.ehrhart_polynomial()
8*t^3 + 12*t^2 + 6*t + 1Note that this computation requires optional package latte_int or normaliz.
There is a new method to obtain the boundary of complex simplicial polytopes:
sage: oc = polytopes.octahedron()
sage: oc.boundary_complex()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 8 facetsHyperplane arrangements have a new method center:
sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: A = H()
sage: A.center()
A 2-dimensional polyhedron in QQ^2 defined as the convex hull of 1 vertex and 2 linesThere are also some bug fixes and other improvements. For more details see the release notes for optimization and polyhedral geometry softwares interactions in Sage.
Group Theory
Cubic Braid Groups
Factor groups of the Artin braid groups such that their generators have order three can easily be declared, now. They show a lot of their known properties:
sage: C4 = CubicBraidGroup(4)
sage: C4Cl = C4.as_classical_group(); C4Cl
Subgroup with 3 generators (
[ E(3)^2 0 0] [ 1 -E(12)^7 0]
[-E(12)^7 1 0] [ 0 E(3)^2 0]
[ 0 0 1], [ 0 -E(12)^7 1],
[ 1 0 0]
[ 0 1 -E(12)^7]
[ 0 0 E(3)^2]
) of General Unitary Group of degree 3 over Universal Cyclotomic Field with respect to positive definite hermitian form
[-E(12)^7 + E(12)^11 -1 0]
[ -1 -E(12)^7 + E(12)^11 -1]
[ 0 -1 -E(12)^7 + E(12)^11]See also the reference manual.
Manifolds
The main novelty is the introduction of vector bundles (ticket #28159). For instance:
sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: TM = M.tangent_bundle(); TM
Tangent bundle TM over the 2-dimensional differentiable manifold M
sage: v = TM.section([-y, x], name='v'); v
Vector field v on the 2-dimensional differentiable manifold M
sage: v.display()
v = -y d/dx + x d/dy
sage: p = M((2,3), name='p'); p
Point p on the 2-dimensional differentiable manifold M
sage: TM.fiber(p)
Tangent space at Point p on the 2-dimensional differentiable manifold M
sage: v.at(p) in TM.fiber(p)
True
sage: v.at(p).display()
v = -3 d/dx + 2 d/dyOther new features are
- characteristic classes (ticket #27784)
- the construction of a vector frame from a family of predefined vector fields (ticket #28716)
- the handling of multiple symmetries and multiple contractions with index notation (ticket #28784)
- more control on the numerical ODE solver for integrated curves and geodesics (ticket #28707)
See the full changelog for details, as well as the list of improvements and bug fixes in this release.
EdgesView for graphs
An EdgesView is a read-only iterable container of edges enabling operations like for e in E and e in E. An EdgesView can be iterated multiple times, and checking membership is done in constant time. It avoids the construction of edge lists and so consumes little memory. It is updated as the graph is updated. Hence, the graph should not be updated while iterating through an EdgesView.
sage: G = Graph([(0, 1, 'C'), (0, 2, 'A'), (1, 2, 'B')])
sage: E = G.edges()
sage: E
[(0, 1, 'C'), (0, 2, 'A'), (1, 2, 'B')]
sage: type(E)
<class 'sage.graphs.views.EdgesView'>
sage: (0, 2) in E
False
sage: (0, 2, 'A') in E
True
sage: (2, 0, 'A') in E
True
sage: for e in E:
....: for ee in E:
....: print(e, ee)
....:
(0, 1, 'C') (0, 1, 'C')
(0, 1, 'C') (0, 2, 'A')
(0, 1, 'C') (1, 2, 'B')
(0, 2, 'A') (0, 1, 'C')
(0, 2, 'A') (0, 2, 'A')
(0, 2, 'A') (1, 2, 'B')
(1, 2, 'B') (0, 1, 'C')
(1, 2, 'B') (0, 2, 'A')
(1, 2, 'B') (1, 2, 'B')See http://doc.sagemath.org/html/en/reference/graphs/sage/graphs/views.html for more details.
Availability of Sage 9.0 and installation help
- Release announcements: See sage-release, sage-announce
- Availability in distributions: see repology.org: sagemath
- sagemath.org download page
Installation FAQ
- See sage-support, sage-devel.
- Debian/Ubuntu: Installation prerequisites, Recommended installation
- Sage 9.0 (and earlier) does not support compilation within a conda environment. Deactivate conda before building Sage.
- Sage 9.0 (and earlier) may not compile from source on some cutting edge Linux distributions such as Ubuntu 20.04 LTS (workaround) and Fedora 32. Use either a distribution package of Sage if that is available in your distribution, a binary distribution of 9.0, or compile a development version of Sage leading to the Sage 9.1 release.
Full Changelog: 8.9...9.0