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10.0

20 May 23:59
@mkoeppe mkoeppe
10.0
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10.0

Release Notes

This release includes

  • support for GCC 13; major package upgrades
  • new optimization solvers via CVXpy, major new features in combinatorics and algebra

What's Changed

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Contributors

roed314, hemmecke, and 54 other contributors
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9.8

11 Feb 16:23
@dimpase dimpase
9.8
c000c95
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9.8

Release Notes

Sage 9.8 was released on Feb 11, 2023 (changelog), 402 tickets/PRs merged. This is the first release after our move from Trac to GitHub and

  • adds support for Python 3.11, removes support for GCC older than 8
  • includes major new solvers, major package upgrades

What's Changed

Merged in sage-9.8.beta0:

#12696: Lorenz Panny: rename .rational_reconstruct() to .rational_reconstruction() for consistency [Reviewed by Kwankyu Lee]
#17965: Frédéric Chapoton: Uniformize the API to compute the inverse of an element [Reviewed by Travis Scrimshaw]
#20467: Frédéric Chapoton: stackoverflow in Maxima integral [Reviewed by Nils Bruin]
#29619: Matthias Koeppe: Matrix and Components should have a sparse iterator [Reviewed by Eric Gourgoulhon]
#30229: Matthias Koeppe: Submodules of TensorFreeModule defined by the symmetries of a Components object [Reviewed by Eric Gourgoulhon]
#30241: Matthias Koeppe: New implementation class FiniteRankDualFreeModule [Reviewed by Eric Gourgoulhon]
#32297: Linden Disney-Hogg: Abel-Jacobi map on Riemann surfaces [Reviewed by Nils Bruin, Travis Scrimshaw]
#32669: Dennis Jahn: Adding upper and lower Bruhat cones of M. Dyer to sage/combinat/root_system/reflection_group_real.py [Reviewed by Frédéric Chapoton, Travis Scrimshaw]
#32921: Gabriel F. Lipnik: k-recursive sequences with inhomogeneities [Reviewed by Clemens Heuberger]
#33002: Sébastien Labbé: Method tikz of polyhedron class can now return an object of type TikzPicture [Reviewed by Laith Rastanawi]
#33062: Matthias Koeppe: GH Actions (docker): Run a job for "make build-local" first, cache image for job "make build" [Reviewed by Dima Pasechnik]
#33093: John Palmieri: 5 doctests failing related to external program octave [Reviewed by Frédéric Chapoton]
#33431: Matthias Koeppe: Category of submodules of sage.modules.free_module.FreeModule_ambient_pid [Reviewed by Travis Scrimshaw]
#33613: Matthias Koeppe: Add pip packages to conda environments, add missing conda.txt files [Reviewed by Dima Pasechnik]
#33640: Frédéric Chapoton: sage fails to factor some easy expressions [Reviewed by David Lowry-Duda]
#33981: Matthias Koeppe: latte_int: ExponentialSubst.h:62:3: error: ISO C++17 does not allow dynamic exception specifications [Reviewed by John Palmieri]
#34118: Oscar Benjamin, Matthias Koeppe: Update sympy to 1.11.1 [Reviewed by Matthias Koeppe, Antonio Rojas, Eric Gourgoulhon]
#34195: Matthias Koeppe: sage.geometry.polyhedron: More # optional - sage.rings.number_field [Reviewed by Jonathan Kliem]
#34203: Matthias Koeppe: Find and uninstall broken installed SPKGs and wheels [Reviewed by John Palmieri]
#34228: Matthias Koeppe: tox -e docker-...-incremental [Reviewed by Dima Pasechnik]
#34235: Matthias Koeppe: Error in comparing two Symbolic Ring elements [Reviewed by Travis Scrimshaw]
#34242: Julian Rüth: Update docker build [Reviewed by MartinPepin]
#34266: Matthias Koeppe: Drop support for GCC < 8 in Sage 9.8 [Reviewed by Dima Pasechnik]
#34282: Sebastian Oehms, Kwankyu Lee: Make feature TeXFile check latex first [Reviewed by Kwankyu Lee, Sebastian Oehms]
#34287: Frédéric Chapoton: modernize super in crystals and asymptotics [Reviewed by Travis Scrimshaw]
#34290: Frédéric Chapoton: fix some W391 [Reviewed by Travis Scrimshaw]
#34313: David Coudert: pycodestyle cleanup in strongly_regular_db.pyx (part 3) [Reviewed by Kwankyu Lee]
#34320: Frédéric Chapoton: fix E251 in schemes [Reviewed by David Coudert]
#34323: Frédéric Chapoton: fix E251 in groups [Reviewed by David Coudert]
#34342: Frédéric Chapoton: fix various suggestions by lgtm [Reviewed by David Coudert]
#34345: Trevor K. Karn: Method to get cells in horizontal/vertical border strip [Reviewed by Travis Scrimshaw]
#34350: Travis Scrimshaw: speed up powers of lazy Taylor series [Reviewed by Martin Pepin]
#34379: Travis Scrimshaw: Implement a hook to access free (graded) resolutions [Reviewed by Kwankyu Lee]
#34393: Frédéric Chapoton: add method "tensor_factors" to tensor products [Reviewed by Matthias Koeppe, Travis Scrimshaw]
#34412: Travis Scrimshaw: q-commuting polynomials [Reviewed by Frédéric Chapoton]
#34427: Matthias Koeppe: TensorFreeModule.isomorphism_with_fixed_basis [Reviewed by Eric Gourgoulhon]
#34449: Matthias Koeppe: Add package pyproject_metadata [Reviewed by François Bissey]
#34451: Matthias Koeppe: sage.tensor: Canonicalize sym, antisym [Reviewed by Eric Gourgoulhon]
#34453: Travis Scrimshaw: Tensor products of commutative algebras do not know they are commutative rings [Reviewed by Frédéric Chapoton]
#34466: Lorenz Panny: fix various linter errors [Reviewed by Frédéric Chapoton]
#34467: Lorenz Panny: fix random doctest failure in EllipticCurveHom_velusqrt [Reviewed by Kwankyu Lee]
#34474: Eric Gourgoulhon: Make FiniteRankFreeModule.tensor_module(0, 1) return the dual [Reviewed by Matthias Koeppe]
#34478: Matthias Koeppe: pynormaliz: Add distros information for conda, arch [Reviewed by Isuru Fernando]
#34484: Matthias Koeppe: Methods quotient, quotient_module are not the same for some modules from sage.modules [Reviewed by John Palmieri, Travis Scrimshaw]
#34490: Thierry Monteil: sage --package clean command to remove outdated source tarballs [Reviewed by David Coudert]
#34491: Matthias Koeppe: Upgrade igraph to 0.9.10 [Reviewed by David Coudert]
#34494: Frédéric Chapoton: better monomials in shuffle algebras [Reviewed by Travis Scrimshaw]
#34495: Matthias Koeppe: Construction for invariant/equivariant submodules [Reviewed by Travis Scrimshaw]
#34498: Matthias Koeppe: Update igraph, python_igraph to 0.10.x [Reviewed by David Coudert]
#34505: Frédéric Chapoton: using items in indexed_elements.pyx [Reviewed by Travis Scrimshaw]
#34509: Matthias Koeppe: Make IndexedFreeModuleElement compatible with collections.abc, change method support to return a SupportView [Reviewed by Travis Scrimshaw]
#34510: Trevor K. Karn: Add to check for fixed total/length in IntegerVector [Reviewed by Travis Scrimshaw]
#34514: David Roe: Incorrect decomposition type for relative number fields [Reviewed by Travis Scrimshaw]
#34519: Marc Mezzarobba: Fix + extend msolve interface [Reviewed by Matthias Koeppe]
#34527: Trevor K. Karn: Make Compositions into a collections.abc.Sequence [Reviewed by Travis Scrimshaw, Matthias Köppe]
#34535: Trevor K. Karn: Segfault when floor dividing in infinite polynomial ring [Reviewed by Travis Scrimshaw]
#34541: Frédéric Chapoton: enhance free dendriform algebras [Reviewed by Travis Scrimshaw]
#34561: Jan Groenewald: Documentation: There is no package in debian/ubuntu called openssh [Reviewed by Matthias Koeppe]
#34568: Frédéric Chapoton: rename example to QuaternionicProjectivePlane [Reviewed by John Palmieri]

Merged in sage-9.8.beta1:

#21129: Paul Fili, Holly Krieger, Jing Guo: Arakelov-Zhang pairing of rational maps [Reviewed by Alexander Galarraga]
#25046: Lorenz Panny: Add warning for is_prime(n) that produce correct but unexpected output [Reviewed by Kwankyu Lee]
#31664: Matthias Koeppe: Add package msolve 0.4.4 (multivariate polynomial system solver) [Reviewed by Marc Mezzarobba]
#32483: Matthias Koeppe: Remove workarounds for packages that use flit_core or poetry_core as build system [Reviewed by John Palmieri]
#33805: Dima Pasechnik: use Homebrew's primecount [Reviewed by Matthias Koeppe]
#33812: Matthias Koeppe: Refactor distributions sagemath-{objects,categories} through sagemath-{environment,repl} [Reviewed by Kwankyu Lee]
#33850: Sophia Elia: Unstable doctests in equivariant Ehrhart methods [Reviewed by Matthias Koeppe]
#33999: Xavier Caruso: Reduced charpoly for skew polynomials [Reviewed by Travis Scrimshaw, Frédéric Chapoton]
#34110: Matthias Koeppe: numpy 1.23.x [Reviewed by Dima Pasechnik, John Palmieri]
#34212: Jing Guo: log_embedding for number field elements [Reviewed by Alexander Galarraga]
#34272: Matthias Koeppe: Update pip to 22.2.2 [Reviewed by Kwankyu Lee]
#34314: David Coudert: pycodestyle cleanup in strongly_regular_db.pyx (part 4) [Reviewed by Matthias Koeppe]
#34344: Matthias Koeppe: Update setuptools 63.4.3, setuptools_wheel 65.x, tomlkit 0.11.4, importlib_resources 5.9.0, importlib_metadata 4.12.0, distlib 0.3.6, hatchling 1.10.0, pathspec 0.10.1 [Reviewed by John Palmieri]
#34358: David Coudert: pycodestyle cleanup in src/sage/graphs/generic_graph.py (part 4) [Reviewed by Kwankyu Lee]
#34375: Travis Scrimshaw: Replace sage.algebras.yangian.GeneratorIndexingSet with cartesian_product [Reviewed by Matthias Koeppe]
#34383: Martin Rubey: revert for LazyTaylorSeries and LazySymmetricFunction is missing [Reviewed by Travis Scrimshaw]
#34392: David Coudert: pycodestyle cleanup in src/sage/graphs/generic_graph.py (part 5) [Reviewed by Matthias Koeppe]
#34409: Lorenz Panny: remove experimental warning for composite elliptic-curve isogenies [Reviewed by Kwankyu Lee]
#34413: Martin Rubey: implement derivatives of lazy series [Reviewed by Travis Scrimshaw]
#34422: Martin Rubey: implement functorial composition of lazy symmetric functions [Reviewed by Travis Scrimshaw]
#34423: Martin Rubey: implement arithmetic product of lazy symmetric functions [Reviewed by Travis Scrimshaw]
#34432: Samuel Lelièvre: Upgrade: jupyter-packaging 0.12.3 [Reviewed by Matthias Koeppe]
#34435: Trevor K. Karn: Add method to trim trailing zeros from IntegerVector [Reviewed by Travis Scrimshaw]
#34448: Matthias Koeppe: Put tensor modules of FiniteRankFreeModule in Modules().TensorProducts() [Reviewed by Eric Gourgoulhon]
#34454: Matthias Koeppe: sage --package create: Add option --source wheel [Reviewed by Kwankyu Lee]
#34462: Kwankyu Lee: Uniformize the headline: finite rings [Reviewed by Matthias Koeppe]
#34465: John Palmieri: Invalid escape sequence in special.py [Reviewed by Dima Pasechnik]
#34468: Frédéric C...

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9.7

01 Oct 08:14
@dimpase dimpase
9.7
627b2bd
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9.7

Release Notes

Sage 9.7 was released on Sep 19, 2022 (changelog), 549 tickets (PRs) merged, 92 contributors. This release

  • removes support for GCC older than 6.3, removes support for Python 3.7
  • adds new code in algebra, number theory, elliptic curves; major package upgrades; new developer tools
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9.6

23 Jul 11:35
@dimpase dimpase
9.6
bf6aeb9
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9.6

Release Tour

Sage 9.6 was released on May 15, 2022 (changelog), 497 tickets (PRs) merged. This release includes

  • support for GCC 12; major package upgrades
  • major improvements to user interface and graphics; new developer tools

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 jupyterlab

Also two new interface variants are provided:

./sage -n nbclassic

and

./sage -n retrolab

LaTeX 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.

editor

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

tikz_arrow

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 file

Another 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()

tikz_graph

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()

tikz_polyhedron

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()

tikz_hello

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()

tikz_tableau

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')

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)

cmap_twilight_contour

sage: complex_plot((x - 5)*sqrt(x), (-10, 10), (-10, 10), cmap='twilight', plot_points=500, tiled=True)

cmap_twilight_tiled

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](https://github.com/sagemath/sage...

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9.5

24 Feb 16:08
@dimpase dimpase
9.5
439907f
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9.5

Release Tour

SageMath version 9.5 was released on Jan 30, 2022 (changelog), 663 tickets (PRs) merged, 94 contributors. This release includes

  • 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

Symbolics

Changes to symbolic expressions

  • symbolic_expression is 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 SR marked optional in sage.tensor.modules (#32712)
  • sage.tensor.modules made independent from sage.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

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](htt...
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9.4

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Release Tour

SageMath 9.4 was released on Aug 22, 2021 (changelog), 442 tickets (PRs) merged, 73 contributors. This release

  • adds support for GCC 11, removes support for Python 3.6
  • includes 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
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. #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_()
EmptySet

See 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
False

ConvexSet_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().un...
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9.3

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Release Tour

SageMath 9.3 was released on May 9, 2021 (changelog), 679 tickets (PRs) merged, 112 contributors. This release includes

  • support for macOS 11 "Big Sur"
  • major package upgrades

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

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:

    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:

    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_prec is 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.<x> = [[PowerSeriesRing]](QQ, default_prec=10)
    sage: sin(x)
    x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 + O(x^10)
    sage: R.<x> = [[PowerSeriesRing]](QQ, default_prec=15)
    sage: sin(x)
    x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^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

    sage: lie_algebras.SymplecticDerivation(QQ, 4)
    Symplectic derivation Lie algebra of rank 4 over Rational Field
    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,...
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9.2

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Release Tour

SageMath 9.2 was released October 24, 2020 (changelog), 727 tickets (PRs) merged, 134 contributors. This release includes

  • 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

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.sws

We 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.sws

Then, 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.html

Moreover, one can use the sage -rst2ipynb script to translate the rst file obtained above to a Jupyter notebook:

`...

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9.1

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Release Tour

SageMath 9.1 was released on May 21, 2020 (changelog), 551 tickets (PRs) merged, 106 contributors. This is the last release supporting both Python 2 and 3 and contains major portability improvements.

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 ot...

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9.0

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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 facets

Sage 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 s

Ehrhart 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 + 1

Note 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 facets

Hyperplane 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 lines

There 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/dy

Other 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

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

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