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10.3

20 Mar 01:40
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10.3

SageMath version 10.3 (released Mar 19, 2024)

Release Notes

  • adds initial support for Python 3.12
  • JupyterLab, Notebook 7, major package upgrades
  • major new developer tools

Sage-10.3-Release-Tour

What's Changed

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saraedum, thecaligarmo, and 54 other contributors
Assets 270
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10.2

03 Dec 11:54
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10.2

SageMath version 10.2 (released Dec 3, 2023)

Release Notes

  • major package upgrades, Cython 3
  • removes support for Python 3.8 and GCC older than 8.4

Sage-10.2-Release-Tour

What's Changed

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rburing, ivmai, and 44 other contributors
Assets 241
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10.1

23 Aug 10:00
@dimpase dimpase
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10.1

SageMath version 10.1 (released Aug 20, 2023)

Release Notes

  • major package upgrades; new development tools

Sage-10.1-Release-Tour

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remyoudompheng, fingolfin, and 46 other contributors
Assets 3
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10.0

20 May 23:59
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10.0

SageMath version 10.0 (released May 20, 2023)

Release Notes

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

Release Tour 10.0

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roed314, hemmecke, and 55 other contributors
Assets 2
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9.8

11 Feb 16:23
@dimpase dimpase
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9.8

SageMath version 9.8 (released Feb 11, 2023)

Release Notes

  • first release after our move from Trac to GitHub
  • adds support for Python 3.11, removes support for GCC older than 8
  • major new solvers, major package upgrades
  • 402 tickets/PRs merged

Sage 9.8 Release Tour

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: Fr...

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9.7

01 Oct 08:14
@dimpase dimpase
9.7
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9.7

SageMath version 9.7 (released Sep 19, 2022)

Release Notes

  • removes support for GCC older than 6.3, removes support for Python 3.7
  • new code in algebra, number theory, elliptic curves; major package upgrades; new developer tools
  • 549 tickets (PRs) merged, 92 contributors

Sage 9.7 Release Tour

changelog

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9.6

23 Jul 11:35
@dimpase dimpase
9.6
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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 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

#...

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9.5

24 Feb 16:08
@dimpase dimpase
9.5
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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_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](https://trac.sagemath...
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9.4

27 Aug 09:49
@dimpase dimpase
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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
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().union(polytopes.tetrahedron())      ...
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9.3

12 May 08:12
@dimpase dimpase
9.3
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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

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_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. = [[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...

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