Sage 9.3 Release Tour

released May 9, 2021

Linear and multilinear algebra

Bär–Faddeev–LeVerrier algorithm for the Pfaffian of skew-symmetric matrices

According to, 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

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

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:

   1 sage: fcube = polytopes.hypercube(4)
   2 sage: tfcube = fcube.face_truncation(fcube.faces(0)[0])
   3 sage: tfcube.facets()[-1]
   4 A 3-dimensional face of a Polyhedron in QQ^4 defined as the convex hull of 8 vertices
   5 sage: sp = tfcube.schlegel_projection(tfcube.facets()[-1])
   6 sage: sp.plot()  # The proper Schlegel diagram is shown

A different values of position changes the projection:

   1 sage: sp = tfcube.schlegel_projection(tfcube.facets()[4],1/2)
   2 sage: sp.plot()
   3 Graphics3d Object
   4 sage: sp = tfcube.schlegel_projection(tfcube.facets()[4],4)
   5 sage: sp.plot()
   6 Graphics3d Object

New features:

Implementation improvements

The zonotope construction got improved:


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


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

Major improvements in the backends:


Power Series Ring

Other additions, improvements, and key bug-fixes

Knot theory

Fixed some wrong answers for the unknot: 31001

Symbolic expressions

The predefined constant I in interactive sessions is now an element of the (embedded) number field ℚ[i] rather than a symbolic expression:

sage: I.parent()
Number Field in I with defining polynomial x^2 + 1 with I = 1*I

This allows it to be used in combination with other Sage objects without the coercion mechanism forcing the result to belong to the symbolic ring. For example, one now has

sage: (1.0 + I).parent()
Complex Field with 53 bits of precision

Expressions such as

sage: I + pi
pi + I

still yield symbolic results. The symbolic imaginary unit I remains available as SR(I) or SR.I(), or, for library code, as sage.symbolic.constants.I. Importing it from sage.symbolic is deprecated. (#18036)


Spheres added to the manifold catalog

Spheres of arbitrary dimension have been added to the manifold catalog, so that they can be initialized via manifolds.Sphere(n) (#30804). For instance:

sage: S3 = manifolds.Sphere(3)
sage: S3
3-sphere S^3 of radius 1 smoothly embedded in the 4-dimensional Euclidean space E^4

By default, a single chart is initialized: that of spherical coordinates:

sage: S3.atlas()
[Chart (A, (chi, theta, phi))]
sage: S3.default_chart() is S3.spherical_coordinates()

Note that A stands for the domain of spherical coordinates, which is a strict subset of S3:

sage: S3.spherical_coordinates().domain()                                                           
Open subset A of the 3-sphere S^3 of radius 1 smoothly embedded in the 4-dimensional Euclidean space E^4

The sphere S3 is considered as an embedded submanifold of the Euclidean space E4:

sage: S3.embedding().display()
iota: S^3 --> E^4
on A: (chi, theta, phi) |--> (x1, x2, x3, x4) = (cos(phi)*sin(chi)*sin(theta), sin(chi)*sin(phi)*sin(theta), cos(theta)*sin(chi), cos(chi))

S3 is a Riemannian manifold, with the metric induced by the Euclidean metric on E4:

sage: g = S3.induced_metric()
sage: g.display()
gamma = dchi*dchi + sin(chi)^2 dtheta*dtheta + sin(chi)^2*sin(theta)^2 dphi*dphi

Beside spherical coordinates, stereographic coordinates are available too. Those from the North pole (NP) are obtained by

sage: S3.stereographic_coordinates()
Chart (S^3-{NP}, (y1, y2, y3))

The above call has augmented the manifold atlas with various charts and subcharts, corresponding to stereographic coordinates from the North pole and from the South pole:

sage: S3.atlas()
[Chart (A, (chi, theta, phi)),
 Chart (S^3-{NP}, (y1, y2, y3)),
 Chart (S^3-{SP}, (yp1, yp2, yp3)),
 Chart (S^3-{NP,SP}, (y1, y2, y3)),
 Chart (S^3-{NP,SP}, (yp1, yp2, yp3)),
 Chart (A, (y1, y2, y3)),
 Chart (A, (yp1, yp2, yp3))]

All relevant change-of-coordinate formulas have been initialized. For instance, those relating spherical coordinates to stereographic ones are obtained as follows:

sage: spher = S3.spherical_coordinates()
sage: stereo = S3.stereographic_coordinates()
sage: A = spher.domain()
sage: stereoA = stereo.restrict(A)
sage: S3.coord_changes()[(spher, stereoA)].display()
y1 = -cos(phi)*sin(chi)*sin(theta)/(cos(chi) - 1)
y2 = -sin(chi)*sin(phi)*sin(theta)/(cos(chi) - 1)
y3 = -cos(theta)*sin(chi)/(cos(chi) - 1)

The change-of-coordinate formulas are automatically invoked to compute the coordinates of a point from previously known ones in a different chart. For instance:

sage: p = S3((pi/2, pi/3, pi/4), chart=spher)                                                       
sage: p                                                                                             
Point on the 3-sphere S^3 of radius 1 smoothly embedded in the 4-dimensional Euclidean space E^4
sage: spher(p)                                                                                      
(1/2*pi, 1/3*pi, 1/4*pi)
sage: stereo(p)                                                                                     
(1/4*sqrt(3)*sqrt(2), 1/4*sqrt(3)*sqrt(2), 1/2)

Use of ambient metric by default for the Hodge dual of a differential form

When asking for the Hodge dual of a differential form on a pseudo-Riemannian manifold without specifying the metric, the manifold's default metric is assumed (#31322). For example:

sage: E.<x,y,z> = EuclideanSpace()
sage: E.cartesian_coordinates().coframe()
Coordinate coframe (E^3, (dx,dy,dz))
sage: dx = E.cartesian_coordinates().coframe()[1]
sage: dx
1-form dx on the Euclidean space E^3
sage: dx.hodge_dual()
2-form *dx on the Euclidean space E^3
sage: dx.hodge_dual().display()
*dx = dy/\dz

Internal code improvements and bug fixes

Various improvements/refactoring of the code have been performed in this release:

* topological part: #30310, #31243

* differentiable part: #30284, #31215, #31255, #31273, #31323.

In addition, various bugs and typos have been fixed: #30174, #30830, #31202.


Automatic conversions from from floating-point numbers and symbolic expressions to real and complex intervals are deprecated and will be removed in the future. Code that relies on the existence of coercions


sage: RIF(1/3) + RR(1)


sage: 1.1*pi*CIF(i)

needs to be updated. (Other similar conversions, e.g. from RealField to RealBallField or from float to RealField, already need to be performed explicitly.) Beside leading to more consistent behavior, the removal of these coercions will make it easier to use interval arithmetic reliably. (#15114)


Three.js viewer

The thickness option for lines is now supported on more platforms, including Windows. It can now also be specified for surfaces in conjunction with mesh=True to control the thickness of the wireframe lines: 26410

   1 line = parametric_plot3d([x*cos(x),x*sin(x),x], (x,0,2*pi), thickness=10)
   2 surface = dodecahedron(mesh=True, thickness=10)
   3 show(line + surface)

A dark theme has been implemented for the viewer. Use theme='dark' for the new theme or theme='light' for the default. If you would prefer the dark theme to be the default, edit your Sage start-up script, adding the following: 30462

   1 sage.plot.plot3d.base.SHOW_DEFAULTS['theme'] = 'dark'

You may now set the title of the HTML pages generated by the viewer, which can be helpful when having several such tabs open at once: 30612

   1 show(dodecahedron(), page_title="My Favorite Polyhedron")

The font and opacity of the axis labels can now be customized similar to what was added in 9.2 for text3d. Pass either a single dictionary of options for all three axes or a list of three dictionaries, one per axis: 30628

   1 style = [dict(color='red', fontweight='bold'),
   2          dict(fontsize=20, fontfamily='Times New Roman, Georgia, serif'),
   3          dict(fontstyle='italic', opacity=0.5)]
   4 show(dodecahedron(), axes_labels_style=style)

Fixed clipping issues for plots that are very small or very large. 30613

New and upgraded packages, system packages


SageMath 9.3 continues to support system Python versions 3.6.x, 3.7.x, 3.8.x, and 3.9.x; support for Python 3.6.x is deprecated and will be removed in the SageMath 9.4 development cycle. If no suitable system Python 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 3.8.5 to 3.9.2. #30589, #31318, #31419

New packages

A number of user packages have been added as optional packages:

Other package upgrades

Upgrade tickets, milestone 9.3

SageMath on

The SageMath distribution is now listed as a repository sagemath_stable on repology (tracking the latest stable SageMath release). In addition, the list of packages of the sagemath_develop repository allows developers to see which packages are in need of updating.

System package information for OpenSUSE

In particular, users of OpenSUSE Tumbleweed will notice the information about system packages that the Sage distribution can use.

Chapter on packages in the Sage reference manual

We have consolidated the information about packages into a single source, build/pkgs/ in the source tree, from which we generate package lists for the reference manual #29655. Also the Sage website now points to this consolidated package directory. Meta-ticket #31164 tracks the task of adding user packages to build/pkgs/.

For developers and packagers: Version constraints for Python packages

For all Python packages in the Sage distribution, build/pkgs/SPKG/install-requires.txt now encodes version constraints (such as lower and upper bounds) #30719. The constraints are in the format of the install_requires key of setup.cfg or

The files may include comments (starting with #) that explain why a particular lower bound is warranted or why we wish to include or reject certain versions. For example:

$ cat build/pkgs/sphinx/package-version.txt 
$ cat build/pkgs/sphinx/install-requires.txt
# gentoo uses 3.2.1
sphinx >=3, <3.3

The comments may include links to Trac tickets, as in the following example:

$ cat build/pkgs/packaging/install-requires.txt
packaging >=18.0
# Trac #30975: packaging 20.5 is known to work but we have to silence "DeprecationWarning: Creating a LegacyVersion"

The currently encoded version constraints are merely a starting point. Downstream packagers and developers are invited to refine the version constraints based on their experience and tests. When a package update is made in order to pick up a critical bug fix from a newer version, then the lower bound should be adjusted.

In Sage 9.3, the new files are included only for documentation purposes and to facilitate development and packaging. An effort is underway in #30913 to use the new files to generate dependency metadata for the Sage library. There is no mechanism yet to use system Python packages; see Meta-ticket #29013 for this task.

For developers: Setting up Python packages from PyPI as Sage packages

Setting up Python packages from PyPI as Sage packages has become easier #30974.

$ ./sage -package create scikit_spatial --pypi
Downloading tarball to /Users/mkoeppe/s/sage/sage-rebasing/worktree-algebraic-2018-spring/upstream/scikit-spatial-5.0.0.tar.gz
$ ls build/pkgs/scikit_spatial/
SPKG.rst                dependencies            package-version.txt     type
checksums.ini           install-requires.txt
$ cat build/pkgs/scikit_spatial/SPKG.rst 
scikit_spatial: Spatial objects and computations based on NumPy arrays


Spatial objects and computations based on NumPy arrays.


BSD license

Upstream Contact

Note that this new command does not check the dependencies of the Python package; this still needs to be done manually.

The above command creates a "normal" package (with tarball information in checksums.ini). To create a package as a "pip" package (which will be obtained directly from PyPI on an install), use ./sage -package create scikit_spatial --pypi --source pip instead.

Cleaning of the Sage codebase to conform to best practices

Major progress has been made since SageMath 9.0 in a long term effort to clean up the Sage codebase so it conforms to best practices. Since SageMath 9.2, various coding style checkers have been available via ./sage -tox.

The command ./sage -tox -e pycodestyle-minimal uses pycodestyle (formerly known as pep8) in a minimal configuration. By passing --count -qq we can reduce the output to only show the number of style violation warnings.

$ ./sage -tox -e pycodestyle-minimal -- --count -qq worktree-9.0/src/sage
pycodestyle-minimal installed: pycodestyle==2.7.0
pycodestyle-minimal run-test-pre: PYTHONHASHSEED='1305087318'
pycodestyle-minimal run-test: commands[0] | pycodestyle --select E401,E701,E702,W605 --count -qq ../worktree-9.0/src/sage
$ ./sage -tox -e pycodestyle-minimal -- --count -qq worktree-9.1/src/sage
$ ./sage -tox -e pycodestyle-minimal -- --count -qq worktree-9.2/src/sage
$ ./sage -tox -e pycodestyle-minimal -- --count -qq worktree-9.3/src/sage

The command ./sage -tox -e pycodestyle runs a more thorough check.

$ ./sage -tox -e pycodestyle -- --count -qq worktree-9.0/src/sage
pycodestyle recreate: /Users/mkoeppe/s/sage/sage-rebasing/src/.tox/pycodestyle
pycodestyle installdeps: pycodestyle
pycodestyle installed: pycodestyle==2.7.0
pycodestyle run-test-pre: PYTHONHASHSEED='906952231'
pycodestyle run-test: commands[0] | pycodestyle --count -qq ../worktree-9.0/src/sage
$ ./sage -tox -e pycodestyle -- --count -qq worktree-9.1/src/sage
$ rm -f worktree-9.2/src/tox.ini
$ ./sage -tox -e pycodestyle -- --count -qq worktree-9.2/src/sage
$ ./sage -tox -e pycodestyle -- --count -qq worktree-9.3/src/sage
565     E111 indentation is not a multiple of four
202     E114 indentation is not a multiple of four (comment)
70      E115 expected an indented block (comment)
184     E116 unexpected indentation (comment)

Finally, there is ./sage -tox -e codespell, which uses codespell to find misspelled words.

$ ./sage -tox -e codespell -- --count worktree-9.0/src/sage
$ ./sage -tox -e codespell -- --count worktree-9.1/src/sage
$ ./sage -tox -e codespell -- --count worktree-9.2/src/sage
$ ./sage -tox -e codespell -- --count worktree-9.3/src/sage
../worktree-9.3/src/sage/ multible ==> multiple
../worktree-9.3/src/sage/misc/ supress ==> suppress
../worktree-9.3/src/sage/misc/ occurance ==> occurrence
../worktree-9.3/src/sage/misc/ instanciation ==> instantiation

Modularization and packaging of sagelib

A source tarball of the Sage distribution (or a worktree of the Sage git repository after running the ./bootstrap script) contains several self-contained source trees of Python distribution packages in build/pkgs/*/src/, which can be packaged, built and installed by standard Python procedures:

The following Python distribution packages exist in Sage 9.3:

In the course of the modularization effort of Meta-ticket #29705, in the Sage 9.4 series we expect to add many more distribution packages in the same format.

Configuration changes

Editable ("in-place", "develop") installs of the Sage library

Use ./configure --enable-editable to configure the Sage distribution to install the Sage library in "develop" ("editable", "in-place") mode instead of using the Sage library's custom incremental build system. #31377

It has the benefit that to try out changes to Python files, one does not need to run ./sage -b any more; restarting Sage is enough. It may also have benefits in certain develop environments that get confused by sagelib's custom build system.

Note that in an editable install, the source directory will be cluttered with build artifacts (but they are .gitignored). This is normal.

Use of system Jupyter notebook / JupyterLab

The Sage Installation Manual now has instructions on how to run the SageMath Jupyter kernel in a system Jupyter notebook or JupyterLab. #30476

If you intend to (a) only use Sage in a terminal, or (b) only use the SageMath kernel with your system Jupyter notebook or JupyterLab, you can now build Sage without the notebook components, by using ./configure --disable-notebook. #30383 #31278

Availability of Sage 9.3 and installation help

Sage 9.3 was released on 2021-05-09.


The Sage source code is available in the sage git repository, and the self-contained source tarballs are available for download.

SageMath 9.3 supports all platforms that were supported by Sage 9.2, and a few additional ones. Notably, Sage 9.3 adds support for building Sage from source on macOS 11 ("Big Sur") (this requires an installation of homebrew's gcc@10 package). Moreover, on macOS, Sage 9.3 no longer refuses to build on systems that have an installation of MacPorts. #31641

Sage 9.3 has been tested to compile from source on a wide variety of platforms, including:

Installation FAQ

See in the source distribution for installation instructions.

See sage-release, sage-devel.

Availability in distributions and as binaries

The easiest way to install Sage 9.3 is through a distribution that provides it, see sagemath.

A binary build of release candidate 9.3.rc2 for macOS as a signed and notarized app is available for testing.

sage 9.3 now available on Apple silicon through conda

Binaries for Linux platforms are available.

An installer for Sage 9.3 on Windows is available.

Alternative installation methods using pip

There are new installation methods using pip.

   $ python3 -m pip install sage_conf

This will download the distribution package (sdist) sage_conf from PyPI, which will build the non-Python components of the Sage distribution in a subdirectory of $HOME/.sage.

After installation of sage_conf, a wheelhouse containing wheels of various Python packages is available; type ls $(sage-config SAGE_SPKG_WHEELS) to list them and python3 -m pip install $(sage-config SAGE_SPKG_WHEELS)/*.whl to install them.

After this, you can install the Sage library, using any of these options:

Also pre-built wheels are available for some platforms. See for more information.

More details

ReleaseTours/sage-9.3 (last edited 2022-05-14 16:52:08 by mkoeppe)