Sage 3.3 Release Tour

Sage 3.3 was released on FIXME. For the official, comprehensive release note, see sage-3.3.txt. The following points are some of the foci of this release:

• Clean up various doctest failures from 3.2.3
• Fix some build issues from 3.2.3 on the new set of supported images
• Merge small to medium sized patches ready to go in

• Switch to eMPIRe for multi-precision integers and rationals

• Upgrade Python to version 2.5.4 upstream release

• Switch to FLINT for univariate polynomial arithmetic over Z/nZ

• Upgrade NetworkX to version 0.99 upstream release

• Upgrade cddlib to version 0.94f upstream release

• Some improvements to the lrs spkg

• PyNaC interface enhancements

• Update the readline spkg

• Update the ATLAS spkg

• Update the NTL spkg

• Upgrade M4RI to version 20090105 upstream release

• Upgrade jsMath to version 3.6 upstream release

• Upgrade GAP to version 4.4.12 upstream release

• Upgrade GUAVA to version 3.9 upstream release

• Upgrade Jmol to version 11.6 upstream release

• Upgrade matplotlib to version 0.98.5.3-svn6910 upstream release

• Upgrade libpng to version 1.2.34 upstream release

All tickets in the 3.3 milestone can be found on the trac server. Here's a summary of features in this release, categorized under various headings.

Algebra

• Transitivity for permutation groups (William Stein) -- In the permutation group module permgroup.py, the query function is_transitive() returns whether or not the group is transitive on [1..G.degree()]. A few surrounding docstrings are fixed and doctest coverage for the module sage.groups.perm_gps.permgroup.py is now 100%.

• Update the ATLAS spkg (Michael Abshoff).

• New function is_unit() for symbolic ring (Florent Hivert) -- The new function sage.calculus.calculus.SymbolicExpressionRing.is_unit() returns whether or not an element of the symbolic ring is a unit.

Algebraic Geometry

• Improved precision and performance when calculating analytic rank (William Stein) -- When calculating the analytic rank of an elliptic curve, the default is to use Cremona's gp script, where the precision is automatically doubled until it doesn't fail. The precision is started at 16 rather than the previous default precision. The computation is now about 3 times faster usually by starting off using this smaller precision. Here's an example:

  # BEFORE
  sage: E = EllipticCurve('5077a')
  sage: time E.analytic_rank()
  CPU times: user 0.01 s, sys: 0.01 s, total: 0.02 s
  Wall time: 0.21 s
  
  # AFTER
  sage: E = EllipticCurve('5077a')
  sage: time E.analytic_rank()
  CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
  Wall time: 0.06 s

  And another:

  # BEFORE
  sage: time elliptic_curves.rank(4)[0].analytic_rank()
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.50 s
  
  # AFTER
  sage: time elliptic_curves.rank(4)[0].analytic_rank()
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.33 s

• Weil pairing (David Moller Hansen, John Cremona) -- A basic framework for Weil pairing on elliptic curves using Miller's algorithm as contained in Proposition 8 of the following paper:
  • Victor S. Miller. "The Weil pairing, and its efficient calculation". Journal of Cryptology, 17(4):235-261, 2004.

Basic Arithmetic

• ivalue field in integer_mod.pyx is no longer public (Craig Citro) -- The ivalue field for IntegerMod_int is no longer public. This gives about a 1.5 to 2X speed-up when multiplying IntegerMod_ints. Here's an example:

  # BEFORE
  sage: R = Integers(100) ; x = R(3) ; y = R(5)
  sage: timeit('x*y')
  625 loops, best of 3: 403 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 370 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 410 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 405 ns per loop
  
  # AFTER
  sage: R = Integers(100) ; x = R(3) ; y = R(5)
  sage: timeit('x*y')
  625 loops, best of 3: 190 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 213 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 174 ns per loop
  sage: timeit('x*y')
  625 loops, best of 3: 191 ns per loop

• Some fixes for is_perfect_power() and bessel_J(0,0) (Craig Citro, Robert Bradshaw, Robert L. Miller) -- A temporary work around for an upstream bug in GMP when using is_perfect_power(). Resolved a Pari interface bug when using bessel_J(0,0).

• Improved performance for generic polynomial rings, and for univariate polynomial arithmetic over Z/nZ[x] (Yann Laigle-Chapuy, Martin Albrecht, Burcin Erocal) -- Improved performance when performing modulo arithmetic between elements of a generic polynomial ring. Univariate polynomial arithmetic over Z/nZ[x] now has considerable speed-up at approximately 20x.

  # BEFORE
  sage: P.<x> = PolynomialRing(GF(7))
  sage: type(x)
  <type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p'>
  sage: f = P.random_element(100)
  sage: g = P.random_element(100)
  sage: %timeit f*g
  1000 loops, best of 3: 445 µs per loop
  
  # AFTER
  sage: P.<x> = PolynomialRing(GF(7))
  sage: type(x)
  <type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
  sage: f = P.random_element(100)
  sage: g = P.random_element(100)
  sage: %timeit f*g
  100000 loops, best of 3: 7.92 µs per loop

• Technical preview of David Harvey's zn_poly library exposed to the Sage library (Martin Albrecht).

  def f(p,n):
      P = PolynomialRing(GF(next_prime(p)),'x')
      f = P.random_element(n)
      g = P.random_element(n)
  
      t0 = cputime()
      r0 = f*g
      t0 = cputime(t0)
  
      t1 = cputime()
      r1 = f._mul_zn_poly(g)
      t1 = cputime(t1)
  
      assert(r0 == r1)
  
      return p,n,t0,t1
  
  for i in range(21): 
     f(2**47,2**i)

  returns on sage.math

  # (140737488355328, 1, 0.0, 0.0)
  # (140737488355328, 2, 0.0, 0.0)
  # (140737488355328, 4, 0.00099999999999766942, 0.0)
  # (140737488355328, 8, 0.0, 0.0)
  # (140737488355328, 16, 0.0, 0.0)
  # (140737488355328, 32, 0.0059990000000027521, 0.0)
  # (140737488355328, 64, 0.0, 0.0)
  # (140737488355328, 128, 0.0, 0.0)
  # (140737488355328, 256, 0.0, 0.0)
  # (140737488355328, 512, 0.0, 0.00099999999999766942)
  # (140737488355328, 1024, 0.00099999999999766942, 0.0)
  # (140737488355328, 2048, 0.0020000000000024443, 0.0019989999999978636)
  # (140737488355328, 4096, 0.0049989999999979773, 0.005000000000002558)
  # (140737488355328, 8192, 0.010998000000000729, 0.011997999999998399)
  # (140737488355328, 16384, 0.023995999999996798, 0.023997000000001378)
  # (140737488355328, 32768, 0.050992000000000814, 0.052991999999996153)
  # (140737488355328, 65536, 0.1149820000000048, 0.10598499999999689)
  # (140737488355328, 131072, 0.29195599999999189, 0.21996599999999944)
  # (140737488355328, 262144, 0.6119070000000022, 0.45393199999999467)
  # (140737488355328, 524288, 1.5217689999999919, 1.0278430000000043)
  # (140737488355328, 1048576, 3.1365230000000111, 2.0966819999999871)

• Deprecate the function sqrt_approx() (David Roe) -- To obtain a numerical approximation of the square root of a ring element (integers, polynomials over GF(2^x), rationals), users are advised to use the function sqrt() with a given number of bits of precision instead.

• Use Pohlig-Hellman for generic discrete logarithm (Yann Laigle-Chapuy) -- This results in significant improvement in performance and less memory foot print. Here's an example with a smooth order:

  sage: factor(5^15-1)
  2^2 * 11 * 31 * 71 * 181 * 1741
  
  # BEFORE
  sage: F.<a>=GF(5^15)
  sage: g=F.gen()
  sage: u=g^123456789
  sage: time log(u,g)
  CPU times: user 271.39 s, sys: 4.72 s, total: 276.11 s
  Wall time: 276.96 s
  123456789
  sage: get_memory_usage()
  378.21875
  
  # AFTER
  sage: F.<a>=GF(5^15)
  sage: g=F.gen()
  sage: u=g^123456789
  sage: time log(u,g)
  CPU times: user 0.14 s, sys: 0.00 s, total: 0.14 s
  Wall time: 0.16 s
  123456789
  sage: get_memory_usage()
  115.8984375

  And here's another example with a not-so-smooth order:

  sage:factor(3^13-1)
  2 * 797161
  
  # BEFORE
  sage: F.<a>=GF(3**13)
  sage: g=F.gen()
  sage: u=g^1234567
  sage: timeit('log(u,g)')
  5 loops, best of 3: 1.54 s per loop
  sage: get_memory_usage()
  155.11328125
  
  # AFTER
  sage: F.<a>=GF(3**13)
  sage: g=F.gen()
  sage: u=g^1234567
  sage: timeit('log(u,g)')
  5 loops, best of 3: 931 ms per loop
  sage: get_memory_usage()
  139.4296875

• Exact division syntax in finite fields of prime order (David Roe) -- Support the division operator // for finite fields of prime order.

• Consistent integer hashing (Craig Citro).

• Unit of least precision for RR and RDF (Robert Bradshaw) -- New function ulp() to get the unit of least precision for a real number defined using RR or RDF. The unit of least precision for such a number is the weight of its least significant bit. Unless the number in question is exactly a power of two, it is gap between this number and the next closest distinct number that can be represented.

• Wrap FLINT's pseudo-division algorithm for univariate polynomials over ZZ (William Stein).

Build

• 64-bit OSX (Michael Abshoff) -- Fixed 64-bit OSX build support for f2c, added 64-bit OSX build support for tachyon, added 64-bit OSX build support for flintqs, and added persistent Sage 64-bit building switch on OSX and Solaris.
• Update readline spkg (Michael Abshoff) -- The updated readline spkg allows readline-5.2 to be built on the 64-bit version of OpenSUSE 11.1.

Calculus

• LaTeX output (Mike Hansen) -- Added LaTeX output for ceiling, floor, and derivative functions, and LaTaX'ing of powers of negative numbers.

• Make bernoulli_polynomial() independent of Maxima (Craig Citro) -- A rewrite of bernoulli_polynomial() to avoid using Maxima completely in computing Bernoulli polynomials. This gives roughly a factor of 10 speed-up.

• Support integration without explicit variable declaration (Karl-Dieter Crisman).

• CPU time of a Maxima session (Martin Albrecht) -- The new function cputime() in sage.interfaces.maxima.py returns the amount of CPU time used by a Maxima session.

• PyNaC interface enhancements (Burcin Erocal) -- New enhancements to the PyNaC interface include:

  • Add the method operator() to sage.symbolic.expression.pyx to return the top-most operator in an expression.

  • Allow PyNaC symbolic variables as arguments to factorial() and binomial().

  • Iterator support to sage.symbolic.expression.Expression.

  • Allow substituting more than one variable in PyNaC expressions.

  • New function collect_common_factors() to sage.symbolic.expression.Expression.

  • LaTeX printing.

  • Allow PyNaC symbolics to be used in CallableSymbolicExpressions.

• Indefinite integration for piecewise functions (Paul Butler).

• New function jacobian() for computing the Jacobian matrix (Jason Grout) -- The new function is sage.calculus.functions.jacobian() which returns the Jacobian matrix consisting of partial derivatives in which the i,j entry of the Jacobian matrix is the partial derivative diff(functions[i], variables[j]).

Coding Theory

• Weight distribution for binary codes (Robert L. Miller) -- A weight distribution algorithm for binary codes using Robert Bradshaw's bitsets. This implementation in Cython gives a 19 to 20 times performance speed-up over the previous GAP/Guava implementation. Here's an example:

  # BEFORE
  sage: time C.spectrum()
  CPU times: user 0.03 s, sys: 0.02 s, total: 0.05 s
  Wall time: 3.36 s
  [1, 0, 0, 7, 7, 0, 0, 1]
  sage: time C.spectrum()
  CPU times: user 0.02 s, sys: 0.01 s, total: 0.03 s
  Wall time: 2.20 s
  [1, 0, 0, 7, 7, 0, 0, 1]
  sage: time C.spectrum()
  CPU times: user 0.02 s, sys: 0.01 s, total: 0.03 s
  Wall time: 3.26 s
  [1, 0, 0, 7, 7, 0, 0, 1]
  sage: time C.spectrum()
  CPU times: user 0.02 s, sys: 0.01 s, total: 0.03 s
  Wall time: 2.74 s
  [1, 0, 0, 7, 7, 0, 0, 1]
  
  # AFTER
  sage: timeit('C.spectrum()')
  625 loops, best of 3: 1.86 ms per loop

• Linear codes decoding algorithms (David Joyner, Robert L. Miller) -- A number of algorithms in the GAP package Guava are moved to Sage. Two decoding methods are implemented, in particular, the methods nearest neighbor and syndrome.

Coercion

• Ring coercion for polynomials over finite fields (William Stein).
• Move univariate polynomial rings to new coercion model (Robert Bradshaw).

Combinatorics

• Bell polynomials (Blair Sutton) -- New function bell_polynomial() in sage.combinat.combinat.py returns the Bell polynomial.

• Covering design (Daniel Gordon) -- New module sage.combinat.designs.covering_design.py adds basic support for covering design. The function best_known_covering_design_www() is an interface to the online database of best known covering design at the La Jolla Covering Repository.

Commutative Algebra

• Multivariate polynomials over residue fields of number fields (Nick Alexander) -- Fixed an infinite loop bug when working with multivariate polynomials over residue fields of number fields. Previously in hashing "large" characteristic residue fields, the hash method would try to hash an ideal of the residue field itself, which in turn would try to hash its parent, and so on ad infinitum. At no point has a residue field with cardinality a very large prime been created in Sage.
• GCD of polynomials over finite fields (Martin Albrecht) -- Previously when using libsingular to compute the GCD of two (multivariate) polynomials over finite fields, Sage would segfault whenever the base rings are not identical.

• Deprecate Ideal.reduced_basis() (John Perry) -- The previous name Ideal.reduced_basis() is misleading as it suggests that it can be used for computing the reduced Groebner basis, when in fact it returns the interreduced basis. Thus Ideal.reduced_basis() is now deprecated and users are encouraged to use Ideal.interreduced_basis() instead.

• Factoring multivariate polynomials over non-prime finite fields (William Stein) -- The factoring algorithm works as follow. If f is a polynomial over a non-prime finite field, factoring f is reduced to factoring over a prime field and using GCD over the non-prime field.

• Multivariate ideal dimensions over any field (John Perry) -- Support for computing the dimension of fields of large prime characteristics via the method dimension() in the module sage.rings.polynomial.multi_polynomial_ideal.py. The default is to use the functionalities of Singular to do so. However, if the characteristic of the field is larger than what Singular can handle, the method falls back on a toy implementation of Buchberger to compute the Groebner basis, and finally using the algorithm described in Chapter 9, Section 1 of the following text:

  • David A. Cox, John B. Little & Donal O'Shea. "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" 3rd edition. Springer, 2007.

Distribution

• GAP configuration file (Matthias Meulien) -- A user's local GAP configuration file is usually named $HOME/.gaprc. When such a file already exists and Sage is compiled from source, using the Sage interface to GAP, e.g. gap._eval_line('1+3;'), can result in gibberish. This is now fixed so that the GAP interface would output a comprehensible message/answer as a result of some GAP calculation.

• An OSX Sage launcher (Ivan Andrus, Karl-Dieter Crisman) -- Support for building a clickable Sage launcher on Mac OSX. The clickable Mac application launcher can be built using -bdist on OSX. A screenshot of a Sage Mac OSX clickable app in action can be found in sage-OSX-app.png.

• Port to 64-bit OSX 10.5 (Michael Abshoff).

Geometry

• Polyhedral improvements (Marshall Hampton) -- Added more built-in Archimedean solids and some new methods such as the Gale transform, bipyramid construction, edge truncation, and perspective projection with (optionally) hidden faces invisible. The Schlegel projection code has also been refactored to make it more general in the future.

Graph Theory

• Equality testing in graphs (Robert L. Miller) -- The "weighted" property or edge label is taken into account when testing for equality in graphs.

• Update NetworkX to version 0.99 upstream release (Robert L. Miller, Michael Abshoff) -- NetworkX is a Python package for studying the structure and dynamics of complex networks. It can be used to analyze large networks including social, biological and technological networks.

• Improve timings for adjacency_matrix, weighted_adjacency_matrix, and kirchoff_matrix (Mike Hansen).

  # BEFORE
  sage: %time m = graphs.GridGraph([50,50]).laplacian_matrix()
  CPU times: user 38.42 s, sys: 0.24 s, total: 38.66 s
  Wall time: 39.02 s
  
  # AFTER
  sage: %time m = graphs.GridGraph([50,50]).laplacian_matrix()
  CPU times: user 0.63 s, sys: 0.06 s, total: 0.69 s
  Wall time: 0.89 s

• Pretty plot of graphs with multiedges (Emily Kirkman) -- A plot of a multiedge graph can be found in multiedge-graph.png. The plot is produced using the following code from the docstring of sage.graphs.graph.GenericGraph.plot():

  sage: g = Graph({}, loops=True, multiedges=True)
  sage: g.add_edges([(0,0,'a'), (0,0,'b'), (0,1,'c'),(0,1,'d'), 
  ...     (0,1,'e'), (0,1,'f'), (0,1,'f'), (2,1,'g'),(2,2,'h')]) 
  sage: g.plot(edge_labels=True, color_by_label=True, edge_style='dashed')

• Improvements to the Graph(...) constructor for input matrices Robert L. Miller) -- This is an overhaul of the Graph and DiGraph initialization functions. Among other things, one can now naturally convert between Graph and DiGraph.

Graphics

• Plotting a region (Arnaud Bergeron) -- New function region_plot() for plotting a region where a system of equations/inequalities holds true. Here's a region plot, concentric-ellipses.png, produced using the following code from the docstring of sage.plot.contour_plot.region_plot():

  sage: x, y = var('x, y') 
  sage: region_plot(cos(x^2 + y^2) < 0, (-3, 3), (-3, 3))

  Here is a more sophisticated plot, yellow-dots.png, produced using the following code from the doctring of sage.plot.contour_plot.region_plot():

  sage: x, y = var('x, y')
  sage: region_plot(sin(x) * sin(y) >= 1/4, (-10,10), (-10,10), incol='yellow', bordercol='black', plot_points=100).show(aspect_ratio=1)

• Consistency in variable range (Mike Hansen, Jason Grout) -- The variable range of parametric_plot() is now consistent with that of plot(), namely (var, min, max).

• Polar plot syntax (Jason Grout) -- Polar plot now accepts the syntax (t, 0, 2*pi) for the interval.

• New function density_plot() and improved colour map handling (Arnaud Bergeron) -- The new function density_plot() takes a function of two variables and plots contour lines of the function over two specified ranges. Also, some improvements on how color map is handled. Here's a density plot in grayscale, density-plot-bw.png, produced using the following code:

  sage: x,y = var('x, y')
  sage: density_plot(sin(x) * sin(y), (-2, 2), (-2, 2))

  A sample plot in colour, density-plot-colour.png, is produced using the following code:

  sage: density_plot(sin(x^2 + y^2) * cos(x) * sin(y), (-4, 4), (-4, 4), cmap='jet', plot_points=100)

• 3-D polygon (Arnaud Bergeron) -- The new function polygon3d() allows for plotting of 3-D polygons.

• Fill option for plot(), polar_plot() and parametric_plot() (Wilfried Huss, Karl-Dieter Crisman, Michael Abshoff) -- Added new options "fill", "fillcolor", and "fillalpha" to the plot family of functions. These new fill options allow users to fill the area between two functions in a plot, or to fill the area between the function and the x-axis. The syntax for the new fill option is similar to what Mathematica uses. Here's a fill plot, fill-plot.png, produced using the following code:

  maxima.load('interpol')
  
  def f(x):
      return RDF(1 / (1 + 25 * x^2))
  
  def runge():
      g = plot(f, -1, 1, rgbcolor='red', thickness=1)
      polynom = []
      for i, n in enumerate([6, 8, 10, 12]):
          data = [(x, f(x)) for x in xsrange(-1, 1, 2 / (n - 1), include_endpoint=True)]
          polynom.append(maxima.lagrange(data).sage())
          g += list_plot(data, rgbcolor='black', pointsize=5)
      g += plot(polynom, -1, 1, fill=f, fillalpha=0.2, thickness=0)
      return g
  
  runge().show(ymin=0, ymax=1)

Group Theory

• New method is_cyclic() (David Joyner) -- The new method is_cyclic() for (finite) groups is currently not as optimized as it should be. Given a finite abelian group, one can test to see whether or not it is cyclic, a test that depends on calculating the elementary divisors of the group. As correctness is the main concern in the current implementation of elementary_divisors(), performance is not taken into account. However, the docstring for elementary_divisors() describes an algorithm communicated by Robert L. Miller for speeding up this method.

Interfaces

• Magma interface (William Stein) -- Fixed a problem in Sage's interface to Magma so that now the Magma .sig files in extcode are no longer written there by Magma itself. The current behaviour is that the first (and only first) time a Magma interface is started in a given session, files in data/extcode/magma are copied to a temporary directory.

• GAP interface (Mike Hansen) -- Ensured that the GAP interface handles keyboard interrupt gracefully, that GAP output would be printed, and that multi-line input and comments are handled as expected.
• Sage preparser (Robert Bradshaw) -- Some simplification of the Sage preparser, together with improved handling of literal numbers.

Linear Algebra

• Multiplication of sparse matrices over finite fields (William Stein, Craig Citro) -- Significant performance improvement when multiplying two sparse matrices over the same finite field. In some cases, performance is about 37 times faster than previously.

  # BEFORE
  sage: m = random_matrix(GF(10007), 100, 100, sparse=True)
  sage: %time m*m
  CPU times: user 3.36 s, sys: 0.03 s, total: 3.39 s
  Wall time: 3.42 s
  100 x 100 sparse matrix over Finite Field of size 10007
  
  # AFTER
  sage: m = random_matrix(GF(10007), 100, 100, sparse=True)
  sage: %time m*m
  CPU times: user 0.09 s, sys: 0.00 s, total: 0.09 s
  Wall time: 0.09 s
  100 x 100 sparse matrix over Finite Field of size 10007

• Minimum polynomials for GF(p) (Alex Ghitza).

• Matrix exponential for general matrices (Jason Grout) -- Added a generic matrix exponential that depends on Maxima's matrix exponentiation function. If the matrix has floating point numbers, these will be rounded automatically to rational numbers during the computation. For numerical approximations to the exponential, one can first convert the matrix to RDF or CDF.

• 100% doctest coverage for the modules sage.matrix.constructor.py, sage.matrix.misc.pyx, sage.matrix.matrix_generic_dense.pyx (William Stein) -- Apart from the full doctest coverage, calculating the lift of a matrix is now 20 times faster than previously.

• Improved performance for method density() in sage.matrix.matrix_modn_sparse.pyx (Craig Citro).

• Added a kernel method for sparse integer matrices (John H. Palmieri).

• Speed up right_nullity() for matrices (John H. Palmieri).

• Considerable (optional) speed-up for row echelon forms of dense matrices over GF(2) due to new M4RI library (Martin Albrecht).

  # BEFORE
  sage: A = random_matrix(GF(2),2*10^4,2*10^4)
  sage: %time A.echelon_form(algorithm='m4ri')
  CPU times: user 15.49 s, sys: 0.05 s, total: 15.54 s
  Wall time: 15.72 s
  20000 x 20000 dense matrix over Finite Field of size 2
  
  # AFTER
  sage: A = random_matrix(GF(2),2*10^4,2*10^4)
  sage: %time A.echelon_form(algorithm='pluq')
  CPU times: user 9.86 s, sys: 0.04 s, total: 9.91 s
  Wall time: 9.97 s
  20000 x 20000 dense matrix over Finite Field of size 2

Miscellaneous

• Rewrite of the function __getitem__ (Jason Grout, Craig Citro) -- A rewrite of the function __getitem__ in sage.matrix.matrix0.pyx to better support slices and negative indices.

• Function get_memory_usage() now returns a float on all platforms (William Stein).

• The function CremonaDatabase().number_of_curves() now works even when the optional Cremona database isn't installed (Alex Ghitza).

• Set iteration for finite sets (Robert L. Miller) -- Set iteration is now implemented for finite sets.

Notebook

• Interact cells in the notebook are not automatically evaluated (Mike Hansen).
• Clear browser cache when restarting the worksheet (Mike Hansen).
• Update jsMath to version 3.6 upstream release (Jason Grout) -- jsMath is a package for including professional quality mathematics typesetting a la LaTeX/TeX in HTML pages. It works across multiple popular browsers and is platform independent.

• In-line WYSIWYG editor for text cells using TinyMCE (Jason Grout) -- TinyMCE is a Javascript library that implements a WYSIWYG editor within a browser.

• Automatic indentation (Alexander Hupfer, Tom Boothby) -- Automatic indentation for Python code after colons and same level identation. Currently, up to four levels of indentation are supported.

Number Theory

• Number field ideal utilities (John Cremona, Maite Aranes) -- New function invertible_residues() for iterating through only the invertible residues modulo an integral ideal. New function element_1_mod() such that A.element_1_mod(B) returns some a in A such that 1 - a is in B.

• New function random_element() for returning a random element of a number field (Alex Ghitza).

• Elliptic curve function integral_points() misses some points (John Cremona) -- Francois Glineur reported that for the elliptic curve 20160bg2, the output of integral_points() misses the points x = 168 and x = 381. This problem has been narrowed down to the function point_preprocessing(), and the bug is now fixed.

• Elliptic curve (Robert Bradshaw) -- Support for the construction of an elliptic curve via a Weierstrass equation. The Weierstrass equation can be passed as an argument to EllipticCurve().

• Separating relative number fields from generic/absolute number fields (Nick Alexander) -- Functionalities in the module sage.rings.number_field.number_field.py that deal with relative number fields are now wrapped inside the new module sage.rings.number_field.number_field_rel.py.

• Cremona's database of elliptic curves (William Stein) -- Improved handling of the case 990h.

• Manin constant (William Stein) -- New function manin_constant() to compute the Manin constant of an elliptic curve. This function only works if the curve is in the installed Cremona database. By default Sage includes a small database, whereas the full database must be installed as an optional package. WARNING: The result is _not_ provably correct in the sense that when the numbers are huge, isogenies could be missed due to precision issues. The newly implemented function can be found in the module sage.schemes.elliptic_curves.ell_rational_field.py.

• Bach bound (William Stein) -- New function bach_bound() to compute the Bach bound associated to a number field. Assuming the General Riemann Hypothesis, the Bach bound is a bound B such that every integral ideal is equivalent modulo principal fractional ideals to an integral ideal of norm at most B. The newly implemented function can be found in the module sage.rings.number_field.number_field_base.pyx.

• Make relative number fields lazy (Nick Alexander) -- Relative number fields are now truly lazy, meaning that they no longer require PARI's nf or bnf structures for the base field nor PARI's rnf structures for the extension. Arithmetic can now be done in huge extensions, those for which there is no hope of finding units, class groups, etc.

Optional Packages

• Modular polynomials database (Alex Ghitza) -- Removed the use of polydict in the module sage.databases.db_modular_polynomials.py.

• lrs.spkg improvements (Marshall Hampton) -- Further tests added to the makefile of the package lrs. This optional package implements the reverse search algorithm for vertex enumeration and convex hull problems.

Packages

• Switch from GMP to MPIR and upgrade to version 0.9.rc3 upstream release (Michael Abshoff) -- MPIR, otherwise known as eMPIRe, is a library for multiprecision integers and rationals based on the GMP project. Among other things, MPIR aims to provide native build capability under Windows.

• Upgrade MPFR to version 2.4.0 upstream release (Michael Abshoff) -- MPFR is a C library for multiprecision floating-point computations that is based on the GMP project.

• Upgrade GMP-ECM to version 6.2.1 upstream release (Michael Abshoff).

• Move jquery into its own spkg (Jason Grout).

• Upgrade Mercurial to version 1.1.2 upstream release (Mike Hansen).

• OS X 64-bit (Michael Abshoff) -- Added 64-bit build support for PyNaC, R, GHMM, and Boehm GC. Also added fortran-OSX64-20090120.spkg to the experimental spkg repository.

• Upgrade IPython to version 0.9.1 upstream release (Mike Hansen).

• Upgrade cddlib to version 0.94f upstream release (Sebastien Barthelemy).

• Upgrade M4RI to version 20090105 upstream release (Martin Albrecht).

• Upgrade GAP/GUAVA to versions 4.4.12/3.9 upstream releases (David Joyner).

• Upgrade Jmol to version 11.6 upstream release (Michael Abshoff).

• Upgrade matplotlib to version 0.98.5.3-svn6910 upstream release (Jason Grout, Michael Abshoff).

• Upgrade Python to version 2.5.4 upstream release (Michael Abshoff).

• Upgrade libpng to version 1.2.34 upstream release (Michael Abshoff).

Porting

• OSX 64-bit (Michael Abshoff) -- Added proper libcsage build support, and Singular is now built using the flag --with-malloc=system.

Solaris

• Upgrade libgcrypt to version 1.4.3 upstream release and force the function get_memory_usage() to fall back to using top when not on Linux (Michael Abshoff).

User Interface

• Added a hg_examples Mercurial wrapper for the examples repository (Mike Hansen).

Website/Wiki

• The trac server now displays the comments of a Mercurial diff (Robert Bradshaw).