Sage 4.1 Release Tour

Sage 4.1 was released on July 09, 2009. For the official, comprehensive release note, please refer to sage-4.1.txt. A nicely formatted version of this release tour can be found at FIXME. The following points are some of the foci of this release:

• Upgrade to Python 2.6.x
• Support for building Singular with GCC 4.4
• Optimized integer division
• Combinatorics: irreducible matrix representations of symmetric groups; and Yang-Baxter Graphs
• Cryptography: Mini Advanced Encryption Standard for educational purposes
• Graph theory: back-end for graph theory with Cython (c_graph); and improve accuracy of graph eigenvalues
• Linear algebra: a general package for finitely generated, not-necessarily free R-modules; and multiplicative order for matrices over finite fields
• Miscellaneous: optimized Sudoku solver; a decorator for declaring abstract methods; and support Unicode in LaTeX cells (notebook)
• Number theory: improved random element generation for number field orders and ideals; support Michael Stoll's ratpoints package; and elliptic exponential
• Numerical: computing numerical values of constants using mpmath
• Update, upgrade 18 packages to latest upstream releases

Algebraic Geometry

• Construct an elliptic curve from a plane curve of genus one (Lloyd Kilford, John Cremona ) -- New function EllipticCurve_from_plane_curve() in the module sage/schemes/elliptic_curves/constructor.py to allow the construction of an elliptic curve from a smooth plane cubic with a rational point. Currently, this function uses Magma and it will not work on machines that do not have Magma installed. Assuming you have Magma installed on your computer, we can use the function EllipticCurve_from_plane_curve() to first check that the Fermat cubic is isomorphic to the curve with Cremona label "27a1":

  sage: x, y, z = PolynomialRing(QQ, 3, 'xyz').gens() # optional - magma  
  sage: C = Curve(x^3 + y^3 + z^3) # optional - magma 
  sage: P = C(1, -1, 0) # optional - magma 
  sage: E = EllipticCurve_from_plane_curve(C, P) # optional - magma 
  sage: E # optional - magma 
  Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field 
  sage: E.label() # optional - magma 
  '27a1'
  

  Here is a quartic example:

  sage: u, v, w = PolynomialRing(QQ, 3, 'uvw').gens() # optional - magma  
  sage: C = Curve(u^4 + u^2*v^2 - w^4) # optional - magma 
  sage: P = C(1, 0, 1) # optional - magma 
  sage: E = EllipticCurve_from_plane_curve(C, P) # optional - magma 
  sage: E # optional - magma 
  Elliptic Curve defined by y^2  = x^3 + 4*x over Rational Field 
  sage: E.label() # optional - magma 
  '32a1'
  

Basic Arithmetic

• Speed-up integer division (Robert Bradshaw ) -- In some cases, integer division is now up to 31% faster than previously. The following timing statistics were obtained using the machine sage.math:

  # BEFORE
  
  sage: a = next_prime(2**31)
  sage: b = Integers(a)(100)
  sage: %timeit a % b;
  1000000 loops, best of 3: 1.12 µs per loop
  sage: %timeit 101 // int(5);
  1000000 loops, best of 3: 215 ns per loop
  sage: %timeit 100 // int(-3)
  1000000 loops, best of 3: 214 ns per loop
  sage: a = ZZ.random_element(10**50)
  sage: b = ZZ.random_element(10**15)
  sage: %timeit a.quo_rem(b)
  1000000 loops, best of 3: 454 ns per loop
  
  
  # AFTER
  
  sage: a = next_prime(2**31)
  sage: b = Integers(a)(100)
  sage: %timeit a % b;
  1000000 loops, best of 3: 1.02 µs per loop
  sage: %timeit 101 // int(5);
  1000000 loops, best of 3: 201 ns per loop
  sage: %timeit 100 // int(-3)
  1000000 loops, best of 3: 194 ns per loop
  sage: a = ZZ.random_element(10**50)
  sage: b = ZZ.random_element(10**15)
  sage: %timeit a.quo_rem(b)
  1000000 loops, best of 3: 313 ns per loop
  

Combinatorics

• Irreducible matrix representations of symmetric groups (Franco Saliola) -- Support for constructing irreducible representations of the symmetric group. This is based on Alain Lascoux's article Young representations of the symmetric group. The following types of representations are supported:

  • Specht representations -- The matrices have integer entries:

    sage: chi = SymmetricGroupRepresentation([3, 2]); chi
    Specht representation of the symmetric group corresponding to [3, 2]
    sage: chi([5, 4, 3, 2, 1])
    
    [ 1 -1  0  1  0]
    [ 0  0 -1  0  1]
    [ 0  0  0 -1  1]
    [ 0  1 -1 -1  1]
    [ 0  1  0 -1  1]
    

  • Young's seminormal representation -- The matrices have rational entries:

    sage: snorm = SymmetricGroupRepresentation([2, 1], "seminormal"); snorm
    Seminormal representation of the symmetric group corresponding to [2, 1]
    sage: snorm([1, 3, 2])
    
    [-1/2  3/2]
    [ 1/2  1/2]
    

  • Young's orthogonal representation (the matrices are orthogonal) -- These matrices are defined over Sage's Symbolic Ring:

    sage: ortho = SymmetricGroupRepresentation([3, 2], "orthogonal"); ortho
    Orthogonal representation of the symmetric group corresponding to [3, 2]
    sage: ortho([1, 3, 2, 4, 5])
    
    [          1           0           0           0           0]
    [          0        -1/2 1/2*sqrt(3)           0           0]
    [          0 1/2*sqrt(3)         1/2           0           0]
    [          0           0           0        -1/2 1/2*sqrt(3)]
    [          0           0           0 1/2*sqrt(3)         1/2]
    

  You can also create the CombinatorialClass of all irreducible matrix representations of a given symmetric group. Then particular representations can be created by providing partitions. For example:

  • sage: chi = SymmetricGroupRepresentations(5); chi
    Specht representations of the symmetric group of order 5! over Integer Ring
    sage: chi([5]) # the trivial representation
    Specht representation of the symmetric group corresponding to [5]
    sage: chi([5])([2, 1, 3, 4, 5])
    [1]
    sage: chi([1, 1, 1, 1, 1]) # the sign representation
    Specht representation of the symmetric group corresponding to [1, 1, 1, 1, 1]
    sage: chi([1, 1, 1, 1, 1])([2, 1, 3, 4, 5])
    [-1]
    sage: chi([3, 2])
    Specht representation of the symmetric group corresponding to [3, 2]
    sage: chi([3, 2])([5, 4, 3, 2, 1])
    
    [ 1 -1  0  1  0]
    [ 0  0 -1  0  1]
    [ 0  0  0 -1  1]
    [ 0  1 -1 -1  1]
    [ 0  1  0 -1  1]
    

  See the documentation of SymmetricGroupRepresentation and SymmetricGroupRepresentations for more information and examples.

• Yang-Baxter graphs (Franco Saliola) -- Besides being used for constructing the irreducible matrix representations of the symmetric group, Yang-Baxter graphs can also be used to construct the Cayley graph of a finite group. For example:

  • sage: def left_multiplication_by(g):
    ....:     return lambda h : h*g
    ....: 
    sage: G = AlternatingGroup(4)
    sage: operators = [ left_multiplication_by(gen) for gen in G.gens() ]
    sage: Y = YangBaxterGraph(root=G.identity(), operators=operators); Y
    Yang-Baxter graph with root vertex ()
    sage: Y.plot(edge_labels=False)
    

• Yang-Baxter graphs can also be used to construct the permutahedron:

  • sage: from sage.combinat.yang_baxter_graph import SwapIncreasingOperator
    sage: operators = [SwapIncreasingOperator(i) for i in range(3)]
    sage: Y = YangBaxterGraph(root=(1,2,3,4), operators=operators); Y
    Yang-Baxter graph with root vertex (1, 2, 3, 4)
    sage: Y.plot()
    

• See the documentation of YangBaxterGraph for more information and examples.

Cryptography

• Mini Advanced Encryption Standard for educational purposes (Minh Van Nguyen) -- New module sage/crypto/block_cipher/miniaes.py to support the Mini Advanced Encryption Standard (Mini-AES) to allow students to explore the working of a block cipher. This is a simplified variant of the Advanced Encryption Standard (AES) to be used for cryptography education. Mini-AES is described in the paper:

  • A. C.-W. Phan. Mini advanced encryption standard (mini-AES): a testbed for cryptanalysis students. Cryptologia, 26(4):283--306, 2002.
  We can encrypt a plaintext using Mini-AES as follows:

  sage: from sage.crypto.block_cipher.miniaes import MiniAES
  sage: maes = MiniAES()
  sage: K = FiniteField(16, "x")
  sage: MS = MatrixSpace(K, 2, 2)
  sage: P = MS([K("x^3 + x"), K("x^2 + 1"), K("x^2 + x"), K("x^3 + x^2")]); P
  
  [  x^3 + x   x^2 + 1]
  [  x^2 + x x^3 + x^2]
  sage: key = MS([K("x^3 + x^2"), K("x^3 + x"), K("x^3 + x^2 + x"), K("x^2 + x + 1")]); key
  
  [    x^3 + x^2       x^3 + x]
  [x^3 + x^2 + x   x^2 + x + 1]
  sage: C = maes.encrypt(P, key); C
  
  [            x       x^2 + x]
  [x^3 + x^2 + x       x^3 + x]
  

  Here is the decryption process:

  sage: plaintxt = maes.decrypt(C, key)
  sage: plaintxt == P
  True
  

  We can also work directly with binary strings:

  sage: from sage.crypto.block_cipher.miniaes import MiniAES
  sage: maes = MiniAES()
  sage: bin = BinaryStrings()
  sage: key = bin.encoding("KE"); key
  0100101101000101
  sage: P = bin.encoding("Encrypt this secret message!")
  sage: C = maes(P, key, algorithm="encrypt")
  sage: plaintxt = maes(C, key, algorithm="decrypt")
  sage: plaintxt == P
  True
  

  Or work with integers n such that 0 <= n <= 15:

  sage: from sage.crypto.block_cipher.miniaes import MiniAES
  sage: maes = MiniAES()
  sage: P = [n for n in xrange(16)]; P
  [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
  sage: key = [2, 3, 11, 0]; key
  [2, 3, 11, 0]
  sage: P = maes.integer_to_binary(P)
  sage: key = maes.integer_to_binary(key)
  sage: C = maes(P, key, algorithm="encrypt")
  sage: plaintxt = maes(C, key, algorithm="decrypt")
  sage: plaintxt == P
  True
  

Graph Theory

• Fast compiled graphs c_graph (Robert Miller) -- The Python package NetworkX version 0.36 is currently the default graph implementation in Sage. The goal of fast compiled graphs, or c_graph, is to be the default implementation of graph theory in Sage. The c_graph implementation is developed using Cython, which allows graph theoretic computations to run at the speed of C. The c_graph backend is implemented in the module sage/graphs/base/c_graph.pyx. This module is called by higher-level frontends in sage/graphs/. Where support is provided for using c_graph, graph theoretic computations is usually more efficient than using NetworkX. For example, the following timing statistics were obtained using the machine sage.math:

  # NetworkX 0.36
  
  sage: time G = Graph(1000000, implementation="networkx")
  CPU times: user 8.74 s, sys: 0.27 s, total: 9.01 s
  Wall time: 9.08 s
  
  
  # c_graph
  
  sage: time G = Graph(1000000, implementation="c_graph")
  CPU times: user 0.01 s, sys: 0.14 s, total: 0.15 s
  Wall time: 0.19 s
  

  Here, we see an efficiency gain of up to 47x using c_graph.

• Improve accuracy of graph eigenvalues (Rob Beezer) -- New routines compute eigenvalues and eigenvectors of integer matrices more precisely than before. Rather than convert adjacency matrices of graphs to computations over the real or complex fields, adjacency matrices are retained as matrices over the integers, yielding more accurate and informative results for eigenvalues, eigenvectors, and eigenspaces. Here is a comparison involving the computation of graph spectrum:

  # BEFORE
  
  sage: g = graphs.CycleGraph(8); g
  Cycle graph: Graph on 8 vertices
  sage: g.spectrum()
  
  [-2.0,
   -1.41421356237,
   -1.41421356237,
   4.02475820828e-18,
   6.70487495185e-17,
   1.41421356237,
   1.41421356237,
   2.0]
  
  
  # AFTER
  
  sage: g = graphs.CycleGraph(8); g
  Cycle graph: Graph on 8 vertices
  sage: g.spectrum()
  [2, 1.414213562373095?, 1.414213562373095?, 0, 0, -1.414213562373095?, -1.414213562373095?, -2]
  

  Integer eigenvalues are now exact, irrational eigenvalues are more precise than previously, making multiplicities easier to determine. Similar comments apply to eigenvectors:

  sage: g.eigenvectors()
  
  [(2, [
  (1, 1, 1, 1, 1, 1, 1, 1)
  ], 1),
   (-2, [
  (1, -1, 1, -1, 1, -1, 1, -1)
  ], 1),
   (0, [
  (1, 0, -1, 0, 1, 0, -1, 0),
  (0, 1, 0, -1, 0, 1, 0, -1)
  ], 2),
   (-1.414213562373095?,
    [(1, 0, -1, 1.414213562373095?, -1, 0, 1, -1.414213562373095?),
     (0, 1, -1.414213562373095?, 1, 0, -1, 1.414213562373095?, -1)],
    2),
   (1.414213562373095?,
    [(1, 0, -1, -1.414213562373095?, -1, 0, 1, 1.414213562373095?),
     (0, 1, 1.414213562373095?, 1, 0, -1, -1.414213562373095?, -1)],
    2)]
  

  Eigenspaces are exact, in that they can be expressed as vector spaces over number fields. When the defining polynomial has several roots, the eigenspaces are not repeated. Previously, eigenspaces were "fractured" owing to slight computational differences in identical eigenvalues. In concert with eigenvectors(), this command illuminates the structure of a graph's eigenspaces more than purely numerical results.

  sage: g.eigenspaces()
  
  [
  (2, Vector space of degree 8 and dimension 1 over Rational Field
  User basis matrix:
  [1 1 1 1 1 1 1 1]),
  (-2, Vector space of degree 8 and dimension 1 over Rational Field
  User basis matrix:
  [ 1 -1  1 -1  1 -1  1 -1]),
  (0, Vector space of degree 8 and dimension 2 over Rational Field
  User basis matrix:
  [ 1  0 -1  0  1  0 -1  0]
  [ 0  1  0 -1  0  1  0 -1]),
  (a3, Vector space of degree 8 and dimension 2 over Number Field in a3 with defining polynomial x^2 - 2
  User basis matrix:
  [  1   0  -1 -a3  -1   0   1  a3]
  [  0   1  a3   1   0  -1 -a3  -1])
  ]
  

  Complex eigenvalues (of digraphs) previously were missing their imaginary parts. This issue has been fixed as part of the improvement in calculating graph eigenvalues.

Graphics

• Plot histogram improvement (David Joyner) -- Some improvements to the plot_histogram() function of the class IndexedSequence in sage/gsl/dft.py. The default colour of the histogram is blue:

  sage: J = range(3)
  sage: A = [ZZ(i^2)+1 for i in J]
  sage: s = IndexedSequence(A, J)
  sage: s.plot_histogram()
  

• You can now change the colour of the histogram with the argument clr:

  sage: s.plot_histogram(clr=(1,0,0))
  

• and even use the argument eps to change the width of the spacing between the bars:

  sage: s.plot_histogram(clr=(1,0,1), eps=0.3)
  

Linear Algebra

• Multiplicative order for matrices over finite fields (Yann Laigle-Chapuy) -- New method multiplicative_order() in the class Matrix of sage/matrix/matrix0.pyx for computing the multiplicative order of a matrix. Here are some examples on using the new method multiplicative_order():

  sage: A = matrix(GF(59), 3, [10,56,39,53,56,33,58,24,55])
  sage: A.multiplicative_order()
  580
  sage: (A^580).is_one()
  True
  sage: B = matrix(GF(10007^3, 'b'), 0)
  sage: B.multiplicative_order()
  1
  sage: E = MatrixSpace(GF(11^2, 'e'), 5).random_element()
  sage: (E^E.multiplicative_order()).is_one()
  True
  

• A general package for finitely generated not-necessarily free R-modules (William Stein, David Loeffler ) -- This consists of the following new Sage modules:

  • sage/modules/fg_pid/fgp_element.py -- Elements of finitely generated modules over a principal ideal domain. Here are some examples:

  sage: V = span([[1/2,1,1], [3/2,2,1], [0,0,1]], ZZ)
  sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
  sage: Q = V/W
  sage: x = Q(V.0-V.1); x
  (0, 3)
  sage: type(x)
  <class 'sage.modules.fg_pid.fgp_element.FGP_Element'>
  sage: x is Q(x)
  True
  sage: x.parent() is Q
  True
  sage: Q
  Finitely generated module V/W over Integer Ring with invariants (4, 12)
  sage: Q.0.additive_order()
  4
  sage: Q.1.additive_order()
  12
  sage: (Q.0+Q.1).additive_order()
  12
  

  • sage/modules/fg_pid/fgp_module.py -- Finitely generated modules over a principal ideal domain. Currently, on the principal ideal domain ZZ of integers is supported. Here are some examples:

  sage: V = span([[1/2,1,1], [3/2,2,1], [0,0,1]], ZZ)
  sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
  sage: import sage.modules.fg_pid.fgp_module
  sage: Q = sage.modules.fg_pid.fgp_module.FGP_Module(V, W)
  sage: type(Q)
  <class 'sage.modules.fg_pid.fgp_module.FGP_Module_class'>
  sage: Q is sage.modules.fg_pid.fgp_module.FGP_Module(V, W, check=False)
  True
  sage: X = ZZ**2 / span([[3,0],[0,2]], ZZ)
  sage: X.linear_combination_of_smith_form_gens([1])
  (1)
  sage: Q
  Finitely generated module V/W over Integer Ring with invariants (4, 12)
  sage: Q.gens()
  ((1, 0), (0, 1))
  sage: Q.coordinate_vector(-Q.0)
  (-1, 0)
  sage: Q.coordinate_vector(-Q.0, reduce=True)
  (3, 0)
  sage: Q.cardinality()
  48
  

  • sage/modules/fg_pid/fgp_morphism.py -- Morphisms between finitely generated modules over a principal ideal domain. Here are some examples:

  sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ)
  sage: W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
  sage: Q = V/W; Q
  Finitely generated module V/W over Integer Ring with invariants (4, 12)
  sage: phi = Q.hom([Q.0+3*Q.1, -Q.1]); phi
  Morphism from module over Integer Ring with invariants (4, 12) to module with invariants (4, 12) that sends the generators to [(1, 3), (0, 11)]
  sage: phi(Q.0) == Q.0 + 3*Q.1
  True
  sage: phi(Q.1) == -Q.1
  True
  sage: Q.hom([0, Q.1]).kernel()
  Finitely generated module V/W over Integer Ring with invariants (4)
  sage: A = Q.hom([Q.0, 0]).kernel(); A
  Finitely generated module V/W over Integer Ring with invariants (12)
  sage: Q.1 in A
  True
  sage: phi = Q.hom([Q.0-3*Q.1, Q.0+Q.1])
  sage: A = phi.kernel(); A
  Finitely generated module V/W over Integer Ring with invariants (4)
  sage: phi(A)
  Finitely generated module V/W over Integer Ring with invariants ()
  

Miscellaneous

• FIXME: summarize #3084
• FIXME: summarize #6097
• FIXME: summarize #6417

Modular Forms

Notebook

• FIXME: summarize #5637

Number Theory

• FIXME: summarize #6273
• FIXME: summarize #5854
• FIXME: summarize #6386

Numerical

• FIXME: summarize #6200

Packages

• Upgrade Singular to version singular-3-1-0-2-20090620 with support for compiling with GCC 4.4.

• FIXME: summarize #6359
• FIXME: summarize #6196
• FIXME: summarize #6276
• FIXME: summarize #5517
• FIXME: summarize #5854
• FIXME: summarize #5866
• FIXME: summarize #5867
• FIXME: summarize #5868
• FIXME: summarize #5869
• FIXME: summarize #5870
• FIXME: summarize #5872
• FIXME: summarize #5874
• FIXME: summarize #5875
• FIXME: summarize #6281
• FIXME: summarize #6470
• FIXME: summarize #6470
• FIXME: summarize #6492
• FIXME: summarize #6408

P-adics

Quadratic Forms

Symbolics

• FIXME: summarize #6421

Topology