Sage 4.3.1 Release Tour

Major features

• Substantial work towards a complete SPARC Solaris 10 port. This is due to the hard work of David Kirkby. The relevant tickets include #6595, #7067, #7138, #7162, #7387, #7505, #7817.

• We're moving closer towards a FreeBSD port, thanks to the work of Peter Jeremy at ticket #7825.

Algebra

• Implement Chinese Remainder Theorem for polynomials, which is needed for general descents on curves: #7595

Basic arithmetics

• Implement conjugate() for RealDoubleElement #7834 (Dag Sverre Seljebotn) --- New method conjugate() in the class RealDoubleElement of the module sage/rings/real_double.pyx for returning the complex conjugate of a real number. This is consistent with conjugate() methods in ZZ and RR. For example,

  sage: ZZ(5).conjugate()
  5
  sage: RR(5).conjugate()
  5.00000000000000
  sage: RDF(5).conjugate()
  5.0

• #7739

Combinatorics

• Weyl group optimizations #7754 (Nicolas M. Thiéry) --- Three major improvements that indirectly also improve efficiency of most Weyl group routines:

  • Faster hash method calling the hash of the underlying matrix (which is set as immutable for that purpose).

  • New __eq__() method.

  • Action on the weight lattice realization: optimization of the matrix multiplication.
  Some operations are now up to 34% faster than previously:

  BEFORE
  
  sage: W = WeylGroup(["F", 4])
  sage: W.cardinality()
  1152
  sage: %time list(W);
  CPU times: user 10.51 s, sys: 0.05 s, total: 10.56 s
  Wall time: 10.56 s
  sage: W = WeylGroup(["E", 8])
  sage: %time W.long_element();
  CPU times: user 1.47 s, sys: 0.00 s, total: 1.47 s
  Wall time: 1.47 s
  
  
  AFTER
  
  sage: W = WeylGroup(["F", 4])
  sage: W.cardinality()
  1152
  sage: %time list(W);
  CPU times: user 6.89 s, sys: 0.04 s, total: 6.93 s
  Wall time: 6.93 s
  sage: W = WeylGroup(["E", 8])
  sage: %time W.long_element();
  CPU times: user 1.21 s, sys: 0.00 s, total: 1.21 s
  Wall time: 1.21 s

• #7301

  • The Gale Ryser theorem asserts that if p_1,p_2 are two partitions of n of respective lengths k_1,k_2, then there is a binary k_1\times k_2 matrix M such that p_1 is the vector of row sums and p_2 is the vector of column sums of M, if and only if p_2 dominates p_1.

This was implemented by Nathann Cohen and David Joyner. T.S. Michael helped a great deal with the refereeing process. Here is an example.

sage: from sage.combinat.integer_vector import gale_ryser_theorem 
sage: p1 = [4,2,2] 
sage: p2 = [3,3,1,1] 
sage: gale_ryser_theorem(p1, p2) 
 [1 1 1 1] 
 [1 1 0 0] 
 [1 1 0 0]         
sage: p1 = [4,2,2,0] 
sage: p2 = [3,3,1,1,0,0] 
sage: gale_ryser_theorem(p1, p2) 
 [1 1 1 1 0 0] 
 [1 1 0 0 0 0] 
 [1 1 0 0 0 0] 
 [0 0 0 0 0 0] 

• #7729

Iwahori Hecke algebras are deformations of the group algebras of Coxeter groups, such as Weyl groups (finite or affine). See: http://wiki.sagemath.org/HeckeAlgebras.

sage: R.<q>=PolynomialRing(QQ)
sage: H = IwahoriHeckeAlgebra("A3",q)
sage: [T1,T2,T3]=H.algebra_generators()
sage: T1*(T2+T3)*T1
T1*T2*T1 + (q-1)*T3*T1 + q*T3

• #7753

The Bruhat order for Coxeter groups is now fully implemented.

Elliptic curves

• Two-isogeny descent over QQ natively using ratpoints #6583 (Robert Miller) --- New module sage/schemes/elliptic_curves/descent_two_isogeny.pyx for descent on elliptic curves over QQ with a 2-isogeny. The relevant user interface function is two_descent_by_two_isogeny() that takes an elliptic curve E with a two-isogeny phi : E --> E' and dual isogeny phi', runs a two-isogeny descent on E, and returns n1, n2, n1' and n2'. Here, n1 is the number of quartic covers found with a rational point and n2 is the number which are ELS. Here are some examples illustrating the use of two_descent_by_two_isogeny():

  sage: from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny
  sage: E = EllipticCurve("14a")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  0
  sage: E = EllipticCurve("65a")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  1
  sage: E = EllipticCurve("1088j1")
  sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E)
  sage: log(n1, 2) + log(n1_prime, 2) - 2  # the rank
  2

  You could also ask two_descent_by_two_isogeny() to be verbose in its computation:

  sage: E = EllipticCurve("14a")
  sage: two_descent_by_two_isogeny(E, verbosity=1)
  2-isogeny
  Results:
  2 <= #E(Q)/phi'(E'(Q)) <= 2
  2 <= #E'(Q)/phi(E(Q)) <= 2
  #Sel^(phi')(E'/Q) = 2
  #Sel^(phi)(E/Q) = 2
  1 <= #Sha(E'/Q)[phi'] <= 1
  1 <= #Sha(E/Q)[phi] <= 1
  1 <= #Sha(E/Q)[2], #Sha(E'/Q)[2] <= 1
  0 <= rank of E(Q) = rank of E'(Q) <= 0
  (2, 2, 2, 2)

• More functions for elliptic curve isogenies #6887 (John Cremona, Jenny Cooley) --- Code for constructing elliptic curve isogenies already existed in Sage 4.1.1. The enhancements here include:

  • For l=2,3,5,7,13 over any field, find all l-isogenies of a given elliptic curve. (These are the l for which X_0(l) has genus 0).

  • Similarly for the remaining l for which l-isogenies exist over QQ.

  • Given an elliptic curve over QQ, find the whole isogeny class in a robust manner.

  • Testing if two curves are isogenous at least over QQ.

  The relevant use interface method is isogenies_prime_degree() in the class EllipticCurve_field of the module sage/schemes/elliptic_curves/ell_field.py. Here are some examples showing isogenies_prime_degree() in action. Examples over finite fields:

  sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8])
  sage: E.isogenies_prime_degree()
  [Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]
  sage: E.isogenies_prime_degree(13)
  [Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003]

  Examples over number fields (other than QQ):

  sage: QQroot2.<e> = NumberField(x^2 - 2)
  sage: E = EllipticCurve(QQroot2, [1,0,1,4,-6])
  sage: E.isogenies_prime_degree(2)
  [Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2]
  sage: E.isogenies_prime_degree(3)
  [Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2]

Graph theory

• An interactive graph editor #1321 (Radoslav Kirov, Mitesh Patel) --- Embed an interactive graph editor into the notebook. The following screenshot shows a graph editor for playing around with the complete graph on 5 vertices:

  

• Breadth/depth first searches and basic connectivity for c_graphs #7724 (Nathann Cohen, Yann Laigle-Chapuy) --- Implementation of the following methods for the class CGraphBackend in the module sage/graphs/base/c_graph.pyx:

  • depth_first_search()

  • breadth_first_search()

  • is_connected()

  • is_strongly_connected()

  In some cases, the c_graphs implementation of these methods provides a 2x speed improvement:

  sage: g = graphs.RandomGNP(1000, 0.01)
  sage: h = g.copy(implementation="c_graph")
  sage: %timeit list(g.depth_first_search(0));
  100 loops, best of 3: 8.17 ms per loop
  sage: %timeit list(h.depth_first_search(0));
  100 loops, best of 3: 3.29 ms per loop
  sage: 
  sage: %timeit list(g.breadth_first_search(0));
  100 loops, best of 3: 6.48 ms per loop
  sage: %timeit list(h.breadth_first_search(0));
  10 loops, best of 3: 34 ms per loop
  sage: 
  sage: %timeit g.is_connected();
  100 loops, best of 3: 8.47 ms per loop
  sage: %timeit h.is_connected();
  100 loops, best of 3: 3.41 ms per loop
  sage:
  sage: g = g.to_directed()
  sage: h = g.copy(implementation="c_graph")
  sage: %timeit g.is_strongly_connected();
  10 loops, best of 3: 23.5 ms per loop
  sage: %timeit h.is_strongly_connected();
  10 loops, best of 3: 25 ms per loop

• Tower of Hanoi graph #7770 (Rob Beezer) --- The Tower of Hanoi puzzle can be described by a graph whose vertices are possible states of the disks on the pegs, with edges representing legitimate moves of a single disk. The new method HanoiTowerGraph() of the class GraphGenerators in the module sage/graphs/graph_generators.py returns the graph whose vertices are the states of the Tower of Hanoi puzzle, with edges representing legal moves between states. See the documentation of this method for details on interpreting the the possible states of this puzzle. The following screenshot shows all the possible states of an instance of the puzzle with 3 pegs and 3 disks, produced using the following code:

  H = graphs.HanoiTowerGraph(3, 3, positions=False)
  show(H, figsize=[8,8])

  

• #7292

• #7590

• #7634

Linear algebra

• Viewing entries of large matrices #5174 (John Palmieri) --- For a small matrix such as 2 x 2, the default is to print the entries of the matrix. This default behaviour is unsuitable for large matrices such as 100 x 100. The string representation of such large matrices now indicate how to view all their entries. Here are some examples illustrating the new way to view the string representation of matrices. If the matrix is too big, all the elements are not displayed by default:

  sage: A = random_matrix(ZZ, 5)
  sage: A
  [ 1 -4 -4  1 -1]
  [-1  1 13 -1 -1]
  [-1  0  0 -1 -1]
  [-8  1 -1  1 -4]
  [ 1 -5 -1  1  2]
  sage: A = random_matrix(ZZ, 100)
  sage: A
  100 x 100 dense matrix over Integer Ring (type 'print A.str()' to see all of the entries)

  If a matrix has several names, refer to the matrix as "obj":

  sage: A = random_matrix(ZZ, 200)
  sage: B = A
  sage: B
  200 x 200 dense matrix over Integer Ring (type 'print obj.str()' to see all of the entries)

  If a matrix doesn't have a name, don't print any name referring to the matrix in its string representation:

  sage: A = random_matrix(ZZ, 150)
  sage: A.transpose()
  150 x 150 dense matrix over Integer Ring
  sage: T = A.transpose(); T
  150 x 150 dense matrix over Integer Ring (type 'print T.str()' to see all of the entries)

• #7728 (Dag Sverre Seljebotn)

Miscellaneous

• Command line access to HTML documentation and docstrings #6820 (John Palmieri, Mitesh Patel) --- Browse the Sage standard documentation from the command line or within the notebook interface. Use the following commands to browse documents in the standard documentation:

  • browse_sage_doc.tutorial() or its alias tutorial()

  • browse_sage_doc.reference(), or its aliases reference() and manual()

  • browse_sage_doc.developer() or its alias developer()

  • browse_sage_doc.constructions() or its alias constructions()

  The following screenshot illustrates viewing the Sage tutorial from the command line interface, activated using the command:

  sage: tutorial()

  This command invoked a terminal-based web browser such as Links to view the tutorial.

  

• A mode for automatic names #7482 (William Stein) --- Provide a mode so that undeclared variables magically spring into existence and object oriented notation is not necessary. The target audience is people wanting to simplify use of Sage for calculus for undergraduate students. This new mode currently only works within the notebook. The following screenshot illustates how to use the mode for automatic names.

• Complete rewrite of the load and attach commands: #7514 (William Stein) --- Now the code is uniform between the command line and notebook. It is also much more flexible and sensible. E.g., you can use load and attach as normal functions now, e.g. load('filename.sage'), attach('filename.sage'). Type load? and attach? for more help.

• Rewrite the @parallel decorate to be vastly more robust, flexible, and usable. #6967 (William Stein) --- Now @parallel uses the exact state of the running Sage session, which allows you to do much more robust parallel computations on a multiprocessor computers. In particular, this works:

# File p.sage
def h(s):
    sleep(1)
    return s*s

def f(n1, n2, cores=24):
    @parallel(cores)
    def g(n):
        return h(n)*h(n)
    return [a for _, a in g([n1..n2])]

#------

sage: load p.sage
sage: time f(1,24)
CPU times: user 0.03 s, sys: 0.22 s, total: 0.25 s
Wall time: 2.28 s
[1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, 14641, 28561, 20736, 
 38416, 50625, 65536, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776]

This rewrite involves replacing the old implementation, which used multiprocessing (or Dsage), by a new one which uses the fork system call (it's about 2 pages of code written using only basic Python).

• #7740

Packages

• Upgrade PolyBoRi to latest upstream release version 0.6.3.r1647-20091028#7271 (Martin Albrecht).

• Upgrade ratpoints to latest upstream release version 2.1.3 #7388 (Robert Miller).

• #7692, #7749 (Steven Sivek)

• Upgrade Maxima to latest upstream release version 5.20.1 #7745 (Karl-Dieter Crisman).

• Upgrade Sage Notebook to latest upstream release version 0.6 #7785 (Tim Dumol and William Stein).

• Update Pari/GP to version 2.3.3.p6 #7825 (Peter Jeremy).

• #7856

• #7067

• #7239

• #7387

• #7761

• #7814