# 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 here. The following points are some of the foci of this release:

• Upgrade to the Python 2.6.x series
• Support for building Singular with GCC 4.4
• FreeBSD support for the following packages: FreeType, gd, libgcrypt, libgpg-error, Linbox, NTL, Readline, Tachyon

• Combinatorics: irreducible matrix representations of symmetric groups; and Yang-Baxter Graphs
• Cryptography: Mini Advanced Encryption Standard for educational purposes
• Graph theory: a backend for graph theory using 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; support Unicode in LaTeX cells (notebook); and optimized integer division
• 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 19 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()


## 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 to compute eigenvalues and eigenvectors of integer matrices more precisely than before. Rather than converting 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)
4
12
12

• sage/modules/fg_pid/fgp_module.py -- Finitely generated modules over a principal ideal domain. Currently, only 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

• An optimized Sudoku solver (Rob Beezer, Tom Boothby) -- Support two algorithms for efficiently solving a Sudoku puzzle: a backtrack algorithm and the DLX algorithm. Generally, the DLX algorithm is very fast and very consistent. The backtrack algorithm is very variable in its performance, on some occasions markedly faster than DLX but usually slower by a similar factor, with the potential to be orders of magnitude slower. The following code compares the performance between the Sudoku solver in Sage 4.0.2 and that in this release. We also compare the performance between the backtrack algorithm and the DLX algorithm. All timing statistics were obtained using the machine sage.math:
# BEFORE

sage: A = matrix(ZZ,9,[5,0,0, 0,8,0, 0,4,9, 0,0,0, 5,0,0, 0,3,0, 0,6,7, \
....: 3,0,0, 0,0,1,  1,5,0, 0,0,0, 0,0,0,  0,0,0, 2,0,8, 0,0,0, \
....: 0,0,0, 0,0,0, 0,1,8, \
....: 7,0,0, 0,0,4, 1,5,0, 0,3,0, 0,0,2, 0,0,0,  4,9,0, 0,5,0, 0,0,3])
sage: %timeit sudoku(A);
10 loops, best of 3: 43.5 ms per loop
sage: from sage.games.sudoku import solve_recursive
sage: B = matrix(ZZ, 9, 9, [ [0,0,0,0,1,0,9,0,0], [8,0,0,4,0,0,0,0,0], \
....: [2,0,0,0,0,0,0,0,0], [0,7,0,0,3,0,0,0,0], [0,0,0,0,0,0,2,0,4], \
....: [0,0,0,0,0,0,0,5,8], [0,6,0,0,0,0,1,3,0], [7,0,0,2,0,0,0,0,0], \
....: [0,0,0,8,0,0,0,0,0] ])
sage: %timeit solve_recursive(B, 8, 5);
1000 loops, best of 3: 325 µs per loop

# AFTER

sage: h = Sudoku('8..6..9.5.............2.31...7318.6.24.....73...........279.1..5...8..36..3......')
sage: %timeit h.solve(algorithm='backtrack').next();
1000 loops, best of 3: 1.12 ms per loop
sage: %timeit h.solve(algorithm='dlx').next();
1000 loops, best of 3: 1.58 ms per loop
sage: # These are the first 10 puzzles in a list of "Top 95" puzzles.
sage: top =['4.....8.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......',\
....: '52...6.........7.13...........4..8..6......5...........418.........3..2...87.....',\
....: '6.....8.3.4.7.................5.4.7.3..2.....1.6.......2.....5.....8.6......1....',\
....: '48.3............71.2.......7.5....6....2..8.............1.76...3.....4......5....',\
....: '....14....3....2...7..........9...3.6.1.............8.2.....1.4....5.6.....7.8...',\
....: '......52..8.4......3...9...5.1...6..2..7........3.....6...1..........7.4.......3.',\
....: '6.2.5.........3.4..........43...8....1....2........7..5..27...........81...6.....',\
....: '.524.........7.1..............8.2...3.....6...9.5.....1.6.3...........897........',\
....: '6.2.5.........4.3..........43...8....1....2........7..5..27...........81...6.....',\
....: '.923.........8.1...........1.7.4...........658.........6.5.2...4.....7.....9.....']
sage: p = [Sudoku(top[i]) for i in xrange(10)]
sage: for i in xrange(10):
....:     %timeit p[i].solve(algorithm='dlx').next();
....:     %timeit p[i].solve(algorithm='backtrack').next();
....:
100 loops, best of 3: 2.26 ms per loop
10 loops, best of 3: 223 ms per loop
100 loops, best of 3: 2.6 ms per loop
10 loops, best of 3: 21.3 ms per loop
100 loops, best of 3: 2.38 ms per loop
10 loops, best of 3: 83.5 ms per loop
1000 loops, best of 3: 1.76 ms per loop
10 loops, best of 3: 43.5 ms per loop
1000 loops, best of 3: 1.86 ms per loop
10 loops, best of 3: 316 ms per loop
1000 loops, best of 3: 1.65 ms per loop
10 loops, best of 3: 145 ms per loop
100 loops, best of 3: 1.84 ms per loop
10 loops, best of 3: 547 ms per loop
1000 loops, best of 3: 1.77 ms per loop
10 loops, best of 3: 255 ms per loop
100 loops, best of 3: 2.08 ms per loop
10 loops, best of 3: 445 ms per loop
1000 loops, best of 3: 1.67 ms per loop
10 loops, best of 3: 266 ms per loop

• A decorator for declaring abstract methods (Nicolas Thiéry) -- Support a decorator that can be used to declare a method that should be implemented by derived classes. This declaration should typically include documentation for the specification for this method. The purpose of the decorator is to enforce a consistent and visual syntax for such declarations. The decorator is also used by the Sage categories framework for automated tests. As an example, here we create a class with an abstract method:
sage: class A(object):
....:     @abstract_method
....:     def my_method(self):
....:         """
....:         The method :meth:my_method computes my_method
....:         """
....:         pass
....:
sage: A.my_method
<abstract method my_method at 0x7f53414a7410>


The current policy is that a NotImplementedError is raised when accessing the method through an instance, even before the method is called:

sage: x = A()
sage: x.my_method
Traceback (most recent call last):
...
NotImplementedError: <abstract method my_method at 0x7f53414a7410>

It is also possible to mark abstract methods as optional:
sage: class A(object):
....:     @abstract_method(optional=True)
....:     def my_method(self):
....:         """
....:         The method :meth:my_method computes my_method
....:         """
....:         pass
....:
sage: A.my_method
<optional abstract method my_method at 0x3b551b8>
sage: x = A()
sage: x.my_method
NotImplemented


## Notebook

• Unicode in %latex cells (Peter Mora) -- One can now enter Unicode characters directly in Notebook cells. Here is a screenshot illustrating this:

• Allow $and$ to delimit math in %html blocks (John Palmieri) -- One can now enter

%html
test
$x^2$


and the expression x^2 is typeset in math mode.

## Number Theory

• Improved random_element() method for number field orders and ideals (John Cremona) -- The new method random_element() of the class NumberFieldIdeal in sage/rings/number_field/number_field_ideal.py returns a random element of a fractional ideal, computed as a random ZZ-linear combination of the basis. A similar method has also been implemented for the class Order in sage/rings/number_field/order.py}. Here are some examples on using this new method:

sage: K.<a> = NumberField(x^3 + 2)
sage: I = K.ideal(1 - a)
sage: I.random_element()
2*a^2 + a + 3
sage: I.random_element(distribution="uniform")
-a^2 + 2*a + 2
sage: I.random_element(-30, 30)
-30*a^2 + 17*a - 11
sage: I.random_element(-30,30).parent() is K
True
sage: K.<a> = NumberField(x^3 + 2)
sage: OK = K.ring_of_integers()
sage: OK.random_element()
2*a^2 + 7*a + 2
sage: OK.random_element(distribution="uniform")
-2*a^2 + a - 1
sage: K.order(a).random_element()
-2*a^2 - a - 5

• Support for Michael Stoll's ratpoints package (Robert Miller, Michael Stoll) -- Stoll's ratpoints package is a program for finding points of bounded height on curves of the form y^2 = a_n x^n + ... + a_1 x + a_0. The library code is contained in the Cython module sage/libs/ratpoints.pyx. Here are some examples for working with ratpoints:

sage: from sage.libs.ratpoints import ratpoints
sage: for x,y,z in ratpoints([1..6], 200):
....:     print -1*y^2 + 1*z^6 + 2*x*z^5 + 3*x^2*z^4 + 4*x^3*z^3 + 5*x^4*z^2 + 6*x^5*z
....:
0
0
0
0
0
0
0
sage: for x,y,z in ratpoints([1..5], 200):
....:     print -1*y^2 + 1*z^4 + 2*x*z^3 + 3*x^2*z^2 + 4*x^3*z + 5*x^4
....:
0
0
0
0
0
0
0
0

• Elliptic exponential (John Cremona) -- New method elliptic_exponential() in the class EllipticCurve_rational_field of sage/schemes/elliptic_curves/ell_rational_field.py for computing the elliptic exponential of a complex number with respect to an elliptic curve. A similar method is also defined for the class PeriodLattice_ell in sage/schemes/elliptic_curves/period_lattice.py. Here are some examples:

sage: E = EllipticCurve([1,1,1,-8,6])
sage: P = E([0,2])
sage: z = P.elliptic_logarithm()
sage: E.elliptic_exponential(z)
(-1.6171648557030742010940435588e-29 : 2.0000000000000000000000000000 : 1.0000000000000000000000000000)
sage: z = E([0,2]).elliptic_logarithm(precision=200)
sage: E.elliptic_exponential(z)
(-1.6490990486332025523931769742517329237564168247111092902718e-59 : 2.0000000000000000000000000000000000000000000000000000000000 : 1.0000000000000000000000000000000000000000000000000000000000)

And here are some torsion examples:
sage: E = EllipticCurve('389a')
sage: w1,w2 = E.period_lattice().basis()
sage: E.two_division_polynomial().roots(CC,multiplicities=False)
[-2.04030220028546, 0.135409240221753, 0.904892960063711]
sage: [E.elliptic_exponential((a*w1+b*w2)/2)[0] for a,b in [(0,1),(1,1),(1,0)]]
[-2.04030220028546, 0.135409240221753, 0.904892960063711]
sage: E.division_polynomial(3).roots(CC,multiplicities=False)

[-2.88288879135334,
1.39292799513138,
0.0783137314443164 - 0.492840991709879*I,
0.0783137314443164 + 0.492840991709879*I]
sage: [E.elliptic_exponential((a*w1+b*w2)/3)[0] for a,b in [(0,1),(1,0),(1,1),(2,1)]]

[-2.88288879135335,
1.39292799513138,
0.0783137314443165 - 0.492840991709879*I,
0.0783137314443168 + 0.492840991709879*I]


## Numerical

• Use mpmath to compute constants (Fredrik Johannson, Mike Hansen) -- Previously the functions khinchin(), mertens() and twinprime() in sage/symbolic/constants.py were LimitedPrecisionConstant. Using mpmath, these functions now support arbitrary precision for the corresponding constants. There is now also support for the Glaisher-Kinkelin constant A = \exp(\frac{1}{12}-\zeta'(-1)) using mpmath. Here are some examples on using these functions with the mpmath backend. The Khinchin constant:

sage: float(khinchin)
2.6854520010653062
sage: khinchin.n(digits=60)
2.68545200106530644530971483548179569382038229399446295305115
sage: khinchin._mpfr_(RealField(100))
2.6854520010653064453097148355
sage: RealField(100)(khinchin)
2.6854520010653064453097148355

The Twin Primes constant:
sage: float(twinprime)
0.66016181584686962
sage: twinprime.n(digits=60)
0.660161815846869573927812110014555778432623360284733413319448
sage: twinprime._mpfr_(RealField(100))
0.66016181584686957392781211001
sage: RealField(100)(twinprime)
0.66016181584686957392781211001

The Mertens constant:
sage: float(mertens)
0.26149721284764277
sage: mertens.n(digits=60)
0.261497212847642783755426838608695859051566648261199206192064
sage: mertens._mpfr_(RealField(100))
0.26149721284764278375542683861
sage: RealField(100)(mertens)
0.26149721284764278375542683861

The Glaisher-Kinkelin constant:
sage: float(glaisher)
1.2824271291006226
sage: glaisher.n(digits=60)
1.28242712910062263687534256886979172776768892732500119206374
sage: a = glaisher + 2
sage: parent(a)
Symbolic Ring
sage: glaisher._mpfr_(RealField(100))
1.2824271291006226368753425689
sage: RealField(100)(glaisher)
1.2824271291006226368753425689


## Packages

• New package mpmath version 0.12 for multiprecision floating-point arithmetic (Fredrik Johannson, Mike Hansen) -- The Python package mpmath is now a standard package of Sage. Functions in mpmath can be called from Sage using the library under sage/libs/mpmath, with automatic data conversion between Sage and mpmath.

• New package Ratpoints version 2.1.2 for computing rational points on hyperelliptic curves (Robert Miller, Michael Stoll) -- The C package Ratpoints is now a standard spkg. The corresponding library file is sage/libs/ratpoints.pyx.

• Upgrade Singular to version singular-3-1-0-2-20090620 with support for compiling with GCC 4.4 (Andrzej Giniewicz, Martin Albrecht, Craig Citro).

• Upgrade Sage's Python spkg to the 2.6.x series (Mike Hansen).

• Upgrade Twisted to version 8.2.0 latest upstream release (Mike Hansen).

• Upgrade SCons to version 1.2.0 latest upstream release (Mike Hansen).

• Update the Pynac spkg to version pynac-0.1.8.p1.spkg (Mike Hansen).

• Update the IPython spkg to version ipython-0.9.1.p0.spkg (Mike Hansen).

• Update the ATLAS spkg to version atlas-3.8.3.p5.spkg (David Kirkby).

• Update the CVXOPT spkg to version cvxopt-0.9.p8.spkg (Gonzalo Tornaria).

• Update the FreeType spkg to version freetype-2.3.5.p1.spkg (Peter Jeremy).

• Update the GD spkg to version gd-2.0.35.p2.spkg (Peter Jeremy).

• Update the libgcrypt spkg to version libgcrypt-1.4.3.p1.spkg (Peter Jeremy).

• Update the libgpg_error spkg to version libgpg_error-1.6.p1.spkg (Peter Jeremy).

• Update the linbox spkg to version linbox-1.1.6.p0.spkg (Peter Jeremy).

• Update the NTL spkg to version ntl-5.4.2.p8.spkg (Peter Jeremy).

• Update the Tachyon spkg to version tachyon-0.98beta (Peter Jeremy).

• Update the Rubik spkg to version rubiks-20070912.p9.spkg (William Stein) -- This adds support for compiling Rubiks in parallel.

• Update the python-gnutls spkg to version python_gnutls-1.1.4.p5.spkg (William Stein).

• Update the ATLAS spkg to version atlas-3.8.3.p5.spkg (David Kirkby).

## Symbolics

• Symbolic arctan2 function (Karl-Dieter Crisman) -- New symbolic trigonometric function arctan2 in sage/functions/trig.py. This symbolic function returns the arctangent (measured in radians) of y/x. Unlike arctan(y/x), the signs of both x and y are considered. For example, note the difference between arctan2() and arctan():

sage: arctan2(1,-1)
3/4*pi
sage: arctan(1/-1)
-1/4*pi


The new symbolic function arctan2() is also consistent with the implementations in Python and Maxima:

sage: arctan2(1,-1)  # the symbolic arctan2
3/4*pi
sage: maxima.atan2(1,-1)  # Maxima implementation
3*%pi/4
sage: math.atan2(1,-1)  # Python implementation
2.3561944901923448

We can also compute an approximation:
sage: arctan2(-.5,1).n(100)
-0.46364760900080611621425623146


ReleaseTours/sage-4.1 (last edited 2009-12-26 15:00:52 by Minh Nguyen)