Releases · sagemath/sage

Release Tour

Sage 4.3.1 was released on Jan 20, 2010 (changelog), 193 tickets (PRs) merged, 55 contributors.

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, #7781, #7815, #7817, #7849
We're moving closer towards a FreeBSD port, thanks to the work of Peter Jeremy at ticket #7825.

Algebra

Chinese Remainder Theorem for polynomials #7595 (Robert Miller) -- An implementation of the Chinese Remainder Theorem is needed for general descents on elliptic curves. Here are some examples for polynomial rings:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<y> = K[]
sage: f = y^2 + 3
sage: g = y^3 - 5
sage: CRT(1,3,f,g)
-3/26*y^4 + 5/26*y^3 + 15/26*y + 53/26
sage: CRT(1,a,f,g)
(-3/52*a + 3/52)*y^4 + (5/52*a - 5/52)*y^3 + (15/52*a - 15/52)*y + 27/52*a + 25/52

This also works for any number of moduli:

sage: K.<a> = NumberField(x^3 - 7)
sage: R.<x> = K[]
sage: CRT([], [])
0
sage: CRT([a], [x])
a
sage: f = x^2 + 3
sage: g = x^3 - 5
sage: h = x^5 + x^2 - 9
sage: k = CRT([1, a, 3], [f, g, h]); k
(127/26988*a - 5807/386828)*x^9 + (45/8996*a - 33677/1160484)*x^8 + (2/173*a - 6/173)*x^7 + (133/6747*a - 5373/96707)*x^6 + (-6/2249*a + 18584/290121)*x^5 + (-277/8996*a + 38847/386828)*x^4 + (-135/4498*a + 42673/193414)*x^3 + (-1005/8996*a + 470245/1160484)*x^2 + (-1215/8996*a + 141165/386828)*x + 621/8996*a + 836445/386828
sage: k.mod(f)
1
sage: k.mod(g)
a
sage: k.mod(h)
3

Basic arithmetic

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

* Improvements to complex arithmetic-geometric mean for real and complex double fields  <a class="http" href="http://trac.sagemath.org/sage_trac/ticket/7739">#7739</a> (Robert Bradshaw, John Cremona) --- Adds an `algorithm` option to the method `agm()` for complex numbers. The values of `algorithm` be can: 
   * "pari" --- Call the agm function from the Pari library. 
   * "optimal" --- Use the AGM sequence such that at each stage `(a,b)` is replaced by `(a_1,b_1) = ((a+b)/2,\pm\sqrt{ab})` where the sign is chosen so that `|a_1-b_1| \le |a_1+b_1|`, or equivalently `\Re(b_1/a_1)\ge0`. The resulting limit is maximal among all possible values. 
   * "principal" --- Use the AGM sequence such that at each stage `(a,b)` is replaced by `(a_1,b_1) = ((a+b)/2,\pm\sqrt{ab})` where the sign is chosen so that `\Re(b_1/a_1)\ge0` (the so-called principal branch of the square root). The following examples illustrate that the returned value depends on the algorithm parameter: ```txt
sage: a = CDF(-0.95, -0.65)
sage: b = CDF(0.683, 0.747)
sage: a.agm(b, algorithm="optimal")
-0.371591652352 + 0.319894660207*I
sage: a.agm(b, algorithm="principal")
0.338175462986 - 0.0135326969565*I
sage: a.agm(b, algorithm="pari")
0.080689185076 + 0.239036532686*I

The same thing for multiprecision real and complex numbers has also been implemented #7719 and will be in the next release.

New decorator coerce_binop #383 (Robert Bradshaw) --- The new decroator coerce_binop can be applied to methods to ensure the arguments have the same parent. For example

@coerce_binop
def quo_rem(self, other):
    ...

will guarantee that self and other have the same parent before this method is called.

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

Implement the Gale Ryser theorem #7301 (Nathann Cohen, David Joyner) --- 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. T.S. Michael helped a great deal with the refereeing process. Here are some examples:

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]

Iwahori Hecke algebras on the T basis #7729 (Daniel Bump, Nicolas M. Thiéry) --- Iwahori Hecke algebras are deformations of the group algebras of Coxeter groups, such as Weyl groups (finite or affine). Here are some examples:

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

Coxeter groups: more Bruhat and weak order features #7753 (Nicolas M. Thiéry, Daniel Bump) --- Four new methods implementing the Bruhat order for Coxeter groups. The method bruhat_le() for Bruhat comparison:

sage: W = WeylGroup(["A", 3])
sage: u = W.from_reduced_word([1, 2, 1])
sage: v = W.from_reduced_word([1, 2, 3, 2, 1])
sage: u.bruhat_le(u)
True
sage: u.bruhat_le(v)
True
sage: v.bruhat_le(u)
False
sage: v.bruhat_le(v)
True
sage: s = W.simple_reflections()
sage: s[1].bruhat_le(W.one())
False

The method weak_le() for comparison in weak order:

sage: W = WeylGroup(["A", 3])
sage: u = W.from_reduced_word([1, 2])
sage: v = W.from_reduced_word([1, 2, 3, 2])
sage: u.weak_le(u)
True
sage: u.weak_le(v)
True
sage: v.weak_le(u)
False
sage: v.weak_le(v)
True

The method bruhat_poset() returns the Bruhat poset of a Weyl group:

sage: W = WeylGroup(["A", 3])
sage: P = W.bruhat_poset()
sage: u = W.from_reduced_word([3, 1])
sage: v = W.from_reduced_word([3, 2, 1, 2, 3])
sage: P(u) <= P(v)
True
sage: len(P.interval(P(u), P(v)))
10
sage: P.is_join_semilattice()
False

The method weak_poset() returns the left (resp. right) poset for weak order:

sage: W = WeylGroup(["A", 2])
sage: P = W.weak_poset(); P
Finite poset containing 6 elements
sage: W = WeylGroup(["B", 3])
sage: P = W.weak_poset(side="left")
sage: P.is_join_semilattice(), P.is_meet_semilattice()
(True, True)

Interval exchange transformations #7145 (Vincent Delecroix) --- New module for manipulating interval exchange transformations and linear involutions. Here, we create an interval exchange transformation:

sage: T = iet.IntervalExchangeTransformation(('a b','b a'),(sqrt(2),1))
sage: print T
Interval exchange transformation of [0, sqrt(2) + 1[ with permutation
a b
b a

It can also be initialized using permutation (group theoretic ones):

sage: p = Permutation([3,2,1])
sage: T = iet.IntervalExchangeTransformation(p, [1/3,2/3,1])
sage: print T
Interval exchange transformation of [0, 2[ with permutation
1 2 3
3 2 1

For the manipulation of permutations of IET, there are special types provided by this module. All of them can be constructed using the con...

Sage 4.3 Release Tour

Sage 4.3 was released on Dec 24, 2009 (changelog), 237 tickets (PRs) merged, 63 contributors.

Major features:

Improvements in the build system.
Major new features in the Sage notebook server.
Numerous new features in the combinatorics module.
Many improvements in the graph theory module, both the backend implementation and new features.
Updates/upgrades to 15 packages.

Algebra

Algebraic Topology

#6099

Calculus

#3587

Combinatorics

Commutative Algebra

Miscellaneous

Numerical

Packages

Statistics

#7197

Symbolics

#6803
#7603

Full Changelog: 4.2.1...4.3

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Release Tour

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4.3.1

Release Tour

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4.3

Sage 4.3 Release Tour

Algebra

Algebraic Topology

Calculus

Categories

Combinatorics

Commutative Algebra

Documentation

Graph Theory

Group Theory

Linear Algebra

Miscellaneous

Numerical

Packages

Statistics

Symbolics

4.2.1