Sage 3.4.1 Release Tour

Sage 3.4.1 was released on FIXME. For the official, comprehensive release note, please refer to sage-3.4.1.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:

• Merging improvements during the Sage Days 13 coding sprint.
• Other bug fixes post Sage 3.4.

Algebra

• Optimized is_primitive() method (Ryan Hinton) -- The method is_primitive() in sage/rings/polynomial/polynomial_element.pyx is used for determining whether or not a polynomial is primitive over a finite field. Prime divisors are calculated during the test for polynomial primitivity. Where n is large, calculating those prime divisors can dominate the running time of the test. The is_primitive() method now has the optional argument n_prime_divs for providing precomputed prime divisors. This optional argument can result in a performance improvement of up to 4x. On the machine sage.math, one has the following timing statistics:

  sage: R.<x> = PolynomialRing(GF(2), 'x')
  sage: nn = 128
  sage: max_order = 2^nn - 1
  sage: pdivs = max_order.prime_divisors()
  sage: poly = R.random_element(nn)
  sage: while not (poly.degree()==nn and poly.is_primitive(max_order, pdivs)):
  ....:     poly = R.random_element(nn)
  ....:     
  sage: %timeit poly.is_primitive()  # without n_prime_divs optional argument
  10 loops, best of 3: 285 ms per loop
  sage: %timeit poly.is_primitive(max_order, pdivs)  # with n_prime_divs optional argument
  10 loops, best of 3: 279 ms per loop
  sage: 
  sage: nn = 256
  sage: max_order = 2^nn - 1
  sage: pdivs = max_order.prime_divisors()
  sage: poly = R.random_element(nn)
  sage: while not (poly.degree()==nn and poly.is_primitive(max_order, pdivs)):
  ....:     poly = R.random_element(nn)
  ....:     
  sage: %timeit poly.is_primitive()  # without n_prime_divs optional argument
  10 loops, best of 3: 3.22 s per loop
  sage: %timeit poly.is_primitive(max_order, pdivs)  # with n_prime_divs optional argument
  10 loops, best of 3: 700 ms per loop

• Speed-up the method order_from_multiple() (John Cremona) -- For groups of prime order n, every non-identity element has order n. The previous implementation of the method order_from_multiple() computes g^n twice when g is not the identity and n is prime. Such double computation is now avoided. Now for each prime p dividing the given multiple of the order, we avoid the last multiplication/powering by p, hence saving some computation time whenever the p-exponent of the order is maximal. The new implementation of order_from_multiple() results in a performance improvement of up to 25%. Here are some timing statistics obtained using the machine sage.math:

  # BEFORE
  sage: F = GF(2^1279, 'a')
  sage: n = F.cardinality() - 1 # Mersenne prime
  sage: order_from_multiple(F.random_element(), n, [n], operation='*') == n
  True
  sage: %timeit order_from_multiple(F.random_element(), n, [n], operation='*') == n
  10 loops, best of 3: 63.7 ms per loop
  
  
  # AFTER
  sage: F = GF(2^1279, 'a')
  sage: n = F.cardinality() - 1 # Mersenne prime
  sage: %timeit order_from_multiple(F.random_element(), n, [n], operation='*') == n
  10 loops, best of 3: 47.2 ms per loop

• Speed-up in irreducibility test (Ryan Hinton) -- For polynomials over the finite field GF(2), the test for irreducibility is now up to 40,000 times faster than previously. On a 64-bit Debian/squeeze machine with Core 2 Duo running at 2.33 GHz, one has the following timing improvements:

  # BEFORE
  sage: P.<x> = GF(2)[]
  sage: f = P.random_element(1000)
  sage: %timeit f.is_irreducible()
  10 loops, best of 3: 948 ms per loop
  sage:
  sage: f = P.random_element(10000)
  sage: %time f.is_irreducible()
  # gave up because it took minutes!
  
  
  # AFTER
  sage: P.<x> = GF(2)[]
  sage: f = P.random_element(1000)
  sage: %timeit f.is_irreducible()
  10000 loops, best of 3: 22.7 µs per loop
  sage:
  sage: f = P.random_element(10000)
  sage: %timeit f.is_irreducible()
  1000 loops, best of 3: 394 µs per loop
  sage:
  sage: f = P.random_element(100000)
  sage: %timeit f.is_irreducible()
  100 loops, best of 3: 10.4 ms per loop

  Furthermore, on Debian 5.0 Lenny with kernel 2.6.24-1-686, an Intel(R) Celeron(R) CPU running at 2.00GHz with 1.0GB of RAM, one has the following timing statistics:

  # BEFORE
  sage: P.<x> = GF(2)[]
  sage: f = P.random_element(1000)
  sage: %timeit f.is_irreducible()
  10 loops, best of 3: 1.14 s per loop
  sage: 
  sage: f = P.random_element(10000)
  sage: %time f.is_irreducible()
  CPU times: user 4972.13 s, sys: 2.83 s, total: 4974.95 s
  Wall time: 5043.02 s
  False
  
  
  # AFTER
  sage: P.<x> = GF(2)[]
  sage: f = P.random_element(1000)
  sage: %timeit f.is_irreducible()
  10000 loops, best of 3: 40.7 µs per loop
  sage: 
  sage: f = P.random_element(10000)
  sage: %timeit f.is_irreducible()
  1000 loops, best of 3: 930 µs per loop
  sage: 
  sage: 
  sage: f = P.random_element(100000)
  sage: %timeit f.is_irreducible()
  10 loops, best of 3: 27.6 ms per loop

Algebraic Geometry

• Refactor dimension() method for schemes (Alex Ghitza) -- Implement methods dimension_absolute() and dimension_relative(), where dimension() is an alias for dimension_absolute(). Here are some examples of using dimension_absolute() and dimension():

  sage: A2Q = AffineSpace(2, QQ)
  sage: A2Q.dimension_absolute()
  2
  sage: A2Q.dimension()
  2

  And here's an example demonstrating the use of dimension_relative():

  sage: S = Spec(ZZ)
  sage: S.dimension_relative()
  0

• Plotting affine and projective curves (Alex Ghitza) -- Improving the plotting usability so it is now easier to plot affine and projective curves. For example, we can plot a 5-nodal curve of degree 11:

  sage: R.<x, y> = ZZ[] 
  sage: C = Curve(32*x^2 - 2097152*y^11 + 1441792*y^9 - 360448*y^7 + 39424*y^5 - 1760*y^3 + 22*y - 1) 
  sage: C.plot((x, -1, 1), (y, -1, 1), plot_points=400)

  Now we plot an elliptic curve:

  sage: E = EllipticCurve('101a') 
  sage: C = Curve(E) 
  sage: C.plot()

Basic Arithmetic

• Speed-up in dividing a polynomial by an integer (Burcin Erocal) -- Dividing a polynomial by an integer is now up to 6x faster than previously. On Debian 5.0 Lenny with kernel 2.6.24-1-686, an Intel(R) Celeron(R) CPU running at 2.00GHz with 1.0GB of RAM, one has the following timing statistics:

  # BEFORE
  sage: R.<x> = ZZ["x"]
  sage: f = 389 * R.random_element(1000)
  sage: timeit("f//389")
  625 loops, best of 3: 312 µs per loop
  
  
  # AFTER
  sage: R.<x> = ZZ["x"]
  sage: f = 389 * R.random_element(1000)
  sage: timeit("f//389")
  625 loops, best of 3: 48.3 µs per loop

• New fast_float supports more datatypes with improved performance (Carl Witty) -- A rewrite of fast_float to support multiple types. Here, we get accelerated evaluation over RealField(k) as well as RDF, real double field. As compared with the previous fast_float, improved performance can range from 2% faster to more than 2x as fast. An extended list of benchmark details is available at ticket 5093.

• Complex double fast callable interpreter (Robert Bradshaw) -- Support for complex double floating point (CDF). The new interpreter is implemented in the class CDFInterpreter of sage/ext/gen_interpreters.py.

• Speed-up the function solve_mod() (Wilfried Huss) -- Performance improvement for the function solve_mod() is now up to 5x when solving an equation or a list of equations modulo a given integer modulus. On the machine sage.math, we have the following timing statistics:

  # BEFORE
  
  sage: x, y = var('x,y')
  sage: time solve_mod([x^2 + 2 == x, x^2 + y == y^2], 14)
  CPU times: user 0.01 s, sys: 0.02 s, total: 0.03 s
  Wall time: 0.18 s
  [(4, 2), (4, 6), (4, 9), (4, 13)]
  sage:
  sage: x,y,z = var('x,y,z')
  sage: time solve_mod([x^5 + y^5 == z^5], 3)
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.10 s
  
  [(0, 0, 0),
   (0, 1, 1),
   (0, 2, 2),
   (1, 0, 1),
   (1, 1, 2),
   (1, 2, 0),
   (2, 0, 2),
   (2, 1, 0),
   (2, 2, 1)]
  
  
  # AFTER
  
  sage: x, y = var('x,y')
  sage: time solve_mod([x^2 + 2 == x, x^2 + y == y^2], 14)
  CPU times: user 0.03 s, sys: 0.01 s, total: 0.04 s
  Wall time: 0.16 s
  [(4, 2), (4, 6), (4, 9), (4, 13)
  sage:
  sage: x,y,z = var('x,y,z')
  sage:  time solve_mod([x^5 + y^5 == z^5], 3)
  CPU times: user 0.01 s, sys: 0.01 s, total: 0.02 s
  Wall time: 0.02 s
  
  [(0, 0, 0),
   (0, 1, 1),
   (0, 2, 2),
   (1, 0, 1),
   (1, 1, 2),
   (1, 2, 0),
   (2, 0, 2),
   (2, 1, 0),
   (2, 2, 1)]

• Optimized binomial function when an input is real or complex floating point (Dan Drake, William Stein) -- The function binomial() for returning the binomial coefficients is now much faster. In some cases, speed efficiency can be up to 4000x. Here are some timing statistics obtained using the machine sage.math:

  # BEFORE
  
  sage: x, y = 1140000.78, 420000
  sage: %timeit binomial(x, y)
  10 loops, best of 3: 1.19 s per loop
  sage: 
  sage: x, y = RR(pi^5), 10
  sage: %timeit binomial(x, y)
  10000 loops, best of 3: 28.2 µs per loop
  sage: 
  sage: x, y = RR(pi^15), 500
  sage: %timeit binomial(x, y)
  1000 loops, best of 3: 799 µs per loop
  sage:
  sage: x, y = RealField(500)(1729000*sqrt(2)), 17000
  sage: %timeit binomial(x, y)
  10 loops, best of 3: 34.4 ms per loop
  
  
  # AFTER
  
  sage: x, y = 1140000.78, 420000
  sage: %timeit binomial(x, y)
  1000 loops, best of 3: 297 µs per loop
  sage: 
  sage: x, y = RR(pi^5), 10
  sage: %timeit binomial(x, y)
  10000 loops, best of 3: 189 µs per loop
  sage: 
  sage: x, y = RR(pi^15), 500
  sage: %timeit binomial(x, y)
  1000 loops, best of 3: 335 µs per loop
  sage: 
  sage: x, y = RealField(500)(1729000*sqrt(2)), 17000
  sage: %timeit binomial(x, y)
  1000 loops, best of 3: 692 µs per loop

• Enhanced nth_root() in ZZ and QQ and related utilities (John Cremona) -- Some consistency in the method nth_root() of ZZ and QQ. There are also some new utility methods for the rational numbers:

  1. prime_to_S_part(self, S=[]) -- Returns self with all powers of all primes in S removed.

  2. is_nth_power(self, int n) -- Returns True if self is an n-th power; else False.

  3. is_S_integral(self, S=[]) -- Determine if the rational number is S-integral.

  4. is_S_unit(self, S=None) -- Determine if the rational number is an S-unit.

Build

Calculus

• Deprecate the calling of symbolic functions with unnamed arguments (Carl Witty, Michael Abshoff) -- Previous releases of Sage supported symbolic functions with "no arguments". This style of constructing symbolic functions is now deprecated. For example, previously Sage allowed for defining a symbolic function in the following way

  f2 = 5 - x^2  # bad; this is deprecated

  But users are encouraged to explicitly declare the variables used in a symolic function. For instance, the following is encouraged:

  sage: x,y = var("x, y")    # explicitly declare your variables
  sage: f(x, y) = x^2 + y^2  # this syntax is encouraged

Coercion

Combinatorics

• Enhancements to the Subsets and Subwords modules (Florent Hivert) -- Numerous enhancements to the modules Subsets and Subwords include:

  1. An implementation of subsets for finite multisets, i.e. sets with repetitions.

  2. Adding the method __contains__ for Subsets and Subwords.

  Here's an example for working with multisets:

  sage: S = Subsets([1, 2, 2], submultiset=True); S
  SubMultiset of [1, 2, 2]
  sage: S.list()
  [[], [1], [2], [1, 2], [2, 2], [1, 2, 2]]
  sage: Set([1,2]) in S  # this uses __contains__ in Subsets
  True
  sage: Set([]) in S
  True
  sage: Set([3]) in S
  False

  And here's an example of using __contains__ with Subwords:

  sage: [] in Subwords([1,2,3,4,3,4,4])
  True
  sage: [2,3,3,4] in Subwords([1,2,3,4,3,4,4])
  True
  sage: [2,3,3,1] in Subwords([1,2,3,4,3,4,4])
  False

• Fix and enhancements to permutations (Sebastien Labbe) -- This corrects the Robinson-Schensted algorithm on trivial permutations. It implements the inverse Robinson-Schensted algorithm:

  sage: Permutation((Tableau([[1,2,4],[3]]), Tableau([[1,3,4],[2]])))
  [3, 1, 2, 4]
  sage: Permutation(([[1,2,4],[3]], [[1,3,4],[2]]))
  [3, 1, 2, 4]

  And it works for arbitrary words (with semi-standard tableaux):

  sage: Permutation(([[1,2,2],[3]], [[1,3,4],[2]]))
  [3, 1, 2, 2]

• First pass of cleanup of the interface of combinatorial classes (Florent Hivert) -- Before the patch, the interface of combinatorial classes had two problems:

  1. There were two redundant ways to get the number of elements len(C) and C.count(). Moreover len must return a plain int where we want an arbitrary large number and even infinity;

  2. There were two redundant ways to get an iterator for the elements C.iterator() and iter(C) (allowing for for c in C: ...) via C.__iter__.

  The patch standardize those issues to:

  1. C.cardinality() which is more explicit and consistent with many other Sage libraries;

  2. iter(C) / for x in C: via C.__iter__ which is clearly more Pythonic.

     The functions iterator() and count() are deprecated (with a warning), but will be removed in a later release. On the other hand, there was no way to keep backward compatibility for len. Indeed, many of function such as list / filter / map try silently to call len, which would have caused miriads of warnings to be issued in seemingly unrelated places. So it was decided to simply remove it and issue an error, suggesting to call cardinality instead.

• New class IntegerListLex for generating integer lists (Nicolas M. Thiery, Florent Hivert) -- The new class provides a Constant Amortized Time iterator through the combinatorial classes of integer lists. For example, we create the combinatorial class of lists of length 3 and sum 2 as follows:

  sage: C = IntegerListsLex(2, length=3); C
  Integer lists of sum 2 satisfying certain constraints
  sage: C.count()
  6
  sage: [p for p in C]
  [[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]

  Here's the list of all compositions of 4:

  sage: list(IntegerListsLex(4, min_part = 1)) 
  [[4], [3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]

• Cleanup of crystal code (Anne Schilling, Nicolas M. Thiery) -- Cartan type is now implemented as the method cartan_type, rather than an attribute as was previously the case.

• Deprecate the function RestrictedPartitions() (Dan Drake) -- The function RestrictedPartitions() in sage/combinat/partition.py is now deprecated and will be removed in a future release. Users are advised to instead consider the function Partitions() with the parts_in keyword, which is functionally equivalent to RestrictedPartitions() but is more memory and time efficient. The timing improvement in Partitions() is up to 5x faster than RestrictedPartitions(). The following memory and timing statistics are produced using the machine sage.math:

  # BEFORE
  
  sage: get_memory_usage()
  721.26171875
  sage: ps = RestrictedPartitions(100, ([1,6..100] + [4,9..100]))
  sage: %time sum(1 for p in ps)
  CPU times: user 27.26 s, sys: 1.06 s, total: 28.32 s
  Wall time: 28.99 s
  74040
  sage: get_memory_usage()
  1807.03515625
  
  sage: get_memory_usage()
  721.265625
  sage: ps = RestrictedPartitions(3000, [10,50,100,500,1000])
  sage: %time sum(1 for p in ps)
  CPU times: user 5.60 s, sys: 0.21 s, total: 5.81 s
  Wall time: 5.95 s
  3506
  sage: get_memory_usage()
  962.54296875
  
  
  # AFTER
  
  sage: get_memory_usage()
  719.3984375
  sage: ps = Partitions(100, parts_in=([1,6..100] + [4,9..100]))
  sage: %time sum(1 for p in ps)
  CPU times: user 5.09 s, sys: 0.01 s, total: 5.10 s
  Wall time: 5.10 s
  74040
  sage: get_memory_usage()
  719.3984375
  
  sage: get_memory_usage()
  719.3984375
  sage: ps = Partitions(3000, parts_in=[10,50,100,500,1000])
  sage: %time sum(1 for p in ps)
  CPU times: user 1.12 s, sys: 0.01 s, total: 1.13 s
  Wall time: 1.13 s
  3506
  sage: get_memory_usage()
  719.3984375

• Speed-up the weyl_characters.py module (Mike Hansen, Daniel Bump, Michael Abshoff) -- The timing efficiency is between 4x to 10x, depending on the operations involved. Here are some timing statistics produced using the machine sage.math:

  # BEFORE
  sage: R = WeylCharacterRing(['B',3], prefix = "R")
  sage: %time r =  R(1,1,0)
  CPU times: user 0.14 s, sys: 0.00 s, total: 0.14 s
  Wall time: 0.14 s
  sage:
  sage: R = WeylCharacterRing(['B',3], prefix = "R")
  sage: %time [R(w) for w in R.lattice().fundamental_weights()]
  CPU times: user 0.25 s, sys: 0.00 s, total: 0.25 s
  Wall time: 0.25 s
  [R(1,0,0), R(1,1,0), R(1/2,1/2,1/2)]
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: %time [A2(0,0,0)+A2(2,1,0), A2(2,1,0)+A2(0,0,0), - A2(0,0,0)+2*A2(0,0,0), 
  -2*A2(0,0,0)+A2(0,0,0), -A2(2,1,0)+2*A2(2,1,0)-A2(2,1,0)]
  CPU times: user 0.18 s, sys: 0.00 s, total: 0.18 s
  Wall time: 0.19 s
  [A2(0,0,0) + A2(2,1,0), A2(0,0,0) + A2(2,1,0), A2(0,0,0), -A2(0,0,0), 0]
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: %time [-x for x in [A2(0,0,0), 2*A2(0,0,0), -A2(0,0,0), -2*A2(0,0,0)]]
  CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
  Wall time: 0.02 s
  [-A2(0,0,0), -2*A2(0,0,0), A2(0,0,0), 2*A2(0,0,0)]
  sage: %timeit [-x for x in [A2(0,0,0), 2*A2(0,0,0), -A2(0,0,0), -2*A2(0,0,0)]]
  10 loops, best of 3: 20 ms per loop
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: chi = A2(0,0,0)+2*A2(1,0,0)+3*A2(2,0,0)
  sage: mu =  3*A2(0,0,0)+2*A2(1,0,0)+A2(2,0,0)
  sage: %time chi - mu
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.01 s
  -2*A2(0,0,0) + 2*A2(2,0,0)
  sage: %timeit chi - mu
  100 loops, best of 3: 8.16 ms per loop
  sage: 
  sage: 
  sage: A2 = WeylCharacterRing(['A',2])
  sage: chi = A2(1,0,0)
  sage: %time [chi^k for k in range(6)]
  CPU times: user 1.05 s, sys: 0.02 s, total: 1.07 s
  Wall time: 1.07 s
  
  [A2(0,0,0),
   A2(1,0,0),
   A2(1,1,0) + A2(2,0,0),
   A2(1,1,1) + 2*A2(2,1,0) + A2(3,0,0),
   3*A2(2,1,1) + 2*A2(2,2,0) + 3*A2(3,1,0) + A2(4,0,0),
   5*A2(2,2,1) + 6*A2(3,1,1) + 5*A2(3,2,0) + 4*A2(4,1,0) + A2(5,0,0)]
  
  
  # AFTER
  
  sage: R = WeylCharacterRing(['B',3], prefix = "R")
  sage: %time r =  R(1,1,0)
  CPU times: user 0.03 s, sys: 0.00 s, total: 0.03 s
  Wall time: 0.03 s
  sage:
  sage: R = WeylCharacterRing(['B',3], prefix = "R")
  sage: %time [R(w) for w in R.lattice().fundamental_weights()]
  CPU times: user 0.05 s, sys: 0.00 s, total: 0.05 s
  Wall time: 0.05 s
  [R(1,0,0), R(1,1,0), R(1/2,1/2,1/2)]
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: %time [A2(0,0,0)+A2(2,1,0), A2(2,1,0)+A2(0,0,0), - A2(0,0,0)+2*A2(0,0,0), -2*A2(0,0,0)+A2(0,0,0), -A2(2,1,0)+2*A2(2,1,0)-A2(2,1,0)]
  CPU times: user 0.04 s, sys: 0.00 s, total: 0.04 s
  Wall time: 0.04 s
  [A2(0,0,0) + A2(2,1,0), A2(0,0,0) + A2(2,1,0), A2(0,0,0), -A2(0,0,0), 0]
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: %time [-x for x in [A2(0,0,0), 2*A2(0,0,0), -A2(0,0,0), -2*A2(0,0,0)]]
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.00 s
  [-A2(0,0,0), -2*A2(0,0,0), A2(0,0,0), 2*A2(0,0,0)]
  sage: %timeit [-x for x in [A2(0,0,0), 2*A2(0,0,0), -A2(0,0,0), -2*A2(0,0,0)]]
  100 loops, best of 3: 3.33 ms per loop
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: chi = A2(0,0,0)+2*A2(1,0,0)+3*A2(2,0,0)
  sage: mu =  3*A2(0,0,0)+2*A2(1,0,0)+A2(2,0,0)
  sage: %time chi - mu
  CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
  Wall time: 0.00 s
  -2*A2(0,0,0) + 2*A2(2,0,0)
  sage: %timeit chi - mu
  1000 loops, best of 3: 771 µs per loop
  sage:
  sage: A2 = WeylCharacterRing(['A',2])
  sage: chi = A2(1,0,0)
  sage: %time [chi^k for k in range(6)]
  CPU times: user 0.20 s, sys: 0.00 s, total: 0.20 s
  Wall time: 0.20 s
  
  [A2(0,0,0),
   A2(1,0,0),
   A2(1,1,0) + A2(2,0,0),
   A2(1,1,1) + 2*A2(2,1,0) + A2(3,0,0),
   3*A2(2,1,1) + 2*A2(2,2,0) + 3*A2(3,1,0) + A2(4,0,0),
   5*A2(2,2,1) + 6*A2(3,1,1) + 5*A2(3,2,0) + 4*A2(4,1,0) + A2(5,0,0)]

Commutative Algebra

• New function weil_restriction() on multivariate ideals (Martin Albrecht) -- The new function weil_restriction() computes the Weil restriction of a multivariate ideal over some extension field. A Weil restriction is also known as a restriction of scalars. Here's an example on computing a Weil restriction:

  sage: k.<a> = GF(2^2) 
  sage: P.<x,y> = PolynomialRing(k, 2)
  sage: I = Ideal([x*y + 1, a*x + 1])
  sage: I.variety() 
  [{y: a, x: a + 1}] 
  sage: J = I.weil_restriction() 
  sage: J 
  Ideal (x1*y0 + x0*y1 + x1*y1, x0*y0 + x1*y1 + 1, x0 + x1, x1 + 1) of 
  Multivariate Polynomial Ring in x0, x1, y0, y1 over Finite Field of size 2

• FIXME: summarize #5146
• FIXME: summarize #5353
• FIXME: summarize #3812

Distribution

Doctest

• FIXME: summarize #5318

Documentation

Geometry

• Improved enumeration of vertices and 0-dimensional faces of LatticePolytope's. There was an inconsistency between indicies of vertices, i.e. columns of the .vertices() matrix, and indicies of 0-dimensional faces, i.e. objects returned by .faces(dim=0). E.g. the 5-th 0-dimensional face could be generated by the 7-th vertex etc. Now the i-th 0-dimensional face is generated by the i-th vertex. (The reason for the old behaviour was the output of the underlying software package PALP, now there is extra sorting.)

Graph Theory

• FIXME: summarize #5623

Graphics

• FIXME: summarize #5606
• FIXME: summarize #5450

Group Theory

• Speed-up in comparing elements of a permutation group (Robert Bradshaw, John H. Palmieri, Rob Beezer) -- For elements of a permutation group, comparison between those elements is now up to 13x faster. On Mac OS X 10.4 with Intel Core 2 duo running at 2.33 GHz, one has the following improvement in timing statistics:

  # BEFORE
  sage: a = SymmetricGroup(20).random_element()
  sage: b = SymmetricGroup(10).random_element()
  sage: timeit("a == b")
  625 loops, best of 3: 3.19 µs per loop
  
  
  # AFTER
  sage: a = SymmetricGroup(20).random_element()
  sage: b = SymmetricGroup(10).random_element()
  sage: time v = [a == b for _ in xrange(2000)]
  CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
  Wall time: 0.00 s
  sage: timeit("a == b")
  625 loops, best of 3: 240 ns per loop

• FIXME: summarize #5264

Interfaces

Linear Algebra

• Deprecate the function invert() (John H. Palmieri) -- The function invert() for calculating the inverse of a dense matrix with rational entries is now deprecated. Instead, users are now advised to use the function inverse(). Here's an example of using the function inverse():

  sage: a = matrix(QQ, 2, [1, 5, 17, 3])
  sage: a.inverse()  
  [-3/82  5/82] 
  [17/82 -1/82] 

• Speed-up in calculating determinants of matrices (John H. Palmieri, William Stein) -- For matrices over Z/nZ with n composite, calculating their determinants is now up to 1500x faster. On Debian 5.0 Lenny with kernel 2.6.24-1-686, an Intel(R) Celeron(R) 2.00GHz CPU with 1.0GB of RAM, one has the following timing statistics:

  # BEFORE
  sage: time random_matrix(Integers(26), 10).determinant()
  CPU times: user 15.52 s, sys: 0.02 s, total: 15.54 s
  Wall time: 15.54 s
  13
  sage: time random_matrix(Integers(256), 10).determinant()
  CPU times: user 15.38 s, sys: 0.00 s, total: 15.38 s
  Wall time: 15.38 s
  144
  
  
  # AFTER
  sage: time random_matrix(Integers(26), 10).determinant()
  CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
  Wall time: 0.01 s
  23
  sage: time random_matrix(Integers(256), 10).determinant()
  CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
  Wall time: 0.00 s

• FIXME: summarize #5715

Miscellaneous

• FIXME: summarize #5638
• FIXME: summarize #5386

Modular Forms

• FIXME: summarize #5520
• FIXME: summarize #5648
• FIXME: summarize #5180

Notebook

FIXME: A number of tickets related to UTF-8 text got merged and should definitely be mentioned! #4547, #5211; #2896 and #1477 got fixed by those tickets. There's also #5564, which may not get merged for 3.4.1 but should get in soon; it pulls together a whole bunch of UTF-8 fixes and improvements.

• FIXME: summarize #5681

Number Theory

• FIXME: summarize #5518
• FIXME: summarize #5508
• FIXME: summarize #793
• FIXME: summarize #4667
• FIXME: summarize #5159
• FIXME: summarize #4990
• FIXME: summarize #3081
• FIXME: summarize #4724
• FIXME: summarize #5673

Numerical

Packages

• FIXME: summarize #4987
• FIXME: summarize #4881
• FIXME: summarize #4880
• FIXME: summarize #4876
• FIXME: summarize #5672
• FIXME: summarize #5240
• FIXME: summarize #5738
• FIXME: summarize #5696
• FIXME: summarize #4987
• FIXME: summarize #5697
• FIXME: summarize #5823

Quadratic Forms

Symbolics

• FIXME: summarize #5737

User Interface

Website / Wiki