3.4.2

Release Tour

Sage 3.4.2 was released on May 05, 2009 (changelog), 74 tickets (PRs) merged, 33 contributors. A nicely formatted version of this release tour can be found here. The following points are some of the foci of this release:

• Improve doctest coverage of the Sage library in anticipation of version 4.0.
• New features for symbolic logic.

Algebra

• Comparison of ring coercion morphisms (Alex Ghitza) -- New comparison method __cmp__() for the class RingHomomorphism_coercion in sage/rings/morphism.pyx. The comparison method __cmp__(self, other) compares a ring coercion morphism self to other. Ring coercion morphisms never compare equal to any other data type. If other is a ring coercion morphism, the parents of self and other are compared. Here are some examples on comparing ring coercion morphisms:

sage: f = ZZ.hom(QQ)
sage: g = ZZ.hom(ZZ)
sage: f == g
False
sage: f > g
True
sage: f < g
False
sage: h = Zmod(6).lift()
sage: f == h
False
sage: f = ZZ.hom(QQ)
sage: g = loads(dumps(f))
sage: f == g
True
 

• Coercing factors into a common universe (Alex Ghitza) -- New method base_change(self, U) in the module sage/structure/factorization.py to allow the factorization self with its factors (including the unit part) coerced into the universe U. Here's an example for working with the new method base_change():

sage: F = factor(2006)
sage: F.universe() 
Integer Ring
sage: P.<x> = ZZ["x"]
sage: F.base_change(P).universe() 
Univariate Polynomial Ring in x over Integer Ring
 

Basic Arithmetic

• Enhancements to symbolic logic (Chris Gorecki) -- This adds a number of utilities for working with symbolic logic:
  1. sage/logic/booleval.py -- For evaluating boolean formulas.
  2. sage/logic/boolformula.py -- For boolean evaluation of boolean formulas.
  3. sage/logic/logicparser.py -- For creating and modifying parse trees of well-formed boolean formulas.
  4. sage/logic/logictable.py -- For creating and printing truth tables associated with logical statements.
  5. sage/logic/propcalc.py -- For propositional calculus. Here are some examples for working with the new symbolic logic modules:

sage: import sage.logic.propcalc as propcalc
sage: f = propcalc.formula("a&((b|c)^a->c)<->b")
sage: g = propcalc.formula("boolean<->algebra")
sage: (f&~g).ifthen(f)
((a&((b|c)^a->c)<->b)&(~(boolean<->algebra)))->(a&((b|c)^a->c)<->b)
sage: f.truthtable()

a      b      c      value
False  False  False  True   
False  False  True   True   
False  True   False  False  
False  True   True   False  
True   False  False  True   
True   False  True   False  
True   True   False  True   
True   True   True   True
 

• New function squarefree_divisors() (Robert Miller) -- The new function squarefree_divisors(x) in the module sage/rings/arith.py allows for iterating over the squarefree divisors (up to units) of the element x. Here, we assume that x is an element of any ring for which the function prime_divisors() works. Below are some examples for working with the new function squarefree_divisors():

sage: list(squarefree_divisors(7))
[1, 7]
sage: list(squarefree_divisors(6))
[1, 2, 3, 6]
sage: list(squarefree_divisors(81))
[1, 3]
 

Combinatorics

• Make cartan_type a method rather than an attribute (Dan Bump) -- For the module sage/combinat/root_system/weyl_characters.py, cartan_type is now a method, not an attribute. For example, one can now invoke cartan_type as a method like so:

sage: A2 = WeylCharacterRing("A2")
sage: A2([1,0,0]).cartan_type()
['A', 2]
 

Commutative Algebra

• Improved performance in MPolynomialRing_libsingular (Simon King) -- This provides some optimization of the method MPolynomialRing_libsingular.__call__(). In some cases, the efficiency is up to 19%. The following timing statistics are obtained using the machine sage.math:

# BEFORE

sage: R = PolynomialRing(QQ,5,"x")
sage: S = PolynomialRing(QQ,6,"x")
sage: T = PolynomialRing(QQ,5,"y")
sage: U = PolynomialRing(GF(2),5,"x")
sage: p = R("x0*x1+2*x4+x3*x1^2")^4
sage: timeit("q = S(p)")
625 loops, best of 3: 321 µs per loop
sage: timeit("q = T(p)")
625 loops, best of 3: 348 µs per loop
sage: timeit("q = U(p)")
625 loops, best of 3: 435 µs per loop


# AFTER

sage: R = PolynomialRing(QQ,5,"x")
sage: S = PolynomialRing(QQ,6,"x")
sage: T = PolynomialRing(QQ,5,"y")
sage: U = PolynomialRing(GF(2),5,"x")
sage: p = R("x0*x1+2*x4+x3*x1^2")^4
sage: timeit("q = S(p)")
625 loops, best of 3: 316 µs per loop
sage: timeit("q = T(p)")
625 loops, best of 3: 281 µs per loop
sage: timeit("q = U(p)")
625 loops, best of 3: 392 µs per loop
 

Graph Theory

• Default edge color is black (Robert Miller) -- If only one edge of a graph is colored red, for example, then the remaining edges should be colored with black by default. Here's an example:

sage: G = graphs.CompleteGraph(5)
sage: G.show(edge_colors={'red':[(0,1)]})
 

Interfaces

• Split off the FriCAS interface from the Axiom interface (Mike Hansen, Bill Page) -- The FriCAS interface is now split off from the Axiom interface and can now be found in the module sage/interfaces/fricas.py.

Modular Forms

• Vast speedup in P1List construction (John Cremona) -- This provides huge improvement in the P1List() constructor for Manin symbols. The efficiency gain can range from 27% up to 6x. Here are some timing statistics obtained using the machine sage.math:

# BEFORE

sage: time P1List(100000)
CPU times: user 4.11 s, sys: 0.08 s, total: 4.19 s
Wall time: 4.19 s
The projective line over the integers modulo 100000
sage: time P1List(1000000)
CPU times: user 192.22 s, sys: 0.60 s, total: 192.82 s
Wall time: 192.84 s
The projective line over the integers modulo 1000000
sage: time P1List(1009*1013)
CPU times: user 31.20 s, sys: 0.05 s, total: 31.25 s
Wall time: 31.25 s
The projective line over the integers modulo 1022117
sage: time P1List(1000003)
CPU times: user 35.92 s, sys: 0.05 s, total: 35.97 s
Wall time: 35.97 s
The projective line over the integers modulo 1000003


# AFTER

sage: time P1List(100000)
CPU times: user 0.78 s, sys: 0.02 s, total: 0.80 s
Wall time: 0.80 s
The projective line over the integers modulo 100000
sage: time P1List(1000000)
CPU times: user 27.82 s, sys: 0.21 s, total: 28.03 s
Wall time: 28.02 s
The projective line over the integers modulo 1000000
sage: time P1List(1009*1013)
CPU times: user 21.59 s, sys: 0.04 s, total: 21.63 s
Wall time: 21.63 s
The projective line over the integers modulo 1022117
sage: time P1List(1000003)
CPU times: user 26.19 s, sys: 0.05 s, total: 26.24 s
Wall time: 26.24 s
The projective line over the integers modulo 1000003
 

Notebook

• Downloading and uploading folders of worksheets (Robert Bradshaw) -- One can now download and upload entire folders of worksheets at once, instead of individual worksheets one at a time. This also allows for downloading only selecting worksheets in one go.
• Reduce the number of actions that trigger taking a snapshot (William Stein, Rob Beezer) -- Snapshots now need to be explicitly requested by clicking the save button. This greatly reduces many unnecessary snapshots.

Number Theory

• Enhanced function prime_pi() for counting primes (R. Andrew Ohana) -- The improved function prime_pi() in sage/functions/prime_pi.pyx implements the prime counting function pi(n). Essentially, prime_pi(n) counts the number of primes less than or equal to n. Here are some examples:

sage: prime_pi(10)
4
sage: prime_pi(100)
25
sage: prime_pi(-10)
0
sage: prime_pi(-0.5)
0
sage: prime_pi(10^10)
455052511
 

• Action of the Galois group on cusps (William Stein) -- New method galois_action() in sage/modular/cusps.py for computing action of the Galois group on cusps for congruence subgroups. The relevant algorithm here is taken from section 1.3 of the following text:
  • S. Glenn. Arithmetic on Modular Curves. Progress in Mathematics, volume 20, Birkhauser, 1982.
    Here are some examples for working with galois_action():

sage: Cusp(1/10).galois_action(3, 50)
1/170
sage: Cusp(oo).galois_action(3, 50)
Infinity
sage: Cusp(0).galois_action(3, 50)
0
 

• Finding elliptic curves with prescribed reduction over QQ (John Cremona) -- New function EllipticCurves_with_good_reduction_outside_S() for constructing elliptic curves with good reduction outside a finite set of primes. This essentially implements the algorithm presented in the paper, but currently only over QQ:
  • J. Cremona and M. Lingham. Finding all elliptic curves with good reduction outside a given set of primes. Experimental Mathematics, 16(3):303--312, 2007. Here are some examples for working with this new function:

sage: EllipticCurves_with_good_reduction_outside_S([])
[]
sage: elist = EllipticCurves_with_good_reduction_outside_S([2])
sage: elist

[Elliptic Curve defined by y^2 = x^3 + 4*x over Rational Field,
 Elliptic Curve defined by y^2 = x^3 - x over Rational Field,
 Elliptic Curve defined by y^2 = x^3 - 11*x - 14 over Rational Field,
 Elliptic Curve defined by y^2 = x^3 - 11*x + 14 over Rational Field,
 Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field,
 Elliptic Curve defined by y^2 = x^3 - 44*x - 112 over Rational Field,
 Elliptic Curve defined by...

3.4.1

Release Tour

Sage 3.4.1 was released on April 22nd, 2009, 226 tickets (PRs) merged. 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 this blog. The following points are some of the foci of this release:

• Upgrade to Cython 0.11.
• Rewrite fast_float to support more data types.
• Improved UTF8/Unicode support in the Notebook.
• Latest upstream versions of MPIR and FLINT.
• Pizer's algorithm for computing Brandt Modules and Brandt Matrices.
• Quadratic twists for p-adic L-functions.
• Overconvergent modular forms for genus 0 primes.
• Many improvements for computing with number field.

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 (William Stein, 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 data types 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 #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) -- 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...

3.4

Release Tour

Sage 3.4 was released on March 11, 2009, 110 tickets (PRs) merged. For the official, comprehensive release note, please refer to sage-3.4.txt. A nicely formatted version of this release tour can be found here. The following points are some of the foci of this release:

• Merging of Jon Hanke's quadratic forms code
• Sphinxifying the Sage documentation and move its content to the main Sage development repository

Here's a summary of features in this release, categorized under various headings.

Algebra

• Rewrite the quaternion algebra module (Jonathan Bober, William Stein, Gonzalo Tornaria, John Voight, Michael Abshoff) -- The quaternion algebra module is completely rewritten to enhance its performance. Note that a number of functions have been removed in the rewritten version.

Graph Theory

• Testing for graph isomorphism (Robert L. Miller) -- Enhanced performance of the graph isomorphism test is_isomorphic(). The improved performance can be up to 3x faster than previously.

Graphics

• Arrowheads in multi-edge digraphs (Emily Kirkman) -- This feature has been in Sage even before this release. However, in version 3.4, Emily worked on enhancing the visualization of multi-edge digraphs. In a multi-edge digraph, the arrowheads pointing to a vertex are now clearly displayed. Here's a plot of a multi-edge digraph, produced using the following code:

sage: S = SupersingularModule(389)
sage: D = DiGraph(S.hecke_matrix(2))
sage: D.plot(vertex_size=50).show(figsize=10)
 

Linear Algebra

• Optimize transpose and antitranspose for dense matrices (Rob Beezer, Yann Laigle-Chapuy) -- A rewrite of sections of the method transpose() in the class sage.matrix.matrix_dense.Matrix_dense, resulting in an improvement of up to 5x, depending on the computer architecture. For a system with architecture

CPU: Intel(R) Core(TM)2 Duo CPU T5450  @ 1.66GHz
RAM: 2066004 KB
Linux kernel: 2.6.24-23
 

one would obtain the following timing and memory statistics for a 3000x3000 identity matrix:

# BEFORE
sage: m=identity_matrix(3000)
sage: time m2=m.transpose(); m3=m.antitranspose()
CPU times: user 14.13 s, sys: 1.11 s, total: 15.44 s
Wall time: 15.44 s
sage: get_memory_usage()
1254.28125

# AFTER
sage: m=identity_matrix(3000)
sage: time m2=m.transpose(); m3=m.antitranspose()
CPU times: user 2.92 s, sys: 0.46 s, total: 3.38 s
Wall time: 3.38 s
sage: get_memory_usage()
835.6171875
 

Furthermore, on KUbuntu 8.10 with architecture

CPU: Intel(R) Core(TM)2 Duo CPU E8500 @ 3.16GHz
RAM: 8 GB
 

for a 5000x5000 identity matrix, the previous time was 11.94s and the new improved time would be about 2.46s.

• Optimize transpose for integer and rational dense matrices (Yann Laigle-Chapuy) -- New methods transpose() and antitranspose() for the classes sage.matrix.matrix_integer_dense.Matrix_integer_dense and sage.matrix.matrix_rational_dense.Matrix_rational_dense. The new method transpose() returns the transpose of an integer (respectively rational) dense matrix without changing the original matrix itself. In addition, the new method antitranspose() returns the antitranspose of an integer (respectively rational) matrix, leaving the original matrix unchanged.

Packages

• Update the libgcrypt spkg to libgcrypt-1.4.3.p0.spkg (Michael Abshoff) -- Originally based on Gnu Privacy Guard (GnuPG), libgcrypt is a general purpose library of cryptographic primitives. As such, it does not provide an implementation of any cryptographic protocols, but rather serves as a collection of cryptographic building blocks.
• Update the Python spkg to python-2.5.2.p9.spkg (Michael Abshoff) -- Python is a general purpose, object oriented programming language. Together with various other open source components, Python serves as a fundamental tool that unify the components of Sage under a common interface.
• Upgrade MPFR to version 2.4.1 upstream release (Michael Abshoff) -- Version 2.4.1 of MPFR fixes critical security vulnerabilities in mpfr_snprintf() and mpfr_vsnprintf(), in particular, buffer overflow or off-by-one vulnerabilities. These vulnerabilities have been reported in previous versions of MPFR, and they can be exploited to compromise an application that uses the MPFR library. Users are advised to upgrade to the new MPFR spkg mpfr-2.4.1.spkg. A Secunia security advisory can be found here and a SecurityFocus security advisory is also available.
• Upgrade PyCrypto to version 2.0.1 upstream release (Michael Abshoff) -- Version 2.0.1 fixes a buffer overflow vulnerability in the hash functions SHA256 and RIPEMD, which previously failed to adequately verify user-supplied input. The affected module is ARC2, an implementation of the RC2 block cipher. A successful exploit may allow an attacker to execute arbitrary code in the context of applications that uses the previously vulnerable ARC2 module. Furthermore, a failed attempt may lead to a denial-of-service attack. Users are advised to upgrade to the new PyCrypto spkg pycrypto-2.0.1.p3.spkg. A SecurityFocus security advisory can be found here.

Quadratic Forms

• Merge Jon Hanke's quadratic forms code (Gonzalo Tornaria, Michael Abshoff) -- John Hanke has written a substantial amount of Sage code for working with quadratic forms. Hanke's code can serve as base for further enhancement to the case of binary quadratic forms, which are quadratic forms involving two variables. There are currently a number of patches on the trac server for enhancing the functionalities of binary quadratic forms.
• Clifford invariant and Clifford conductor (Gonzalo Tornaria) -- New functions clifford_invariant() and clifford_conductor() for computing Clifford invariants and conductors. The Clifford invariant is the class in the Brauer group of the Clifford algebra for even dimension. The new function clifford_invariant() computes the Clifford invariant of the even Clifford algebra for odd dimension. The new function clifford_conductor() computes the Clifford conductor, i.e. the product of all primes where the Clifford invariant is -1. See the following text for the definition of the Clifford invariant and p.119 for the formula relating it to the Hasse invariant:
  * T.Y. Lam. "Introduction to Quadratic Forms over Fields". Graduate Studies in Mathematics, vol.67. American Mathematical Society, 2005.

Full Changelog: 3.3...3.4

3.3

Release Tour

Sage 3.3 was released on February 21st, 2009 (changelog), 385 tickets (PRs) merged, 57 contributors. There's also a beautifully formatted version of this release tour. The following points are some of the foci of this release:

• Clean up various doctest failures from 3.2.3
• Fix some build issues from 3.2.3 on the new set of supported images
• Merge small to medium sized patches ready to go in
• Switch to MPIR for multi-precision integers and rationals
• Update the gmp-mpir spkg
• Switch to FLINT for univariate polynomial arithmetic over Z/nZ
• Upgrade NetworkX to version 0.99 upstream release
• Upgrade cddlib to version 0.94f upstream release
• Some improvements to the lrs spkg
• Pynac interface enhancements
• Update the readline spkg
• Update the ATLAS spkg
• Upgrade ATLAS to version 3.8.3 upstream release
• Update the NTL spkg
• Upgrade M4RI to version 20090105 upstream release
• Upgrade jsMath to version 3.6 upstream release
• Upgrade GAP to version 4.4.12 upstream release
• Upgrade GUAVA to version 3.9 upstream release
• Upgrade Jmol to version 11.6 upstream release
• Upgrade matplotlib to version 0.98.5.3-svn6910 upstream release
• Upgrade libpng to version 1.2.34 upstream release
• Upgrade Sphinx to version 0.5.1 upstream release
• Upgrade bzip2 to version 1.0.5 upstream release
• Upgrade trac to version 0.11 upstream release
  All tickets in the 3.3 milestone can be found on the trac server. Here's a summary of features in this release, categorized under various headings.

Algebra

• Transitivity for permutation groups (William Stein) -- In the permutation group module permgroup.py, the query function is_transitive() returns whether or not the group is transitive on [1..G.degree()]. A few surrounding docstrings are fixed and doctest coverage for the module sage.groups.perm_gps.permgroup.py is now 100%.
• Update the ATLAS spkg (Michael Abshoff).
• New function is_unit() for symbolic ring (Florent Hivert) -- The new function sage.calculus.calculus.SymbolicExpressionRing.is_unit() returns whether or not an element of the symbolic ring is a unit.

Algebraic Geometry

• Improved precision and performance when calculating analytic rank (William Stein) -- When calculating the analytic rank of an elliptic curve, the default is to use Cremona's gp script, where the precision is automatically doubled until it doesn't fail. The precision is started at 16 rather than the previous default precision. The computation is now about 3 times faster usually by starting off using this smaller precision. Here's an example:

# BEFORE
sage: E = EllipticCurve('5077a')
sage: time E.analytic_rank()
CPU times: user 0.01 s, sys: 0.01 s, total: 0.02 s
Wall time: 0.21 s

# AFTER
sage: E = EllipticCurve('5077a')
sage: time E.analytic_rank()
CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s
Wall time: 0.06 s
 

And another:

# BEFORE
sage: time elliptic_curves.rank(4)[0].analytic_rank()
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.50 s

# AFTER
sage: time elliptic_curves.rank(4)[0].analytic_rank()
CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s
Wall time: 0.33 s
 

• Weil pairing (David Moller Hansen, John Cremona) -- A basic framework for Weil pairing on elliptic curves using Miller's algorithm as contained in Proposition 8 of the following paper:
  * Victor S. Miller. "The Weil pairing, and its efficient calculation". Journal of Cryptology, 17(4):235-261, 2004.

Basic Arithmetic

• ivalue field in integer_mod.pyx is no longer public (Craig Citro) -- The ivalue field for IntegerMod_int is no longer public. This gives about a 1.5 to 2X speed-up when multiplying IntegerMod_ints. Here's an example:

# BEFORE
sage: R = Integers(100) ; x = R(3) ; y = R(5)
sage: timeit('x*y')
625 loops, best of 3: 403 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 370 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 410 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 405 ns per loop

# AFTER
sage: R = Integers(100) ; x = R(3) ; y = R(5)
sage: timeit('x*y')
625 loops, best of 3: 190 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 213 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 174 ns per loop
sage: timeit('x*y')
625 loops, best of 3: 191 ns per loop
 

• Some fixes for is_perfect_power() and bessel_J(0,0) (Craig Citro, Robert Bradshaw, Robert L. Miller) -- A temporary work around for an upstream bug in GMP when using is_perfect_power(). Resolved a Pari interface bug when using bessel_J(0,0).
• Improved performance for generic polynomial rings, and for univariate polynomial arithmetic over Z/nZ[x] (Yann Laigle-Chapuy, Martin Albrecht, Burcin Erocal) -- Improved performance when performing modulo arithmetic between elements of a generic polynomial ring. Univariate polynomial arithmetic over Z/nZ[x] now has considerable speed-up at approximately 20x.

# BEFORE
sage: P.<x> = PolynomialRing(GF(7))
sage: type(x)
<type 'sage.rings.polynomial.polynomial_modn_dense_ntl.Polynomial_dense_mod_p'>
sage: f = P.random_element(100)
sage: g = P.random_element(100)
sage: %timeit f*g
1000 loops, best of 3: 445 µs per loop

# AFTER
sage: P.<x> = PolynomialRing(GF(7))
sage: type(x)
<type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'>
sage: f = P.random_element(100)
sage: g = P.random_element(100)
sage: %timeit f*g
100000 loops, best of 3: 7.92 µs per loop
 

• Technical preview of David Harvey's zn_poly library exposed to the Sage library (Martin Albrecht).

def f(p,n):
    P = PolynomialRing(GF(next_prime(p)),'x')
    f = P.random_element(n)
    g = P.random_element(n)

    t0 = cputime()
    r0 = f*g
    t0 = cputime(t0)

    t1 = cputime()
    r1 = f._mul_zn_poly(g)
    t1 = cputime(t1)

    assert(r0 == r1)

    return p,n,t0,t1

for i in range(21): 
   f(2**47,2**i)
 

returns on sage.math

# (140737488355328, 1, 0.0, 0.0)
# (140737488355328, 2, 0.0, 0.0)
# (140737488355328, 4, 0.00099999999999766942, 0.0)
# (140737488355328, 8, 0.0, 0.0)
# (140737488355328, 16, 0.0, 0.0)
# (140737488355328, 32, 0.0059990000000027521, 0.0)
# (140737488355328, 64, 0.0, 0.0)
# (140737488355328, 128, 0.0, 0.0)
# (140737488355328, 256, 0.0, 0.0)
# (140737488355328, 512, 0.0, 0.00099999999999766942)
# (140737488355328, 1024, 0.00099999999999766942, 0.0)
# (140737488355328, 2048, 0.0020000000000024443, 0.0019989999999978636)
# (140737488355328, 4096, 0.0049989999999979773, 0.005000000000002558)
# (140737488355328, 8192, 0.010998000000000729, 0.011997999999998399)
# (140737488355328, 16384, 0.023995999999996798, 0.023997000000001378)
# (140737488355328, 32768, 0.050992000000000814, 0.052991999999996153)
# (140737488355328, 65536, 0.1149820000000048, 0.10598499999999689)
# (140737488355328, 131072, 0.29195599999999189, 0.21996599999999944)
# (140737488355328, 262144, 0.6119070000000022, 0.45393199999999467)
# (140737488355328, 524288, 1.5217689999999919, 1.0278430000000043)
# (140737488355328, 1048576, 3.1365230000000111, 2.0966819999999871)
 

• Deprecate the function sqrt_approx() (David Roe) -- To obtain a numerical approximation of the square root of a ring element (integers, polynomials over GF(2^x), rationals), users are advised to use the function sqrt() with a given number of bits of precision instead.
• Use Pohlig-Hellman for generic discrete logarithm (Yann Laigle-Chapuy) -- This results in significant improvement in performance and less memory foot print. Here's an example with a smooth order:

sage: factor(5^15-1)
2^2 * 11 * 31 * 71 * 181 * 1741

# BEFORE
sage: F.<a>=GF(5^15)
sage: g=F.gen()
sage: u=g^123456789
sage: time log(u,g)
CPU times: user 271.39 s, sys: 4.72 s, total: 276.11 s
Wall time: 276.96 s
123456789
sage: get_memory_usage()
378.21875

# AFTER
sage: F.<a>=GF(5^15)
sage: g=F.gen()
sage: u=g^123456789
sage: time log(u,g)
CPU times: user 0.14 s, sys: 0.00 s, total: 0.14 s
Wall time: 0.16 s
123456789
sage: get_memory_usage()
115.8984375
 

And here's another example with a not-so-smooth order:

sage:factor(3^13-1)
2 * 797161

# BEFORE
sage: F.<a>=GF(3**13)
sage: g=F.gen()
sage: u=g^1234567
sage: timeit('log(u,g)')
5 loops, best of 3: 1.54 s per loop
sage: get_memory_usage()
155.11328125
...

3.2.3

Release Tour

Sage 3.2.3 was released on January 5th, 2009 (changelog), 40 tickets (PRs) merged, 17 contributors. The following points are some of the foci of this release:

• Fixed performance regression in eisenstein_submodule.py introduced in Sage 3.2.2.
• Fixed readline build troubles on OpenSUSE 11.1.
• Disabled DSage tests.
• Merged some last minute non-invasive small tickets.
  Here's a summary of features in this release, categorized under various headings.

Algebra

• Powers of polynomial variables (Alex Ghitza) -- Report an error message when a determinate of a (multivariate) polynomial is raised to a fractional exponent. Previously, raising a polynomial determinate to a fractional power has the effect of rounding the exponent to an integer. As yet, fractional powers are not supported.
• Extensions of finite fields (Alex Ghitza) -- Implements methods random_element() and order() for quotients of polynomial rings. The method order() returns the number of elements of a quotient ring, and random_element() returns a random element of a quotient ring.
• Conjugates for integer, rational and real numbers (Alex Ghitza) -- Implements trivial conjugate() methods for elements over the integers, rationals, and reals. Conjugates work (trivially) for matrices over rings that embed canonically into the real numbers.
• Square roots of Galois field elements (John Cremona, William Stein) -- Improve the square root of an element of a Galois field GF(2^e), where e > 15. Previously, this works fine except for the square root of 1, where 1 is an element of GF(2^e) for e > 15.

Build

• Upgrade ATLAS in Sage to version 3.8.2 (Michael Abshoff) -- An update of the ATLAS spkg to the upstream version 3.8.2. This upstream version now provides: (1) better detection of Pentium D and E; (2) detect more Core 2 Duo cores; and (3) properly detect Dunnington cores. Versions 3.8.x for x < 2 sometimes detect a modern CPU architecture as an older architecture, hence causing a massive blow up in the time it takes to compile ATLAS on systems like Xeon core 2 quad, Itanium 2, and Xeon E5420.
• Update optional Sage package polymake (Michael Abshoff) -- The updated optional Sage package is polymake-2.2.p5. Earlier versions hard coded spkg versions of cddlib and gmp, and could cause polymake to break in Sage versions 3.0.3 and 3.0.4.

Coercion

• Fix performance regression in eisenstein_submodule.py (Robert Bradshaw) -- Performance regression in eisenstein_submodule.py was due to cyclotomic coercion. Previously, it would take about 73.3 seconds to run all doctests in eisenstein_submodule.py. Now, the performance is substantially increased such that all dotests in eisenstein_submodule.py should now take about 3.4 seconds.

Doctest

• Disable DSage doctests (Michael Abshoff) -- Doctesting DSage is disabled for now due to a number of problems in the doctests. This issue is expected to be revisited in the 3.4.x series, the earliest one probably is 3.4.alpha0. It will not be considered in Sage 3.3 since that release is mainly concerned with sphinxifying the Sage documentation.
• Numerical noise in Sage 3.2.2 (Michael Abshoff) -- Compiling with GCC 4.3.2, the module sage/rings/number_field/number_field_morphisms.pyx exhibited numerical noise during doctesting.
• matrix1.pyx reference related doctest crash (William Stein).

Graphics

• Some fixes to matrix_plot() and the plotting of gamma(x) (Mike Hansen, Robert Bradshaw).
• Fix fallout in refactoring the plotting module (William Stein, Mike Hansen) -- Sage 3.2.1 refactored plot.py so that it was splitted up into multiple modules. However, the functions xmin/xmax/ymin/ymax were all removed without deprecating them. These are now added back exactly as before, since they are depended upon by a lot of plotting code.

Linear Algebra

• Upgrade ATLAS to version 3.8.2 (Michael Abshoff).

Miscellaneous

• Remove ancient files (Michael Abshoff) -- The following files are now removed from Sage
  1. sage/functions/elementary.py
  1. sage/rings/interval.py
  1. sage/schemes/elliptic_curves/heegner.py

Notebook

• By default, the twisted package is no longer imported at startup (William Stein).
• Support "application shortcut" in chrome and gears (Alexander Hupfer, Timothy Clemans).

Packages

• Upgrade FLINT to version 1.0.21 (Michael Abshoff).
• Upgrade ECM to version 6.2.1 (Michael Abshoff).

Full Changelog: 3.2.2...3.2.3

3.2.2

Release Tour

Sage 3.2.2 was released on December 30, 2008 (changelog), 105 tickets (PRs) merged, 41 contributors.

Algebra

• Using Python's (version 2.5) pickling protocols (Burcin Erocal) -- Changed sage.structure.element.Element to use Python's pickling protocol via __getstate__() and __setstate__(). The previous pickling implementation in sage.structure.element.make_element is retained for backward compatibility.
• Method injvar() is deprecated (John Palmieri) -- The method injvar() in sage/structure/category_object.pyx is now deprecated. One should instead use inject_variables() in order to make variable names available for interactive use.

Basic Arithmetic

• Fraction fields (Burcin Erocal) -- Updated the sage.rings.fraction_field.FractionField_generic class to the new coercion model, and Cythonize the sage.rings.fraction_field_element.FractionFieldElement class. Homomorphisms of fraction fields now work, and the random_element() method of fraction fields returns sensible results.

Build

• Controlling the number of threads used for parallel testing (Dan Drake) -- Added the NUM_THREADS variable to the file SAGE_ROOT/makefile to make it easier to control the number of threads used during parallel testing. Previously the number of threads was hard coded into SAGE_ROOT/makefile at various places, which made the file rather difficult to maintain.

Calculus

• Derivative of a vector and a matrix (Jason Grout) -- Given a vector or matrix of differentiable expressions, the entries in that vector or matrix can be differentiated. This is handy for working with differential equations when we want to do differentiation and integration of matrices and vectors, with the exact same answer as obtained by using the apply_map method.
• Cleaned up implementation of piecewise-defined functions (Mike Hansen, Paul Butler) -- Some updating of the class sage/functions/piecewise.py. This includes not explicitly using Maxima where not necessary in order to take advantage of pynac in the future. When differentiating piecewise functions where some piece uses multiplication, the expression that is passed to Maxima is properly formatted for Maxima to work with that expression.

Coercion

• A factory and pickling framework (Robert Bradshaw) -- Uniqueness of parents makes Sage operate much more smoothly. This leads to an enormous amount of nearly identical caching code scattered throughout the library. This factory handles all the caching and also provides a good pickling mechanism.

Words

• A library for studying and manipulating words (Arnaud Bergeron, Amy Glen, Sebastien Labbe, Franco Saliola) -- This adds lots of functionality for combinatorics on words. The new features are highlighted in this Sage worksheet:
  WordsWorksheet.pdf

Full Changelog: 3.2.1...3.2.2

3.2.1

Release Tour

Sage 3.2.1 was released on December 1, 2008 (changelog), 120 tickets (PRs) merged, 43 contributors.

Algebra

• Divisors over integers (Robert Bradshaw) -- A much simpler and faster algorithm for the divisors function over integers. The new optimized code is faster than a similar integer divisor function in the version of PARI/GP that's bundled with Sage 3.2.1, as well as outperforming a similar integer divisor function found in the version of Magma that Sage 3.2.1 interfaces with.
• Finite field operations (John Palmieri) -- A few methods for finite field elements including additive order, p-th power, and p-th root where p is the characteristic of the field.

Basic arithmetic

• Polynomials over a field (Burcin Erocal) -- Improving the user interface of polynomial classes.
• Polynomial square roots (John Palmieri, Carl Witty) -- A method to test whether a polynomial is square over the field it is defined. If the polynomial is square, then the method has the option of returning a square root.

Build

• Improve sage -upgrade (William Stein, Michael Abshoff) -- The Sage upgrade command can now take an optional URL from which it will pull all spkg's, and this URL can be a Sage install. The upgrade command lists packages that will be upgraded before upgrading them, and autodownloads a new version of any spkg that hasn't successfully been installed before upgrading it.
• Problematic CPU flags (William Stein, Michael Abshoff) -- Binary distributions of Sage for Linux (e.g. Ubuntu) may not work properly once installed. The following CPU flags are known to prevent Sage from running properly: sse, 3d, mmx, pni, and cmov.

Calculus

• Gamma and factorial functions (Mike Hansen, Burcin Erocal, Wilfried Huss) -- Symbolic gamma and factorial functions.
• Update to sympy-0.6.3 (Ondrej Certik) -- Update to the latest upstream of SymPy (sympy-0.6.3), which is a Python library for symbolic mathematics. For more information about SymPy, please visit http://code.google.com/p/sympy/.
• Numerical trigonometry (Robert Bradshaw) -- Optimized floating point evaluation of trigonometric functions such as sine and cosine. For example, numerical calculation of sine via fast_float is now twice as fast as math.sin.
• Floating point calculation (Robert Bradshaw) -- Changing the parsing code for numerical computation to use RDF, which is a better reflection of the underlying precision. For calculus expressions involving real numbers, redundant trailing zeros are removed.

Coercion

• Coercion API (Robert Bradshaw) -- Some simplification of the coercion interface.

Combinatorics

• Coding theory (David Joyner) -- Several changes in linear_codes.py which should speed up (and in some cases do:-) some coding theory computations considerably. It adds interfaces to Cython and C functions of Robert Miller, CJ Tjhal, and Jeffery Leon. Speed up of minimum_distance (for codes over GF(2) and GF(3)), the spectrum (=weight_distribution), and permutation_automorphism_group are expected and in most cases achieved. (Also a new function is_permutation_equivalent was added, which interfaces with Robert Miller's double coset partition refinement code.)
• Incidence structures and block designs (David Joyner) -- Beginning of an incidence structure class and an implementation of some basic block design algorithms. A few functions require GAP's Design package (which is included in gap_packages-4.4.10_6.spkg) but calling GAP or GAP's Design was only done when the corresponding Sage functionality was missing. Robert Miller's recent code on computing the automorphism group of a non-linear binary code was used to implement the automorphism group of a block design.

Finance

• Obtaining high resolution financial data (Chris Swierczewski, Brett Nakashima, William Stein) -- Refined ability to obtain weekly and daily financial data from Google Finance and Yahoo Finance. Input options, such as start date and end date, are more user-friendly. More elaborate documentation. Some support for international (read: non-NASDAQ or NYSE) stock exchanges.

Testing

• Added only_optional doctest option (William Stein) -- Added a new option sage -t -only_optional=component that allows one to run only the optional doctests that depend on a given component. Thus much of the optional functionality of Sage will now be much easier to automatically test.

Full Changelog: 3.2...3.2.1

3.2

Release Tour

Sage 3.2 was released on November 20th, 2008 (changelog); 183 tickets (PRs) merged, 39 contributors. For the official, comprehensive release notes, see the HISTORY.txt file that comes with the release.

Subspace generation

Robert Miller: generate all subspaces of a vector space/projective space

New Symbolics

William Stein, Burcin Erocal: high level integration of pynac into Sage

General group algebras class

David Loeffler: general group algebras class

Elliptic Curve Doctesting

Paul Zimmermann: much improved elliptic curve doctests and some bug fixes

Modular Forms

Craig Citro: huge number of small fixes to modular forms code. David Loeffler: eta product modular functions. Craig Citro: massively speed up Victor Miller basis code.

Magma Interface

William Stein: much improved Magma interface with 100% doctests

Generalized Bernoulli Numbers

William Stein: massively optimized generalized Bernoulli numbers

Modular Composition

Martin Albrecht, Paul Zimmermann: improve modular composition in GF(2)[x]

Polyhedral Improvements

Marshall Hampton: Schlegel diagrams, standard polytopes, multiplication, polars

Notebook

Timothy Clemans: notebook templatization work

Sage Build

William Stein: make it so "sage -br" does the cythonization in parallel using pyprocessing. Robert Bradshaw, Gonzalo Tonaria, Craig Citro: massively cleaned up and faster setup.py. Robert Bradshaw: update to Cython 0.10.

libSingular

Simon King, Martin Albrecht: fix memory leaks in libsingular's reduce()

Numerical Linear Algebra

Jason Grout: make numpy the backend for matrices over CDF and RDF

Graph Theory

Jason Grout: much more robust planarity testing code for graphs

Full Changelog: 3.1.2...3.2

3.1.4

Release Tour

Sage 3.1.4 was released on October 20th, 2008 (changelog), 6 tickets (PRs) merged, 5 contributors. For the official, comprehensive release notes, see the HISTORY.txt file that comes with the release.

IPython vs. init.sage startup issue

Besides some small fixes the main problem was that due to IPython changes in 3.1.3, Sage no longer started when a user had an init.sage.

Full Changelog: 3.1.3...3.1.4

3.1.3

Release Tour

Sage 3.1.3 was released on October 14th, 2008 (changelog), 125 tickets (PRs) merged, 41 contributors. For the official, comprehensive release notes, see the HISTORY.txt file that comes with the release.

ReST Documentation Preparation

Mike Hansen: inclusion of Sphinx, Docutils, Jinja, and Pygments as a step toward ReST documentation

coercion improvements

Robert Bradshaw: Move over more rings to the new coercion model.

Sage-combinat Integration

Nicolas Thiery, Mike Hansen: tighter integration with sage-combinat, i.e. the 2144 server is now installable with vanilla Sage

libSingular Improvements

Martin Albrecht: MPolynomial_libsingular improvements for number fields and ZZ

Gröbner bases over Z and Z/nZ

Martin Albrecht: Add a toy implementation for Gröbner bases over Z and Z/nZ as well as optional binding to M2's Gröbner base engine.

New Symbolics

William Stein, Burcin Erocal: add initial pynac-0.1.p0 package

Modular Symbols

William Stein, Craig Citro: optimize fast computation of hecke eigenvalues on weight 2 modular symbols for gamma0

Build Improvements

Michael Abshoff: numerous Solaris build fixes

Full Changelog: 3.1.2...3.1.3