Differences between revisions 22 and 23

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

FIXME: summarize ticket #5535.
FIXME: summarize ticket #5658.

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.
FIXME: summarize #5622
FIXME: summarize #5735

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

FIXME: summarize #5685

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:

An implementation of subsets for finite multisets, i.e. sets with repetitions.
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]]
```
FIXME: summarize #5729
FIXME: summarize #5478
FIXME: summarize #5721

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

Diff for "ReleaseTours/sage-3.4.1"