Differences between revisions 11 and 21 (spanning 10 versions)
⇤ ← Revision 11 as of 2009-05-19 23:00:53 →
Size: 5071
Editor: Minh Nguyen
Comment: Reminders to showcase features
← Revision 21 as of 2009-06-01 11:51:45 → ⇥
Size: 20876
Editor: MarshallHampton
Comment:
Deletions are marked like this. Additions are marked like this.
Line 14: Line 14:
 * FIXME: summarize #6052  * Partial fraction decomposition for irreducible denominators (Gonzalo Tornaria) -- For example, over the field {{{ZZ[x]}}} you can do
 {{{
sage: R.<x> = ZZ["x"]
sage: q = x^2 / (x - 1)
sage: q.partial_fraction_decomposition()
(x + 1, [1/(x - 1)])
sage: q = x^10 / (x - 1)^5
sage: whole, parts = q.partial_fraction_decomposition()
sage: whole + sum(parts)
x^10/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)
sage: whole + sum(parts) == q
True
 }}}
 and over the finite field {{{GF(2)[x]}}}:
 {{{
sage: R.<x> = GF(2)["x"]
sage: q = (x + 1) / (x^3 + x + 1)
qsage: q.partial_fraction_decomposition()
(0, [(x + 1)/(x^3 + x + 1)])
 }}}
Line 28: Line 47:
 * FIXME: summarize #6036

 * FIXME: summarize #6080
 * Utility methods for integer arithmetics (Fredrik Johansson) -- New methods {{{trailing_zero_bits()}}} and {{{sqrtrem()}}} for the class {{{Integer}}} in {{{sage/rings/integer.pyx}}}:
  * {{{trailing_zero_bits(self)}}} -- Returns the number of trailing zero bits in {{{self}}}, i.e. the exponent of the largest power of 2 dividing {{{self}}}.
  * {{{sqrtrem(self)}}} -- Returns a pair {{{(s, r)}}} where {{{s}}} is the integer square root of {{{self}}} and {{{r}}} is the remainder such that {{{self = s^2 + r}}}.
 Here are some examples for working with these new methods:
 {{{
sage: 13.trailing_zero_bits()
0
sage: (-13).trailing_zero_bits()
0
sage: (-13 >> 2).trailing_zero_bits()
2
sage: (-13 >> 3).trailing_zero_bits()
1
sage: (-13 << 3).trailing_zero_bits()
3
sage: (-13 << 2).trailing_zero_bits()
2
sage: 29.sqrtrem()
(5, 4)
sage: 25.sqrtrem()
(5, 0)
 }}}
Line 39: Line 77:
 * FIXME: summarize #6111
Line 42: Line 83:
 * Coercion from float to rationals (Robert Bradshaw) -- One can now coerce a number of type float to {{{QQ}}}. Here's an example:
 {{{
sage: a = float(1.0)
sage: QQ(a)
1
sage: type(a); type(QQ(a))
<type 'float'>
<type 'sage.rings.rational.Rational'>
 }}}
Line 44: Line 96:
 * FIXME: summarize #5502

 * FIXME: summarize #5586
 * ASCII art output for Dynkin diagrams (Dan Bump) -- Support for ASCII art representation of [[http://en.wikipedia.org/wiki/Dynkin_diagram|Dynkin diagrams]] of a finite Cartan type. Here are some examples:
 {{{
sage: DynkinDiagram("E6")

        O 2
        |
        |
O---O---O---O---O
1 3 4 5 6
E6
sage: DynkinDiagram(['E',6,1])

        O 0
        |
        |
        O 2
        |
        |
O---O---O---O---O
1 3 4 5 6
E6~
 }}}


 * Crystal of letters for type E (Brant Jones, Anne Schilling) -- Support crystal of letters for type E corresponding to the highest weight crystal {{{B(\Lambda_1)}}} and its dual {{{B(\Lambda_6)}}} (using the Sage labeling convention of the Dynkin nodes). Here are some examples:
 {{{
sage: C = CrystalOfLetters(['E',6])
sage: C.list()

[[1],
 [-1, 3],
 [-3, 4],
 [-4, 2, 5],
 [-2, 5],
 [-5, 2, 6],
 [-2, -5, 4, 6],
 [-4, 3, 6],
 [-3, 1, 6],
 [-1, 6],
 [-6, 2],
 [-2, -6, 4],
 [-4, -6, 3, 5],
 [-3, -6, 1, 5],
 [-1, -6, 5],
 [-5, 3],
 [-3, -5, 1, 4],
 [-1, -5, 4],
 [-4, 1, 2],
 [-1, -4, 2, 3],
 [-3, 2],
 [-2, -3, 4],
 [-4, 5],
 [-5, 6],
 [-6],
 [-2, 1],
 [-1, -2, 3]]
sage: C = CrystalOfLetters(['E',6], element_print_style="compact")
sage: C.list()

[+, a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z]
 }}}
Line 51: Line 162:
 * FIXME: summarize #5576

 * FIXME: summarize #5609

 * FIXME: summarize #5566
 * Improved performance for {{{SR}}} (Martin Albrecht) -- The speed-up gain for {{{SR}}} is up to 6x. The following timing statistics were obtained using the machine sage.math:
 {{{
# BEFORE

sage: sr = mq.SR(4, 4, 4, 8, gf2=True, polybori=True, allow_zero_inversions=True)
sage: %time F,s = sr.polynomial_system()
CPU times: user 21.65 s, sys: 0.03 s, total: 21.68 s
Wall time: 21.83 s


# AFTER

sage: sr = mq.SR(4, 4, 4, 8, gf2=True, polybori=True, allow_zero_inversions=True)
sage: %time F,s = sr.polynomial_system()
CPU times: user 3.61 s, sys: 0.06 s, total: 3.67 s
Wall time: 3.67 s
 }}}


 * Symmetric Groebner bases and infinitely generated polynomial rings (Simon King, Mike Hansen) -- The new modules {{{sage/rings/polynomial/infinite_polynomial_element.py}}} and {{{sage/rings/polynomial/infinite_polynomial_ring.py}}} support computation in polynomial rings with a countably infinite number of variables. Here are some examples for working with these new modules:
 {{{
sage: from sage.rings.polynomial.infinite_polynomial_element import InfinitePolynomial
sage: X.<x> = InfinitePolynomialRing(QQ)
sage: a = InfinitePolynomial(X, "(x1 + x2)^2"); a
x2^2 + 2*x2*x1 + x1^2
sage: p = a.polynomial()
sage: b = InfinitePolynomial(X, a.polynomial())
sage: a == b
True
sage: InfinitePolynomial(X, int(1))
1
sage: InfinitePolynomial(X, 1)
1
sage: Y.<x,y> = InfinitePolynomialRing(GF(2), implementation="sparse")
sage: InfinitePolynomial(Y, a)
x2^2 + x1^2

sage: X.<x,y> = InfinitePolynomialRing(QQ, implementation="sparse")
sage: A.<a,b> = InfinitePolynomialRing(QQ, order="deglex")
sage: f = x[5] + 2; f
x5 + 2
sage: g = 3*y[1]; g
3*y1
sage: g._p.parent()
Univariate Polynomial Ring in y1 over Rational Field
sage: f2 = a[5] + 2; f2
a5 + 2
sage: g2 = 3*b[1]; g2
3*b1
sage: A.polynomial_ring()
Multivariate Polynomial Ring in b5, b4, b3, b2, b1, b0, a5, a4, a3, a2, a1, a0 over Rational Field
sage: f + g
3*y1 + x5 + 2
sage: p = x[10]^2 * (f + g); p
3*y1*x10^2 + x10^2*x5 + 2*x10^2
 }}}
 Furthermore, the new module {{{sage/rings/polynomial/symmetric_ideal.py}}} supports ideals of polynomial rings in a countably infinite number of variables that are invariant under variable permuation. Symmetric reduction of infinite polynomials is provided by the new module {{{sage/rings/polynomial/symmetric_reduction.pyx}}}.
Line 68: Line 232:


 * Simplicial complex method for polytopes (Marshall Hampton) -- New method {{{simplicial_complex()}}} in the class {{{Polyhedron}}} of {{{sage/geometry/polyhedra.py}}} for computing the simplicial complex from a triangulation of the polytope. Here's an example:
 {{{
sage: p = polytopes.cuboctahedron()
sage: p.simplicial_complex()
Simplicial complex with 13 vertices and 20 facets
 }}}


 * Face lattices and f-vectors for polytopes (Marshall Hampton) -- New methods {{{face_lattice()}}} and {{{f_vector()}}} in the class {{{Polyhedron}}} of {{{sage/geometry/polyhedra.py}}}:
  * {{{face_lattice()}}} -- Returns the face-lattice poset. Elements are tuples of (vertices, facets) which keeps track of both the vertices in each face, and all the facets containing them. This method implements the results from the following paper:
   * V. Kaibel and M.E. Pfetsch. Computing the face lattice of a polytope from its vertex-facet incidences. Computational Geometry, 23(3):281--290, 2002.
  * {{{f_vector()}}} -- Returns the f-vector of a polytope as a list.
 Here are some examples:
 {{{
sage: c5_10 = Polyhedron(vertices = [[i,i^2,i^3,i^4,i^5] for i in xrange(1,11)])
sage: c5_10_fl = c5_10.face_lattice()
sage: [len(x) for x in c5_10_fl.level_sets()]
[1, 10, 45, 100, 105, 42, 1]
sage: p = Polyhedron(vertices = [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], [0, 0, 0]])
sage: p.f_vector()
[1, 7, 12, 7, 1]
 }}}
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 * FIXME: summarize #5940

 * FIXME: summarize #6086
Line 97: Line 281:
 * FIXME: summarize #5249
 * Implicit Surfaces (Bill Cauchois, Carl Witty) -- implicit_plot3d plots level sets of 3D functions.


 * Fixed bug in rendering 2D polytopes embedded in 3D (Arnauld Bergeron, Bill Cauchois, Marshall Hampton).
Line 102: Line 290:
 * FIXME: summarize #5664

 * FIXME: summarize #5844
 * Improved efficiency of {{{is_subgroup}}} (Simon King) -- Testing whether a group is a subgroup of another group is now up to 2x faster than previously. The following timing statistics were obtained using the machine sage.math:
 {{{
# BEFORE

sage: G = SymmetricGroup(7)
sage: H = SymmetricGroup(6)
sage: %time H.is_subgroup(G)
CPU times: user 4.12 s, sys: 0.53 s, total: 4.65 s
Wall time: 5.51 s
True
sage: %timeit H.is_subgroup(G)
10000 loops, best of 3: 118 µs per loop


# AFTER

sage: G = SymmetricGroup(7)
sage: H = SymmetricGroup(6)
sage: %time H.is_subgroup(G)
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
True
sage: %timeit H.is_subgroup(G)
10000 loops, best of 3: 56.3 µs per loop
 }}}
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 * FIXME: summarize #5974

 * FIXME: summarize #5557

 * FIXME: summarize #5381

 * FIXME: summarize #5554
 * Improved performance for the generic {{{linear_combination_of_rows}}} and {{{linear_combination_of_columns}}} functions for matrices (William Stein) -- The speed-up for the generic functions {{{linear_combination_of_rows}}} and {{{linear_combination_of_columns}}} is up to 4x. The following timing statistics were obtained using the machine sage.math:
 {{{
# BEFORE

sage: A = random_matrix(QQ, 50)
sage: v = [1..50]
sage: %timeit A.linear_combination_of_rows(v);
1000 loops, best of 3: 1.99 ms per loop
sage: %timeit A.linear_combination_of_columns(v);
1000 loops, best of 3: 1.97 ms per loop


# AFTER

sage: A = random_matrix(QQ, 50)
sage: v = [1..50]
sage: %timeit A.linear_combination_of_rows(v);
1000 loops, best of 3: 436 µs per loop
sage: %timeit A.linear_combination_of_columns(v);
1000 loops, best of 3: 457 µs per loop
 }}}


 * Massively improved performance for {{{4 x 4}}} determinants (Tom Boothby) -- The efficiency of computing the determinants of {{{4 x 4}}} matrices can range from 16x up to 58,083x faster than previously, depending on the base ring. The following timing statistics were obtained using the machine sage.math:
 {{{
# BEFORE

sage: S = MatrixSpace(ZZ, 4)
sage: M = S.random_element(1, 10^8)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 53 µs per loop
sage: M = S.random_element(1, 10^10)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 54.1 µs per loop
sage:
sage: M = S.random_element(1, 10^200)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 121 ms per loop
sage: M = S.random_element(1, 10^300)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 338 ms per loop
sage: M = S.random_element(1, 10^1000)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 9.7 s per loop


# AFTER

sage: S = MatrixSpace(ZZ, 4)
sage: M = S.random_element(1, 10^8)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 3.17 µs per loop
sage: M = S.random_element(1, 10^10)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 3.44 µs per loop
sage:
sage: M = S.random_element(1, 10^200)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 15.3 µs per loop
sage: M = S.random_element(1, 10^300)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 27 µs per loop
sage: M = S.random_element(1, 10^1000)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 167 µs per loop
 }}}


 * Refactor matrix kernels (Rob Beezer) -- The core section of kernel computation for each (specialized) class is now moved into the method {{{right_kernel()}}}. Mostly these would replace {{{kernel()}}} methods that are computing left kernels. A call to {{{kernel()}}} or {{{left_kernel()}}} should arrive at the top of the hierarchy where it would take a transpose and call the (specialized) {{{right_kernel()}}}. So there wouldn't be a change in behavior in routines currently calling {{{kernel()}}} or {{{left_kernel()}}}, and Sage's preference for the left is retained by having the vanilla {{{kernel()}}} give back a left kernel. The speed-up for the computation of left kernels is up to 5% faster, and the computation of right kernels is up to 31% by eliminating paired transposes. The followingn timing statistics were obtained using sage.math:
 {{{
# BEFORE

sage: n = 2000
sage: entries = [[1/(i+j+1) for i in srange(n)] for j in srange(n)]
sage: mat = matrix(QQ, entries)
sage: %time mat.left_kernel();
CPU times: user 21.92 s, sys: 3.22 s, total: 25.14 s
Wall time: 25.26 s
sage: %time mat.right_kernel();
CPU times: user 23.62 s, sys: 3.32 s, total: 26.94 s
Wall time: 26.94 s


# AFTER

sage: n = 2000
sage: entries = [[1/(i+j+1) for i in srange(n)] for j in srange(n)]
sage: mat = matrix(QQ, entries)
sage: %time mat.left_kernel();
CPU times: user 20.87 s, sys: 2.94 s, total: 23.81 s
Wall time: 23.89 s
sage: %time mat.right_kernel();
CPU times: user 18.43 s, sys: 0.00 s, total: 18.43 s
Wall time: 18.43 s
 }}}


 * Cholesky decomposition for matrices other than {{{RDF}}} (Nick Alexander) -- The method {{{cholesky()}}} of the class {{{Matrix_double_dense}}} in {{{sage/matrix/matrix_double_dense.pyx}}} is now deprecated and will be removed in a future release. Users are advised to use {{{cholesky_decomposition()}}} instead. The new method {{{cholesky_decomposition()}}} in the class {{{Matrix}}} of {{{sage/matrix/matrix2.pyx}}} can be used to compute the Cholesky decomposition of matrices with entries over arbitrary precision real and complex fields. Here's an example over the real double field:
 {{{
sage: r = matrix(RDF, 5, 5, [ 0,0,0,0,1, 1,1,1,1,1, 16,8,4,2,1, 81,27,9,3,1, 256,64,16,4,1 ])
sage: m = r * r.transpose(); m

[ 1.0 1.0 1.0 1.0 1.0]
[ 1.0 5.0 31.0 121.0 341.0]
[ 1.0 31.0 341.0 1555.0 4681.0]
[ 1.0 121.0 1555.0 7381.0 22621.0]
[ 1.0 341.0 4681.0 22621.0 69905.0]
sage: L = m.cholesky_decomposition(); L

[ 1.0 0.0 0.0 0.0 0.0]
[ 1.0 2.0 0.0 0.0 0.0]
[ 1.0 15.0 10.7238052948 0.0 0.0]
[ 1.0 60.0 60.9858144589 7.79297342371 0.0]
[ 1.0 170.0 198.623524155 39.3665667796 1.72309958068]
sage: L.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
sage: L*L.transpose()

[ 1.0 1.0 1.0 1.0 1.0]
[ 1.0 5.0 31.0 121.0 341.0]
[ 1.0 31.0 341.0 1555.0 4681.0]
[ 1.0 121.0 1555.0 7381.0 22621.0]
[ 1.0 341.0 4681.0 22621.0 69905.0]
sage: ( L*L.transpose() - m ).norm(1) < 2^-30
True
 }}}
 Here's an example over a higher precision real field:
 {{{
sage: r = matrix(RealField(100), 5, 5, [ 0,0,0,0,1, 1,1,1,1,1, 16,8,4,2,1, 81,27,9,3,1, 256,64,16,4,1 ])
sage: m = r * r.transpose()
sage: L = m.cholesky_decomposition()
sage: L.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Field with 100 bits of precision
sage: ( L*L.transpose() - m ).norm(1) < 2^-50
True
 }}}
 Here's a Hermitian example:
 {{{
sage: r = matrix(CDF, 2, 2, [ 1, -2*I, 2*I, 6 ]); r

[ 1.0 -2.0*I]
[ 2.0*I 6.0]
sage: r.eigenvalues()
[0.298437881284, 6.70156211872]
sage: ( r - r.conjugate().transpose() ).norm(1) < 1e-30
True
sage: L = r.cholesky_decomposition(); L

[ 1.0 0]
[ 2.0*I 1.41421356237]
sage: ( r - L*L.conjugate().transpose() ).norm(1) < 1e-30
True
sage: L.parent()
Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field
 }}}
 Note that the implementation uses a standard recursion that is not known to be numerically stable. Furthermore, it is potentially expensive to ensure that the input is positive definite. Therefore this is not checked and it is possible that the output matrix is not a valid Cholesky decomposition of a matrix.


 * FIXME: summarize #6115
Line 138: Line 502:
Line 141: Line 506:
 * FIXME: summarize #5796


 * FIXME: summarize #5120
Line 143: Line 514:
 * FIXME: summarize #4337

 * FIXME: summarize #4357

 * FIXME: summarize #5262

 * FIXME: summarize #5792

 * FIXME: summarize #5796
 * Action of Hecke operators on {{{Gamma_1(N)}}} modular forms (David Loeffler) -- Here's an example:
 {{{
sage: ModularForms(Gamma1(11), 2).hecke_matrix(2)

[ -2 0 0 0 0 0 0 0 0 0]
[ 0 -381 0 -360 0 120 -4680 -6528 -1584 7752]
[ 0 -190 0 -180 0 60 -2333 -3262 -789 3887]
[ 0 -634/11 1 -576/11 0 170/11 -7642/11 -10766/11 -231 12555/11]
[ 0 98/11 0 78/11 0 -26/11 1157/11 1707/11 30 -1959/11]
[ 0 290/11 0 271/11 0 -50/11 3490/11 5019/11 99 -5694/11]
[ 0 230/11 0 210/11 0 -70/11 2807/11 3940/11 84 -4632/11]
[ 0 122/11 0 120/11 1 -40/11 1505/11 2088/11 48 -2463/11]
[ 0 42/11 0 46/11 0 -30/11 554/11 708/11 21 -970/11]
[ 0 10/11 0 12/11 0 7/11 123/11 145/11 7 -177/11]
 }}}
Line 193: Line 572:
 * FIXME: summarize #6145

 * FIXME: summarize #5218
Line 208: Line 591:


 * FIXME: summarize #5777

 * FIXME: summarize #5930

Sage 4.0 Release Tour

Sage 4.0 was released on FIXME. For the official, comprehensive release note, please refer to sage-4.0.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:

Algebra

  • Deprecate the order() method on elements of rings (John Palmieri) -- The method order() of the class sage.structure.element.RingElement is now deprecated and will be removed in a future release. For additive or multiplicative order, use the additive_order or multiplicative_order method respectively.

  • Partial fraction decomposition for irreducible denominators (Gonzalo Tornaria) -- For example, over the field ZZ[x] you can do 

    sage: R.<x> = ZZ["x"]
sage: q = x^2 / (x - 1)
sage: q.partial_fraction_decomposition()
(x + 1, [1/(x - 1)])
sage: q = x^10 / (x - 1)^5
sage: whole, parts = q.partial_fraction_decomposition()
sage: whole + sum(parts)
x^10/(x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1)
sage: whole + sum(parts) == q
True

    and over the finite field GF(2)[x]: 

    sage: R.<x> = GF(2)["x"]
sage: q = (x + 1) / (x^3 + x + 1)
qsage: q.partial_fraction_decomposition()
(0, [(x + 1)/(x^3 + x + 1)])

Algebraic Geometry

  • Various invariants for genus 2 hyperelliptic curves (Nick Alexander) -- The following invariants for genus 2 hyperelliptic curves are implemented in the module sage/schemes/hyperelliptic_curves/hyperelliptic_g2_generic.py:

    • the Clebsch invariants
    • the Igusa-Clebsch invariants
    • the absolute Igusa invariants

Basic Arithmetic

  • Utility methods for integer arithmetics (Fredrik Johansson) -- New methods trailing_zero_bits() and sqrtrem() for the class Integer in sage/rings/integer.pyx:

    • trailing_zero_bits(self) -- Returns the number of trailing zero bits in self, i.e. the exponent of the largest power of 2 dividing self.

    • sqrtrem(self) -- Returns a pair (s, r) where s is the integer square root of self and r is the remainder such that self = s^2 + r.

    Here are some examples for working with these new methods: 
    sage: 13.trailing_zero_bits()
0
sage: (-13).trailing_zero_bits()
0
sage: (-13 >> 2).trailing_zero_bits()
2
sage: (-13 >> 3).trailing_zero_bits()
1
sage: (-13 << 3).trailing_zero_bits()
3
sage: (-13 << 2).trailing_zero_bits()
2
sage: 29.sqrtrem()
(5, 4)
sage: 25.sqrtrem()
(5, 0)

Build

Calculus

  • FIXME: summarize #6111

Coercion

  • Coercion from float to rationals (Robert Bradshaw) -- One can now coerce a number of type float to QQ. Here's an example: 

    sage: a = float(1.0)
sage: QQ(a)
1
sage: type(a); type(QQ(a))
<type 'float'>
<type 'sage.rings.rational.Rational'>

Combinatorics

  • ASCII art output for Dynkin diagrams (Dan Bump) -- Support for ASCII art representation of Dynkin diagrams of a finite Cartan type. Here are some examples: 

    sage: DynkinDiagram("E6")

        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6   
E6
sage: DynkinDiagram(['E',6,1])

        O 0
        |
        |
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6~

  • Crystal of letters for type E (Brant Jones, Anne Schilling) -- Support crystal of letters for type E corresponding to the highest weight crystal B(\Lambda_1) and its dual B(\Lambda_6) (using the Sage labeling convention of the Dynkin nodes). Here are some examples: 

    sage: C = CrystalOfLetters(['E',6])
sage: C.list()

[[1],
 [-1, 3],
 [-3, 4],
 [-4, 2, 5],
 [-2, 5],
 [-5, 2, 6],
 [-2, -5, 4, 6],
 [-4, 3, 6],
 [-3, 1, 6],
 [-1, 6],
 [-6, 2],
 [-2, -6, 4],
 [-4, -6, 3, 5],
 [-3, -6, 1, 5],
 [-1, -6, 5],
 [-5, 3],
 [-3, -5, 1, 4],
 [-1, -5, 4],
 [-4, 1, 2],
 [-1, -4, 2, 3],
 [-3, 2],
 [-2, -3, 4],
 [-4, 5],
 [-5, 6],
 [-6],
 [-2, 1],
 [-1, -2, 3]]
sage: C = CrystalOfLetters(['E',6], element_print_style="compact")
sage: C.list()

[+, a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z]

Commutative Algebra

  • Improved performance for SR (Martin Albrecht) -- The speed-up gain for SR is up to 6x. The following timing statistics were obtained using the machine sage.math: 

    # BEFORE

sage: sr = mq.SR(4, 4, 4, 8, gf2=True, polybori=True, allow_zero_inversions=True)
sage: %time F,s = sr.polynomial_system()
CPU times: user 21.65 s, sys: 0.03 s, total: 21.68 s
Wall time: 21.83 s


# AFTER

sage: sr = mq.SR(4, 4, 4, 8, gf2=True, polybori=True, allow_zero_inversions=True)
sage: %time F,s = sr.polynomial_system()
CPU times: user 3.61 s, sys: 0.06 s, total: 3.67 s
Wall time: 3.67 s

  • Symmetric Groebner bases and infinitely generated polynomial rings (Simon King, Mike Hansen) -- The new modules sage/rings/polynomial/infinite_polynomial_element.py and sage/rings/polynomial/infinite_polynomial_ring.py support computation in polynomial rings with a countably infinite number of variables. Here are some examples for working with these new modules: 

    sage: from sage.rings.polynomial.infinite_polynomial_element import InfinitePolynomial
sage: X.<x> = InfinitePolynomialRing(QQ)
sage: a = InfinitePolynomial(X, "(x1 + x2)^2"); a
x2^2 + 2*x2*x1 + x1^2
sage: p = a.polynomial()
sage: b = InfinitePolynomial(X, a.polynomial())
sage: a == b
True
sage: InfinitePolynomial(X, int(1))
1
sage: InfinitePolynomial(X, 1)
1
sage: Y.<x,y> = InfinitePolynomialRing(GF(2), implementation="sparse")
sage: InfinitePolynomial(Y, a)
x2^2 + x1^2

sage: X.<x,y> = InfinitePolynomialRing(QQ, implementation="sparse")
sage: A.<a,b> = InfinitePolynomialRing(QQ, order="deglex")
sage: f = x[5] + 2; f
x5 + 2
sage: g = 3*y[1]; g
3*y1
sage: g._p.parent()
Univariate Polynomial Ring in y1 over Rational Field
sage: f2 = a[5] + 2; f2
a5 + 2
sage: g2 = 3*b[1]; g2
3*b1
sage: A.polynomial_ring()
Multivariate Polynomial Ring in b5, b4, b3, b2, b1, b0, a5, a4, a3, a2, a1, a0 over Rational Field
sage: f + g
3*y1 + x5 + 2
sage: p = x[10]^2 * (f + g); p
3*y1*x10^2 + x10^2*x5 + 2*x10^2

    Furthermore, the new module sage/rings/polynomial/symmetric_ideal.py supports ideals of polynomial rings in a countably infinite number of variables that are invariant under variable permuation. Symmetric reduction of infinite polynomials is provided by the new module sage/rings/polynomial/symmetric_reduction.pyx.

Distribution

Doctest

Documentation

Geometry

  • Simplicial complex method for polytopes (Marshall Hampton) -- New method simplicial_complex() in the class Polyhedron of sage/geometry/polyhedra.py for computing the simplicial complex from a triangulation of the polytope. Here's an example: 

    sage: p = polytopes.cuboctahedron()
sage: p.simplicial_complex()
Simplicial complex with 13 vertices and 20 facets

  • Face lattices and f-vectors for polytopes (Marshall Hampton) -- New methods face_lattice() and f_vector() in the class Polyhedron of sage/geometry/polyhedra.py:

    • face_lattice() -- Returns the face-lattice poset. Elements are tuples of (vertices, facets) which keeps track of both the vertices in each face, and all the facets containing them. This method implements the results from the following paper:

      • V. Kaibel and M.E. Pfetsch. Computing the face lattice of a polytope from its vertex-facet incidences. Computational Geometry, 23(3):281--290, 2002.

    • f_vector() -- Returns the f-vector of a polytope as a list.

    Here are some examples: 
    sage: c5_10 = Polyhedron(vertices = [[i,i^2,i^3,i^4,i^5] for i in xrange(1,11)]) 
sage: c5_10_fl = c5_10.face_lattice()
sage: [len(x) for x in c5_10_fl.level_sets()]
[1, 10, 45, 100, 105, 42, 1]
sage: p = Polyhedron(vertices = [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], [0, 0, 0]])
sage: p.f_vector()
[1, 7, 12, 7, 1]

Graph Theory

  • Graph colouring (Robert Miller) -- New method coloring() of the class sage.graphs.graph.Graph for obtaining the first (optimal) coloring found on a graph. Here are some examples on using this new method: 

    sage: G = Graph("Fooba")
sage: P = G.coloring()
sage: G.plot(partition=P)
sage: H = G.coloring(hex_colors=True)
sage: G.plot(vertex_colors=H)

graph-colour-1.png

graph-colour-2.png

  • FIXME: summarize #6066
  • FIXME: summarize #3932

Graphics

  • Implicit Surfaces (Bill Cauchois, Carl Witty) -- implicit_plot3d plots level sets of 3D functions.
  • Fixed bug in rendering 2D polytopes embedded in 3D (Arnauld Bergeron, Bill Cauchois, Marshall Hampton).

Group Theory

  • Improved efficiency of is_subgroup (Simon King) -- Testing whether a group is a subgroup of another group is now up to 2x faster than previously. The following timing statistics were obtained using the machine sage.math: 

    # BEFORE

sage: G = SymmetricGroup(7)
sage: H = SymmetricGroup(6)
sage: %time H.is_subgroup(G)
CPU times: user 4.12 s, sys: 0.53 s, total: 4.65 s
Wall time: 5.51 s
True
sage: %timeit H.is_subgroup(G)
10000 loops, best of 3: 118 µs per loop


# AFTER

sage: G = SymmetricGroup(7)
sage: H = SymmetricGroup(6)
sage: %time H.is_subgroup(G)
CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s
Wall time: 0.00 s
True
sage: %timeit H.is_subgroup(G)
10000 loops, best of 3: 56.3 µs per loop

Interfaces

  • Viewing Sage objects with a PDF viewer (Nicolas Thiery) -- Implements the option viewer="pdf" for the command view() so that one can invoke this command in the form view(object, viewer="pdf") in order to view object using a PDF viewer. Typical uses of this new optional argument include:

    • You prefer to use a PDF viewer rather than a DVI viewer.
    • You want to view LaTeX snippets which are not displayed well in DVI viewers (e.g. graphics produced using tikzpicture).

  • Change name of Pari's sum function when imported (Craig Citro) -- When Pari's sum function is imported, it is renamed to pari_sum in order to avoid conflict Python's sum function.

Linear Algebra

  • Improved performance for the generic linear_combination_of_rows and linear_combination_of_columns functions for matrices (William Stein) -- The speed-up for the generic functions linear_combination_of_rows and linear_combination_of_columns is up to 4x. The following timing statistics were obtained using the machine sage.math: 

    # BEFORE

sage: A = random_matrix(QQ, 50)
sage: v = [1..50]
sage: %timeit A.linear_combination_of_rows(v);
1000 loops, best of 3: 1.99 ms per loop
sage: %timeit A.linear_combination_of_columns(v);
1000 loops, best of 3: 1.97 ms per loop


# AFTER

sage: A = random_matrix(QQ, 50)
sage: v = [1..50]
sage: %timeit A.linear_combination_of_rows(v);
1000 loops, best of 3: 436 µs per loop
sage: %timeit A.linear_combination_of_columns(v);
1000 loops, best of 3: 457 µs per loop

  • Massively improved performance for 4 x 4 determinants (Tom Boothby) -- The efficiency of computing the determinants of 4 x 4 matrices can range from 16x up to 58,083x faster than previously, depending on the base ring. The following timing statistics were obtained using the machine sage.math: 

    # BEFORE

sage: S = MatrixSpace(ZZ, 4)
sage: M = S.random_element(1, 10^8)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 53 µs per loop
sage: M = S.random_element(1, 10^10)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 54.1 µs per loop
sage: 
sage: M = S.random_element(1, 10^200)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 121 ms per loop
sage: M = S.random_element(1, 10^300)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 338 ms per loop
sage: M = S.random_element(1, 10^1000)
sage: timeit("M.det(); M._clear_cache()")
5 loops, best of 3: 9.7 s per loop


# AFTER

sage: S = MatrixSpace(ZZ, 4)
sage: M = S.random_element(1, 10^8)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 3.17 µs per loop
sage: M = S.random_element(1, 10^10)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 3.44 µs per loop
sage: 
sage: M = S.random_element(1, 10^200)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 15.3 µs per loop
sage: M = S.random_element(1, 10^300)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 27 µs per loop
sage: M = S.random_element(1, 10^1000)
sage: timeit("M.det(); M._clear_cache()")
625 loops, best of 3: 167 µs per loop

  • Refactor matrix kernels (Rob Beezer) -- The core section of kernel computation for each (specialized) class is now moved into the method right_kernel(). Mostly these would replace kernel() methods that are computing left kernels. A call to kernel() or left_kernel() should arrive at the top of the hierarchy where it would take a transpose and call the (specialized) right_kernel(). So there wouldn't be a change in behavior in routines currently calling kernel() or left_kernel(), and Sage's preference for the left is retained by having the vanilla kernel() give back a left kernel. The speed-up for the computation of left kernels is up to 5% faster, and the computation of right kernels is up to 31% by eliminating paired transposes. The followingn timing statistics were obtained using sage.math: 

    # BEFORE

sage: n = 2000
sage: entries = [[1/(i+j+1) for i in srange(n)] for j in srange(n)]
sage: mat = matrix(QQ, entries)
sage: %time mat.left_kernel();
CPU times: user 21.92 s, sys: 3.22 s, total: 25.14 s
Wall time: 25.26 s
sage: %time mat.right_kernel();
CPU times: user 23.62 s, sys: 3.32 s, total: 26.94 s
Wall time: 26.94 s


# AFTER

sage: n = 2000
sage: entries = [[1/(i+j+1) for i in srange(n)] for j in srange(n)]
sage: mat = matrix(QQ, entries)
sage: %time mat.left_kernel();
CPU times: user 20.87 s, sys: 2.94 s, total: 23.81 s
Wall time: 23.89 s
sage: %time mat.right_kernel();
CPU times: user 18.43 s, sys: 0.00 s, total: 18.43 s
Wall time: 18.43 s

  • Cholesky decomposition for matrices other than RDF (Nick Alexander) -- The method cholesky() of the class Matrix_double_dense in sage/matrix/matrix_double_dense.pyx is now deprecated and will be removed in a future release. Users are advised to use cholesky_decomposition() instead. The new method cholesky_decomposition() in the class Matrix of sage/matrix/matrix2.pyx can be used to compute the Cholesky decomposition of matrices with entries over arbitrary precision real and complex fields. Here's an example over the real double field: 

    sage: r = matrix(RDF, 5, 5, [ 0,0,0,0,1, 1,1,1,1,1, 16,8,4,2,1, 81,27,9,3,1, 256,64,16,4,1 ])
sage: m = r * r.transpose(); m

[    1.0     1.0     1.0     1.0     1.0]
[    1.0     5.0    31.0   121.0   341.0]
[    1.0    31.0   341.0  1555.0  4681.0]
[    1.0   121.0  1555.0  7381.0 22621.0]
[    1.0   341.0  4681.0 22621.0 69905.0]
sage: L = m.cholesky_decomposition(); L

[          1.0           0.0           0.0           0.0           0.0]
[          1.0           2.0           0.0           0.0           0.0]
[          1.0          15.0 10.7238052948           0.0           0.0]
[          1.0          60.0 60.9858144589 7.79297342371           0.0]
[          1.0         170.0 198.623524155 39.3665667796 1.72309958068]
sage: L.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Double Field
sage: L*L.transpose()

[    1.0     1.0     1.0     1.0     1.0]
[    1.0     5.0    31.0   121.0   341.0]
[    1.0    31.0   341.0  1555.0  4681.0]
[    1.0   121.0  1555.0  7381.0 22621.0]
[    1.0   341.0  4681.0 22621.0 69905.0]
sage: ( L*L.transpose() - m ).norm(1) < 2^-30
True
    Here's an example over a higher precision real field: 
    sage: r = matrix(RealField(100), 5, 5, [ 0,0,0,0,1, 1,1,1,1,1, 16,8,4,2,1, 81,27,9,3,1, 256,64,16,4,1 ])
sage: m = r * r.transpose()
sage: L = m.cholesky_decomposition()
sage: L.parent()
Full MatrixSpace of 5 by 5 dense matrices over Real Field with 100 bits of precision
sage: ( L*L.transpose() - m ).norm(1) < 2^-50
True
    Here's a Hermitian example: 
    sage: r = matrix(CDF, 2, 2, [ 1, -2*I, 2*I, 6 ]); r

[   1.0 -2.0*I]
[ 2.0*I    6.0]
sage: r.eigenvalues()
[0.298437881284, 6.70156211872]
sage: ( r - r.conjugate().transpose() ).norm(1) < 1e-30
True
sage: L = r.cholesky_decomposition(); L

[          1.0             0]
[        2.0*I 1.41421356237]
sage: ( r - L*L.conjugate().transpose() ).norm(1) < 1e-30
True
sage: L.parent()
Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field
    Note that the implementation uses a standard recursion that is not known to be numerically stable. Furthermore, it is potentially expensive to ensure that the input is positive definite. Therefore this is not checked and it is possible that the output matrix is not a valid Cholesky decomposition of a matrix.
  • FIXME: summarize #6115

Miscellaneous

  • Allow use of pdflatex instead of latex (John Palmieri) -- One can now use pdflatex instead of latex in two different ways:

    • Use a %pdflatex cell in a notebook; or

    • Call latex.pdflatex(True)

    after which any use of latex (in a %latex cell or using the view command) will use pdflatex. One visually appealing aspect of this is that if you have the most recent version of pgf installed, as well as the tkz-graph package, you can produce images like the following:

pgf-graph.png

  • FIXME: summarize #5783
  • FIXME: summarize #5796
  • FIXME: summarize #5120

Modular Forms

  • Action of Hecke operators on Gamma_1(N) modular forms (David Loeffler) -- Here's an example: 

    sage: ModularForms(Gamma1(11), 2).hecke_matrix(2)

[       -2         0         0         0         0         0         0         0         0         0]
[        0      -381         0      -360         0       120     -4680     -6528     -1584      7752]
[        0      -190         0      -180         0        60     -2333     -3262      -789      3887]
[        0   -634/11         1   -576/11         0    170/11  -7642/11 -10766/11      -231  12555/11]
[        0     98/11         0     78/11         0    -26/11   1157/11   1707/11        30  -1959/11]
[        0    290/11         0    271/11         0    -50/11   3490/11   5019/11        99  -5694/11]
[        0    230/11         0    210/11         0    -70/11   2807/11   3940/11        84  -4632/11]
[        0    122/11         0    120/11         1    -40/11   1505/11   2088/11        48  -2463/11]
[        0     42/11         0     46/11         0    -30/11    554/11    708/11        21   -970/11]
[        0     10/11         0     12/11         0      7/11    123/11    145/11         7   -177/11]
  • FIXME: summarize #6019
  • FIXME: summarize #5924

Notebook

Number Theory

  • FIXME: summarize #5250
  • FIXME: summarize #6013
  • FIXME: summarize #6008
  • FIXME: summarize #6004
  • FIXME: summarize #6059
  • FIXME: summarize #6064

Numerical

Packages

  • FIXME: summarize #4223
  • FIXME: summarize #6031
  • FIXME: summarize #5934
  • FIXME: summarize #1338
  • FIXME: summarize #6032
  • FIXME: summarize #6024
  • FIXME: summarize #6145
  • FIXME: summarize #5218

P-adics

  • FIXME: summarize #5105
  • FIXME: summarize #5236

Quadratic Forms

  • FIXME: summarize #6037

Symbolics

  • FIXME: summarize #5777
  • FIXME: summarize #5930

Topology

  • Random simplicial complexes (John Palmieri) -- New method RandomComplex() in the module sage/homology/examples.py for producing a random d-dimensional simplicial complex on n vertices. Here's an example: 

    sage: simplicial_complexes.RandomComplex(6,12)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and facets {(0, 1, 2, 3, 4, 5, 6, 7)}

User Interface

Website / Wiki