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| = Sage 4.0 Release Tour = Sage 4.0 was released on FIXME. For the official, comprehensive release note, please refer to [[http://www.sagemath.org/src/announce/sage-4.0.txt|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 [[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] }}} == 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) }}} {{attachment:graph-colour-1.png}} {{attachment:graph-colour-2.png}} * Optimize the construction of large Sage graphs (Radoslav Kirov) -- The construction of large Sage graphs is now up to 19x faster than previously. The following timing statistics were obtained using the machine sage.math: {{{ # BEFORE sage: D = {} sage: for i in xrange(10^3): ....: D[i] = [i+1, i-1] ....: sage: timeit("g = Graph(D)") 5 loops, best of 3: 1.02 s per loop # AFTER sage: D = {} sage: for i in xrange(10^3): ....: D[i] = [i+1, i-1] ....: sage: timeit("g = Graph(D)") 5 loops, best of 3: 51.2 ms per loop }}} * Generate size {{{n}}} trees in linear time (Ryan Dingman) -- The speed-up can be up to 3400x. However, the efficiency gain is greater as {{{n}}} becomes larger. The following timing statistics were produced using the maching sage.math: {{{ # BEFORE sage: %time L = list(graphs.trees(2)) CPU times: user 0.13 s, sys: 0.02 s, total: 0.15 s Wall time: 0.18 s sage: %time L = list(graphs.trees(4)) CPU times: user 0.02 s, sys: 0.00 s, total: 0.02 s Wall time: 0.02 s sage: %time L = list(graphs.trees(6)) CPU times: user 0.08 s, sys: 0.00 s, total: 0.08 s Wall time: 0.07 s sage: %time L = list(graphs.trees(8)) CPU times: user 0.59 s, sys: 0.00 s, total: 0.59 s Wall time: 0.60 s sage: %time L = list(graphs.trees(10)) CPU times: user 34.48 s, sys: 0.02 s, total: 34.50 s Wall time: 34.51 s # AFTER sage: %time L = list(graphs.trees(2)) CPU times: user 0.11 s, sys: 0.02 s, total: 0.13 s Wall time: 0.15 s sage: %time L = list(graphs.trees(4)) CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s Wall time: 0.00 s sage: %time L = list(graphs.trees(6)) CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s Wall time: 0.00 s sage: %time L = list(graphs.trees(8)) CPU times: user 0.00 s, sys: 0.00 s, total: 0.00 s Wall time: 0.00 s sage: %time L = list(graphs.trees(10)) CPU times: user 0.01 s, sys: 0.00 s, total: 0.01 s Wall time: 0.01 s sage: %time L = list(graphs.trees(12)) CPU times: user 0.06 s, sys: 0.00 s, total: 0.06 s Wall time: 0.05 s sage: %time L = list(graphs.trees(14)) CPU times: user 0.51 s, sys: 0.01 s, total: 0.52 s Wall time: 0.52 s }}} == Graphics == * Implicit Surfaces (Bill Cauchois, Carl Witty) -- New function {{{implicit_plot3d}}} for plotting level sets of 3-D functions. The documentation contains many examples. Here's a sphere contained inside a tube-like sphere: {{{ sage: x, y, z = var("x, y, z") sage: T = 1.61803398875 sage: p = 2 - (cos(x + T*y) + cos(x - T*y) + cos(y + T*z) + cos(y - T*z) + cos(z - T*x) + cos(z + T*x)) sage: r = 4.77 sage: implicit_plot3d(p, (-r, r), (-r, r), (-r, r), plot_points=40, zoom=1.2).show() }}} {{attachment:sphere-inside-tube.png}} Here's a Klein bottle: {{{ sage: x, y, z = var("x, y, z") sage: implicit_plot3d((x^2+y^2+z^2+2*y-1)*((x^2+y^2+z^2-2*y-1)^2-8*z^2)+16*x*z*(x^2+y^2+z^2-2*y-1), (x, -3, 3), (y, -3.1, 3.1), (z, -4, 4), zoom=1.2) }}} {{attachment:klein-bottle.png}} This example shows something resembling a water droplet: {{{ sage: x, y, z = var("x, y, z") sage: implicit_plot3d(x^2 +y^2 -(1-z)*z^2, (x, -1.5, 1.5), (y, -1.5, 1.5), (z, -1, 1), zoom=1.2) }}} {{attachment:water-droplet.png}} * 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. * Make symbolic matrices use pynac symbolics (Mike Hansen, Jason Grout) -- Using Pynac symbolics, calculating the determinant of a symbolic matrix can be up to 2500x faster than previously. The following timing statistics were obtained using the machine sage.math: {{{ # BEFORE sage: x00, x01, x10, x11 = var("x00, x01, x10, x11") sage: a = matrix(2, [[x00,x01], [x10,x11]]) sage: %timeit a.det() 100 loops, best of 3: 8.29 ms per loop # AFTER sage: x00, x01, x10, x11 = var("x00, x01, x10, x11") sage: a = matrix(2, [[x00,x01], [x10,x11]]) sage: %timeit a.det() 100000 loops, best of 3: 3.2 µs per loop }}} == 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 [[http://pgf.sourceforge.net|pgf]] installed, as well as the {{{tkz-graph}}} package, you can produce images like the following: {{attachment:pgf-graph.png}} == 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] }}} * Slopes of {{{U_p}}} operator acting on a space of overconvergent modular forms (Lloyd Kilford) -- New method {{{slopes}}} of the class {{{OverconvergentModularFormsSpace}}} in {{{sage/modular/overconvergent/genus0.py}}} for computing the slopes of the {{{U_p}}} operator acting on a space of overconvergent modular forms. Here are some examples of using this new method: {{{ sage: OverconvergentModularForms(5,2,1/3,base_ring=Qp(5),prec=100).slopes(5) [0, 2, 5, 6, 9] sage: OverconvergentModularForms(2,1,1/3,char=DirichletGroup(4,QQ).0).slopes(5) [0, 2, 4, 6, 8] }}} == Notebook == == Number Theory == * Fixed bug in multiplicative_generator function for Z/NZ (#5250) * Rewritten relativization of number fields relativize to be much faster (#6013) * Improved efficiency of elliptic curve torsion computation (#6008) * Added odd_degree_model function to hyperelliptic curves (#6004) * Fixed speed regresion in hilbert_symbol (#6059) * Rational arguments are now allowed in kronecker_symbol and legendre_symbol (#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 == |
