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| = Sage 4.3.1 Release Tour = == Major features == * Substantial work towards a complete SPARC Solaris 10 port. This is due to the hard work of David Kirkby. The relevant tickets include [[http://trac.sagemath.org/sage_trac/ticket/6595 | #6595]], [[http://trac.sagemath.org/sage_trac/ticket/7138 | #7138]], [[http://trac.sagemath.org/sage_trac/ticket/7162 | #7162]], [[http://trac.sagemath.org/sage_trac/ticket/7505 | #7505]], [[http://trac.sagemath.org/sage_trac/ticket/7817 | #7817]]. * We're moving closer towards a FreeBSD port, thanks to the work of Peter Jeremy at ticket [[http://trac.sagemath.org/sage_trac/ticket/7825 | #7825]]. == Basic arithmetics == * Implement `conjugate()` for `RealDoubleElement` [[http://trac.sagemath.org/sage_trac/ticket/7834 | #7834]] (Dag Sverre Seljebotn) --- New method `conjugate()` in the class `RealDoubleElement` of the module `sage/rings/real_double.pyx` for returning the complex conjugate of a real number. This is consistent with `conjugate()` methods in `ZZ` and `RR`. For example, {{{ sage: ZZ(5).conjugate() 5 sage: RR(5).conjugate() 5.00000000000000 sage: RDF(5).conjugate() 5.0 }}} == Combinatorics == * Weyl group optimizations [[http://trac.sagemath.org/sage_trac/ticket/7754 | #7754]] (Nicolas M. Thiéry) --- Three major improvements that indirectly also improve efficiency of most Weyl group routines: * Faster hash method calling the hash of the underlying matrix (which is set as immutable for that purpose). * New `__eq__()` method. * Action on the weight lattice realization: optimization of the matrix multiplication. Some operations are now up to 34% faster than previously: {{{ BEFORE sage: W = WeylGroup(["F", 4]) sage: W.cardinality() 1152 sage: %time list(W); CPU times: user 10.51 s, sys: 0.05 s, total: 10.56 s Wall time: 10.56 s sage: W = WeylGroup(["E", 8]) sage: %time W.long_element(); CPU times: user 1.47 s, sys: 0.00 s, total: 1.47 s Wall time: 1.47 s AFTER sage: W = WeylGroup(["F", 4]) sage: W.cardinality() 1152 sage: %time list(W); CPU times: user 6.89 s, sys: 0.04 s, total: 6.93 s Wall time: 6.93 s sage: W = WeylGroup(["E", 8]) sage: %time W.long_element(); CPU times: user 1.21 s, sys: 0.00 s, total: 1.21 s Wall time: 1.21 s }}} == Elliptic curves == * Two-isogeny descent over `QQ` natively using ratpoints [[http://trac.sagemath.org/sage_trac/ticket/6583 | #6583]] (Robert Miller) --- New module `sage/schemes/elliptic_curves/descent_two_isogeny.pyx` for descent on elliptic curves over `QQ` with a 2-isogeny. The relevant user interface function is `two_descent_by_two_isogeny()` that takes an elliptic curve `E` with a two-isogeny `phi : E --> E'` and dual isogeny `phi'`, runs a two-isogeny descent on `E`, and returns `n1`, `n2`, `n1'` and `n2'`. Here, `n1` is the number of quartic covers found with a rational point and `n2` is the number which are ELS. Here are some examples illustrating the use of `two_descent_by_two_isogeny()`: {{{ sage: from sage.schemes.elliptic_curves.descent_two_isogeny import two_descent_by_two_isogeny sage: E = EllipticCurve("14a") sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1, 2) + log(n1_prime, 2) - 2 # the rank 0 sage: E = EllipticCurve("65a") sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1, 2) + log(n1_prime, 2) - 2 # the rank 1 sage: E = EllipticCurve("1088j1") sage: n1, n2, n1_prime, n2_prime = two_descent_by_two_isogeny(E) sage: log(n1, 2) + log(n1_prime, 2) - 2 # the rank 2 }}} You could also ask `two_descent_by_two_isogeny()` to be verbose in its computation: {{{ sage: E = EllipticCurve("14a") sage: two_descent_by_two_isogeny(E, verbosity=1) 2-isogeny Results: 2 <= #E(Q)/phi'(E'(Q)) <= 2 2 <= #E'(Q)/phi(E(Q)) <= 2 #Sel^(phi')(E'/Q) = 2 #Sel^(phi)(E/Q) = 2 1 <= #Sha(E'/Q)[phi'] <= 1 1 <= #Sha(E/Q)[phi] <= 1 1 <= #Sha(E/Q)[2], #Sha(E'/Q)[2] <= 1 0 <= rank of E(Q) = rank of E'(Q) <= 0 (2, 2, 2, 2) }}} * More functions for elliptic curve isogenies [[http://trac.sagemath.org/sage_trac/ticket/6887 | #6887]] (John Cremona, Jenny Cooley) --- Code for constructing elliptic curve isogenies already existed in Sage 4.1.1. The enhancements here include: * For `l=2,3,5,7,13` over any field, find all `l`-isogenies of a given elliptic curve. (These are the `l` for which `X_0(l)` has genus 0). * Similarly for the remaining `l` for which `l`-isogenies exist over `QQ`. * Given an elliptic curve over `QQ`, find the whole isogeny class in a robust manner. * Testing if two curves are isogenous at least over `QQ`. The relevant use interface method is `isogenies_prime_degree()` in the class `EllipticCurve_field` of the module `sage/schemes/elliptic_curves/ell_field.py`. Here are some examples showing `isogenies_prime_degree()` in action. Examples over finite fields: {{{ sage: E = EllipticCurve(GF(next_prime(1000000)), [7,8]) sage: E.isogenies_prime_degree() [Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 970389*x + 794257 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 29783*x + 206196 over Finite Field of size 1000003, Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 999960*x + 78 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003] sage: E.isogenies_prime_degree(13) [Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 878063*x + 845666 over Finite Field of size 1000003, Isogeny of degree 13 from Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 1000003 to Elliptic Curve defined by y^2 = x^3 + 375648*x + 342776 over Finite Field of size 1000003] }}} Examples over number fields (other than `QQ`): {{{ sage: QQroot2.<e> = NumberField(x^2 - 2) sage: E = EllipticCurve(QQroot2, [1,0,1,4,-6]) sage: E.isogenies_prime_degree(2) [Isogeny of degree 2 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-36)*x + (-70) over Number Field in e with defining polynomial x^2 - 2] sage: E.isogenies_prime_degree(3) [Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-171)*x + (-874) over Number Field in e with defining polynomial x^2 - 2, Isogeny of degree 3 from Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x + (-6) over Number Field in e with defining polynomial x^2 - 2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + (-128/3)*x + 5662/27 over Number Field in e with defining polynomial x^2 - 2] }}} == Graph theory == * An interactive graph editor [[http://trac.sagemath.org/sage_trac/ticket/1321 | #1321]] (Radoslav Kirov, Mitesh Patel) --- Embed an interactive graph editor into the notebook. The following screenshot shows a graph editor for playing around with the complete graph on 5 vertices: {{attachment:graph-editor.png}} * Breadth/depth first searches and basic connectivity for c_graphs [[http://trac.sagemath.org/sage_trac/ticket/7724 | #7724]] (Nathann Cohen, Yann Laigle-Chapuy) --- Implementation of the following methods for the class `CGraphBackend` in the module `sage/graphs/base/c_graph.pyx`: * `depth_first_search()` * `breadth_first_search()` * `is_connected()` * `is_strongly_connected()` In some cases, the c_graphs implementation of these methods provides a 2x speed improvement: {{{ sage: g = graphs.RandomGNP(1000, 0.01) sage: h = g.copy(implementation="c_graph") sage: %timeit list(g.depth_first_search(0)); 100 loops, best of 3: 8.17 ms per loop sage: %timeit list(h.depth_first_search(0)); 100 loops, best of 3: 3.29 ms per loop sage: sage: %timeit list(g.breadth_first_search(0)); 100 loops, best of 3: 6.48 ms per loop sage: %timeit list(h.breadth_first_search(0)); 10 loops, best of 3: 34 ms per loop sage: sage: %timeit g.is_connected(); 100 loops, best of 3: 8.47 ms per loop sage: %timeit h.is_connected(); 100 loops, best of 3: 3.41 ms per loop sage: sage: g = g.to_directed() sage: h = g.copy(implementation="c_graph") sage: %timeit g.is_strongly_connected(); 10 loops, best of 3: 23.5 ms per loop sage: %timeit h.is_strongly_connected(); 10 loops, best of 3: 25 ms per loop }}} * Tower of Hanoi graph [[http://trac.sagemath.org/sage_trac/ticket/7770 | #7770]] (Rob Beezer) --- The Tower of Hanoi puzzle can be described by a graph whose vertices are possible states of the disks on the pegs, with edges representing legitimate moves of a single disk. The new method `HanoiTowerGraph()` of the class `GraphGenerators` in the module `sage/graphs/graph_generators.py` returns the graph whose vertices are the states of the Tower of Hanoi puzzle, with edges representing legal moves between states. See the documentation of this method for details on interpreting the the possible states of this puzzle. The following screenshot shows all the possible states of an instance of the puzzle with 3 pegs and 3 disks, produced using the following code: {{{ H = graphs.HanoiTowerGraph(3, 3, positions=False) show(H, figsize=[8,8]) }}} {{attachment:tower-hanoi-graph.png}} == Linear algebra == * Viewing entries of large matrices [[http://trac.sagemath.org/sage_trac/ticket/5174 | #5174]] (John Palmieri) --- For a small matrix such as `2 x 2`, the default is to print the entries of the matrix. This default behaviour is unsuitable for large matrices such as `100 x 100`. The string representation of such large matrices now indicate how to view all their entries. Here are some examples illustrating the new way to view the string representation of matrices. If the matrix is too big, all the elements are not displayed by default: {{{ sage: A = random_matrix(ZZ, 5) sage: A [ 1 -4 -4 1 -1] [-1 1 13 -1 -1] [-1 0 0 -1 -1] [-8 1 -1 1 -4] [ 1 -5 -1 1 2] sage: A = random_matrix(ZZ, 100) sage: A 100 x 100 dense matrix over Integer Ring (type 'print A.str()' to see all of the entries) }}} If a matrix has several names, refer to the matrix as "obj": {{{ sage: A = random_matrix(ZZ, 200) sage: B = A sage: B 200 x 200 dense matrix over Integer Ring (type 'print obj.str()' to see all of the entries) }}} If a matrix doesn't have a name, don't print any name referring to the matrix in its string representation: {{{ sage: A = random_matrix(ZZ, 150) sage: A.transpose() 150 x 150 dense matrix over Integer Ring sage: T = A.transpose(); T 150 x 150 dense matrix over Integer Ring (type 'print T.str()' to see all of the entries) }}} * [[http://trac.sagemath.org/sage_trac/ticket/7728 | #7728]] (Dag Sverre Seljebotn) == Miscellaneous == * Command line access to HTML documentation and docstrings [[http://trac.sagemath.org/sage_trac/ticket/6820 | #6820]] (John Palmieri, Mitesh Patel) --- Browse the Sage standard documentation from the command line or within the notebook interface. Use the following commands to browse documents in the standard documentation: * `browse_sage_doc.tutorial()` or its alias `tutorial()` * `browse_sage_doc.reference()`, or its aliases `reference()` and `manual()` * `browse_sage_doc.developer()` or its alias `developer()` * `browse_sage_doc.constructions()` or its alias `constructions()` The following screenshot illustrates viewing the Sage tutorial from the command line interface, activated using the command: {{{ sage: tutorial() }}} This command invoked a terminal-based web browser such as Links to view the tutorial. {{attachment:browse-doc-cmd.png}} * [[http://trac.sagemath.org/sage_trac/ticket/7482 | #7482]] (William Stein) * [[http://trac.sagemath.org/sage_trac/ticket/7514 | #7514]] (William Stein) == Packages == * [[http://trac.sagemath.org/sage_trac/ticket/7271 | #7271]] (Martin Albrecht) * [[http://trac.sagemath.org/sage_trac/ticket/7388 | #7388]] (Robert Miller) * [[http://trac.sagemath.org/sage_trac/ticket/7483 | #7483]] (William Stein) * [[http://trac.sagemath.org/sage_trac/ticket/7692 | #7692]], [[http://trac.sagemath.org/sage_trac/ticket/7749 | #7749]] (Steven Sivek) * [[http://trac.sagemath.org/sage_trac/ticket/7745 | #7745]] (Karl-Dieter Crisman) * [[http://trac.sagemath.org/sage_trac/ticket/7825 | #7825]] (Peter Jeremy) * [[http://trac.sagemath.org/sage_trac/ticket/7840 | #7840]] (William Stein) |
