{{{id=302| /// }}} {{{id=301| /// }}}

Follow along: http://sagenb.org/home/pub/3702

Sage:  Creating a Viable Open Source Alternative to
Magma, Maple, Matlab, and Mathematica

Combinatorial Potlach 2011 (Seattle U)

William Stein (University of Washington, Seattle)









Testing, testing, ...

 

{{{id=130| 1 + 2 + 3 /// 6 }}} {{{id=194| /// }}} {{{id=193| /// }}} {{{id=192| /// }}} {{{id=191| /// }}} {{{id=175| /// }}} {{{id=174| /// }}} {{{id=177| /// }}}



Prelude

 

 

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Potlatch

Pot·latch — n. Chinook for the Native spirit of gift-giving (potlatchfund.org)

 

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Sage Project Mission Statement

"Create A Viable Free Open Source Alternative to Magma, Maple, Mathematica, and Matlab"

 

Sage is not a drop-in replacement: does not run programs written in the custom languages of the Ma's.

Sage is not like Octave (versus Matlab).  

Sage's culture, architecture, programming language, and feel are different than the Ma's. 

{{{id=272| /// }}} {{{id=271| /// }}} {{{id=158| /// }}} {{{id=133| /// }}} {{{id=132| /// }}} {{{id=94| /// }}}

Why not Magma, Maple, Matlab, Mathematica?

  1. Commercial: Expensive for my collaborators and students.  Not community owned.

  2. Closed: Implementation of algorithms often secret

  3. Frustrating: Tight control of development

  4. Copy protection: Hard to run on supercomputer or my new laptop or after my 1-year license expires.

  5. Programming language: All use a special math-only language

  6. Bugs: Bug tracking is secret

  7. Compiler: Lack of compilers for their math-only languages
{{{id=242| /// }}} {{{id=241| /// }}} {{{id=247| /// }}} {{{id=246| /// }}} {{{id=245| /// }}} {{{id=250| /// }}}

Have you ever seen this?  I did this morning when preparing this talk:

 

Last login: Fri Nov 18 16:16:32 on ttys008
You have mail.
deep:~ wstein\$ math
Mathematica 7.0 for Mac OS X x86 (64-bit)
Copyright 1988-2009 Wolfram Research, Inc.

/Users/wstein/Library/Mathematica/Licensing/mathpass:1:
	The Mathematica license you are using has expired.
	Please contact Wolfram Research or an authorized
	Mathematica distributor to extend your license and
	obtain a new password.

You will need to get a password from your
license certificate or from Wolfram Research
(http://register.wolfram.com).
Machine name:	deep.local
MathID:	5108-53088-04270

You will need a valid license ID and password in order
to proceed. Go to http://register.wolfram.com or
consult your Getting Started documentation.

Enter your name:
{{{id=249| /// }}} {{{id=248| /// }}} {{{id=244| /// }}} {{{id=243| /// }}} {{{id=153| /// }}} {{{id=105| /// }}} {{{id=93| /// }}} {{{id=106| /// }}}

  1. Python: a mainstream general purpose programming language (with a compiler: Cython)

  2. Distribution: about 100 open source packages (written by you and your colleagues!)

  3. Interfaces: smoothly tie together all these libraries and packages

  4. New library: implements novel algorithms; over a half million lines; worldwide community of several hundred people.

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Distribution

{{{id=134| /// }}} {{{id=91| /// }}}

Hundreds of Sage Developers

(There are at least 242 contributors in 164 different places from all around the world.)

 

{{{id=8| bernoulli?? /// }}} {{{id=135| /// }}} {{{id=83| /// }}}

History

See this article for more details about the (pre-)history of Sage.

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Question Break

????

{{{id=275| factorial(3) /// 6 }}} {{{id=274| A = random_matrix(QQ, 4); b = A^(-1); b /// [ 0 -1/2 0 -1/2] [-2/3 -1 -1/3 -1/2] [ 0 1/2 0 0] [-1/3 -1 1/3 -1/2] }}} {{{id=305| show(b) ///
\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr} 0 & -\frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{2}{3} & -1 & -\frac{1}{3} & -\frac{1}{2} \\ 0 & \frac{1}{2} & 0 & 0 \\ -\frac{1}{3} & -1 & \frac{1}{3} & -\frac{1}{2} \end{array}\right)
}}} {{{id=273| latex(b) /// \left(\begin{array}{rrrr} 0 & -\frac{1}{2} & 0 & -\frac{1}{2} \\ -\frac{2}{3} & -1 & -\frac{1}{3} & -\frac{1}{2} \\ 0 & \frac{1}{2} & 0 & 0 \\ -\frac{1}{3} & -1 & \frac{1}{3} & -\frac{1}{2} \end{array}\right) }}} {{{id=304| show(integrate(sin(x)*cos(x), x)) ///
\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{2} \, \cos\left(x\right)^{2}
}}} {{{id=190| G = plot(sin(x)*cos(x), (0, 4)) G /// }}} {{{id=260| G.save('a.pdf') /// }}} {{{id=259| sage.interfaces. /// }}} {{{id=258| /// }}} {{{id=262| /// }}} {{{id=276| /// }}} {{{id=261| /// }}}

Interfaces

Use a single Mathematica session from within Sage.  Here, Mathematica is running on a computer at UW, since my laptop license expired. 

{{{id=189| mathematica = Mathematica(server='sage.math.washington.edu', server_tmpdir='/tmp/') f = mathematica('Integrate[Sin[x],x]'); f /// ^C __SAGE__ }}} {{{id=188| f.Integrate(x) /// -Sin[x] }}} {{{id=187| f + mathematica(sin(x)*cos(x^2)) /// -Cos[x] + Cos[x^2]*Sin[x] }}} {{{id=257| /// }}} {{{id=256| /// }}} {{{id=255| /// }}} {{{id=254| /// }}} {{{id=202| /// }}} {{{id=204| /// }}}

Interactive Image Compression

(using numpy) 

{{{id=29| import pylab, numpy X = pylab.imread(DATA + 'wyckoff.png') A_image = numpy.mean(X, 2) u,s,v = numpy.linalg.svd(A_image) S = numpy.zeros(A_image.shape) S[:len(s),:len(s)] = numpy.diag(s) n = A_image.shape[0] @interact def svd_image(i = ("Eigenvalues (quality)", (20,(1..A_image.shape[0]//2)))): A_approx = numpy.dot(numpy.dot(u[:,:i], S[:i,:i]), v[:i,:]) g = graphics_array([matrix_plot(A_approx), matrix_plot(A_image)]) show(g, axes=False, figsize=(6,3)) html("%sx%s image compressed to %.1f%% of size using %s eigenvalues."%( A_image.shape[0], A_image.shape[1],100*(2.0*i*n+i)/(n*n), i)) /// }}} {{{id=278| /// }}} {{{id=277| /// }}} {{{id=82| /// }}}

Number Theory

{{{id=115| factor(2009201020112012) # uses PARI /// 2^2 * 43 * 2269 * 5148259709 }}} {{{id=11| # Jon Bober - Rademacher's formula time number_of_partitions(10^6) /// 1471684986358223398631004760609895943484030484439142125334612747351666117418918618276330148873983597555842015374130600288095929387347128232270327849578001932784396072064228659048713020170971840761025676479860846908142829356706929785991290519899445490672219997823452874982974022288229850136767566294781887494687879003824699988197729200632068668735996662273816798266213482417208446631027428001918132198177180646511234542595026728424452592296781193448139994664730105742564359154794989181485285351370551399476719981691459022015599101959601417474075715430750022184895815209339012481734469448319323280150665384042994054179587751761294916248142479998802936507195257074485047571662771763903391442495113823298195263008336489826045837712202455304996382144601028531832004519046591968302787537418118486000612016852593542741980215046267245473237321845833427512524227465399130174076941280847400831542217999286071108336303316298289102444649696805395416791875480010852636774022023128467646919775022348562520747741843343657801534130704761975530375169707999287040285677841619347472368171772154046664303121315630003467104673818 Time: CPU 0.03 s, Wall: 0.03 s }}} {{{id=102| @interact def _(n=(25..10000)): plot(prime_pi, 0, n, gridlines='minor').show(figsize=[8,3]) /// }}} {{{id=116| E = EllipticCurve('37a'); E /// Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field }}} {{{id=103| v = E.integral_points(both_signs=True); v # mwrank, new code /// [(-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 1), (1 : 0 : 1), (2 : -3 : 1), (2 : 2 : 1), (6 : -15 : 1), (6 : 14 : 1)] }}} {{{id=119| plot(E, xmax=7, gridlines=True) + points([z[:2] for z in v], color='red', pointsize=50) /// }}} {{{id=117| /// }}} {{{id=279| /// }}} {{{id=6| /// }}}

Graph Theory

"Sage's graph theory crushes anything I have met from the point of view of methods implemented.  I would say: 'if you found a proprietary graph library and you are convinced that it is better for your needs than Sage, your license is on me.' But those licenses are really expensive :-D"

{{{id=306| graphs.ClawGraph( /// }}} {{{id=216| G = graphs.BuckyBall() G.plot().show(figsize=8) /// }}} {{{id=76| set_random_seed(1) G = graphs.RandomLobster(8, .6, .3) show(G, figsize=7) /// }}} {{{id=77| G.automorphism_group() /// Permutation Group with generators [(12,26)] }}} {{{id=159| G.chromatic_number() /// 2 }}} {{{id=196| G.shortest_path(13,20) /// [13, 0, 1, 2, 3, 4, 5, 19, 20] }}} {{{id=74| /// }}} {{{id=113| /// }}} {{{id=288| /// }}} {{{id=287| /// }}} {{{id=286| /// }}} {{{id=292| /// }}} {{{id=291| /// }}}

Algebraic Combinatorics

The sage-combinat project: "improve the open source mathematical system Sage as an extensible toolbox for computer exploration in (algebraic) combinatorics, and foster code sharing between researchers in this area."

For example, see Combinatorics Chapter of the Reference Manual

{{{id=295| s = SFASchur(QQ) h = SFAHomogeneous(QQ) p = SFAPower(QQ) e = SFAElementary(QQ) m = SFAMonomial(QQ) /// }}} {{{id=296| s /// Symmetric Function Algebra over Rational Field, Schur symmetric functions as basis }}} {{{id=290| a = s([3,1]); a /// s[3, 1] }}} {{{id=307| a*a /// s[3, 3, 1, 1] + s[3, 3, 2] + s[4, 2, 1, 1] + s[4, 2, 2] + 2*s[4, 3, 1] + s[4, 4] + s[5, 1, 1, 1] + 2*s[5, 2, 1] + s[5, 3] + s[6, 1, 1] + s[6, 2] }}} {{{id=281| p(a) /// 1/8*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] - 1/8*p[2, 2] - 1/4*p[4] }}} {{{id=298| a.inner_plethysm(a) /// s[1, 1, 1, 1] + 2*s[2, 1, 1] + s[2, 2] + 2*s[3, 1] }}} {{{id=299| /// }}} {{{id=297| /// }}}

Cython

{{{id=111| def python_sum(n): s = int(0) for i in xrange(1, n+1): s += i*i return s /// }}} {{{id=123| python_sum(3) /// 14 }}} {{{id=142| time python_sum(2*10^6) /// 2666668666667000000 Time: CPU 0.16 s, Wall: 0.16 s }}} {{{id=228| timeit('python_sum(2*10^6)') /// 5 loops, best of 3: 164 ms per loop }}} {{{id=229| def python_sum2(n): return sum(i*i for i in xrange(1,n+1)) /// }}} {{{id=230| time python_sum2(2*10^6) /// 2666668666667000000 Time: CPU 0.19 s, Wall: 0.19 s }}} {{{id=120| %cython def cython_sum(long n): cdef long long i, s = 0 for i in range(1, n+1): s += i*i return s /// }}} {{{id=124| cython_sum(3) /// 14L }}} {{{id=141| time cython_sum(2*10^6) /// 2666668666667000000L Time: CPU 0.00 s, Wall: 0.00 s }}} {{{id=109| timeit('cython_sum(2*10^6)') /// 625 loops, best of 3: 671 ns per loop }}} {{{id=122| 165/.663 /// 248.868778280543 }}} {{{id=140| %cython def cython_sum2(long n): cdef long long i return sum(i*i for i in range(1,n+1)) /// }}} {{{id=233| time cython_sum2(2*10^6) /// 2666668666667000000L Time: CPU 0.16 s, Wall: 0.16 s }}} {{{id=232| /// }}} Of course, it is better to choose a different algorithm: {{{id=234| var('k, n') factor(sum(k^2, k, 1, n)) /// 1/6*(n + 1)*(2*n + 1)*n }}} {{{id=121| def sum2(n): return n*(2*n+1)*(n+1)/6 /// }}} {{{id=146| sum2(2*10^6) /// 2666668666667000000 }}}

Even then, Cython provides a speedup:

{{{id=143| %cython def c_sum2(long long n): return n*(2*n+1)*(n+1)/6 /// }}} {{{id=197| c_sum2(3) /// 14L }}} {{{id=236| c_sum2(2*10^6) /// -407788678951258603L }}} {{{id=237| n = 2*10^6 timeit('sum2(n)') /// 625 loops, best of 3: 2.01 µs per loop }}} {{{id=238| timeit('c_sum2(n)') /// 625 loops, best of 3: 218 ns per loop }}} {{{id=235| 2.01/.218 /// 9.22018348623853 }}} {{{id=240| /// }}} {{{id=239| /// }}}

But at a cost!

{{{id=145| c_sum2(2*10^6) # WARNING: overflow -- it's just like C... /// -407788678951258603L }}} {{{id=147| n=2*10^6; n*(2*n+1)*(n+1) > 2^63 /// True }}} {{{id=284| /// }}} {{{id=283| /// }}} {{{id=137| /// }}}

Solving Equations

Solve a cubic equation:

{{{id=15| x = var('x'); show(solve(x^3 + x - 1==0, x)[0]) ///
\newcommand{\Bold}[1]{\mathbf{#1}}x = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} + \frac{-i \, \sqrt{3} + 1}{6 \, {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}}
}}}

Solve a system of two linear equations with one unknown coefficient $\alpha$:

{{{id=17| var('alpha, y') show(solve([3*x + 7*y == 2, alpha*x + 3*y == 8], x,y)[0]) ///
\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \frac{50}{7 \, \alpha - 9}, y = \frac{2 \, {\left(\alpha - 12\right)}}{7 \, \alpha - 9}\right]
}}}

Solve a system of 500 linear equations exactly over the rational numbers:

{{{id=127| n = 500 A = random_matrix(QQ,n,n,num_bound=10, den_bound=10) v = random_matrix(QQ,n,1,num_bound=10, den_bound=10) A.visualize_structure() /// }}} {{{id=148| # IML is used -- http://www.cs.uwaterloo.ca/~astorjoh/iml.html time w = A \ v /// Time: CPU 3.08 s, Wall: 2.04 s }}} {{{id=150| print w.str() /// WARNING: Output truncated! full_output.txt [ 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}}} {{{id=149| /// }}} {{{id=151| /// }}}

Symbolic Calculus

Symbolic Calculus makes use of Maxima and Ginac under the hood.

{{{id=26| var('x') f = sqrt(1 - x^2) plot(f, thickness=3, color='red', aspect_ratio=1, fill=True) /// }}} {{{id=163| var('t') assume(t+1 > 0) g = integrate(f, (x, -1, t)); show(g) ///
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, \pi + \frac{1}{2} \, \sqrt{-t^{2} + 1} t + \frac{1}{2} \, \arcsin\left(t\right)
}}} {{{id=162| show(g(t=1) - g(t=-1)) ///
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{2} \, \pi
}}} {{{id=160| /// }}} {{{id=161| /// }}}

Plotting in 2D

{{{id=71| plot(sin(1/x^2), (x,.1,.5)) /// }}} {{{id=166| G = sum(circle((random(), random()), random()/3, color=hue(random())) for i in range(25)) G.show(aspect_ratio=1, frame=True) /// }}} {{{id=165| /// }}} {{{id=32| /// }}}

Plotting in 3D

{{{id=30| f(x,y) = sin(x - y) * y * cos(x) plot3d(f, (x,-3,3), (y,-3,3), opacity=.9, color='red') /// }}} {{{id=168| G = sum(sphere((random(), random(), random()), random()/4, color=hue(random()), opacity=.6) for i in range(20)) G.show(aspect_ratio=1, frame=True) /// }}} {{{id=167| /// }}} {{{id=65| /// }}} {{{id=64| /// }}}

Plotting a 3D Model

See http://www.davidson.edu/math/chartier/Starwars/

{{{id=49| # Yoda! (53,756 vertices) from scipy import io yoda = io.loadmat(DATA + 'yodapose.mat') from sage.plot.plot3d.index_face_set import IndexFaceSet V = yoda['V']; F3=yoda['F3']-1; F4=yoda['F4']-1 Y = IndexFaceSet(F3,V,color=Color('#444444')) + IndexFaceSet(F4,V,color=Color('#007700')) Y = Y.rotateX(-1) Y.show(aspect_ratio=1, frame=False, zoom=1.2) /// }}} {{{id=96| /// }}}

Questions

????

{{{id=131| /// }}}