Follow along: http://sagenb.org/home/pub/2909

Sage:  A Community Owned

Foundation for Computational Mathematics

FoCM 2011 Plenary

William Stein (University of Washington, Seattle)

Testing, testing, ...



Prelude

A Dream Come True: Community-Owned Knowledge

• Which do you use more?   Wikipedia or Encyclopædia Britannica?  
  
  
• Which are you more likely to contribute to?
  
  
• The English Wikipedia alone has over 1 billion words, which is over 25 times as many as Encyclopædia Britannica.
  
  

Sage Project Mission Statement

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

• Mathematical features: Of Magma, Maple, Mathematica, and Matlab, with comparable speed
  
  
• Graphics: 2d and 3d
  
  
• Notebook: Interactive graphical user interface
  
  
• Documentation: Books, papers, curriculum, etc.

Sage is not a drop-in replacement; it does not run programs written in the custom languages of the Ma's. It's nothing like Octave (versus Matlab).  It's culture, architecture, and feel are all very different than the Ma's. 

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
   
   
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

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; well over a half million lines; written by a worldwide community of several hundred people.
   
   

= PolynomialRing(QQ,2) /// }}} {{{id=179| R /// Multivariate Polynomial Ring in x, y over Rational Field }}} {{{id=178| /// }}} {{{id=154| /// }}} {{{id=92| /// }}}

Distribution

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

Hundreds of Sage Developers

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

---

History

• 2005:SAGE-0.1 released February 1, 2005; SAGE=Software for Arithmetic Geometry Experimentation
  
  
• 2006: (2 Sage Days workshops); Sage is not just for number theory
  
  
• 2007: (4 Sage Days) 100% test requirements; peer review of all new code (see trac); industry funding; NSF; Trophees du Libre
  
  
• 2008: (7 Sage Days) Release managers besides me.
  
  
• 2009: (8 Sage Days) Better foundations; 3d graphics; more developers (e.g., sage-combinat)
  
  
• 2010: (13 Sage Days) More devs and users; nontrivial NSF grants
  
  
• 2011: (>= 9 Sage Days) Faster http://sagenb.org; undergrad curriculum development taking off.

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

{{{id=98| /// }}} {{{id=157| /// }}} {{{id=156| /// }}} {{{id=181| /// }}} {{{id=185| /// }}} {{{id=184| /// }}}

Question Break 1 (of 2)

????

{{{id=190| reduce('2+3') /// }}} {{{id=189| f = mathematica('Integrate[Sin[x],x]'); f /// -Cos[x] }}} {{{id=188| f.Integrate(x) /// -Sin[x] }}} {{{id=187| f + mathematica(sin(x)*cos(x^2)) /// -Cos[x] + Cos[x^2]*Sin[x] }}} {{{id=202| a = float(2^64); a /// 1.8446744073709552e+19 }}} {{{id=183| math.sin(a) /// 0.023598509904439558 }}} {{{id=182| mathematica('N[Sin[2^64], 100]') /// 0.023598509904439558634365922876134775188797125047081359505208257512787036719571480535326817566662323544 }}} {{{id=87| a = sin(2^64); a /// sin(18446744073709551616) }}} {{{id=203| N(a, digits=100) /// 0.02359850990443955863436592287613477518879712504708135950520825751278703671957148053532681756666232354 }}} {{{id=86| a = [bernoulli(2*n) for n in [1..25]]; a /// [1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798, -174611/330, 854513/138, -236364091/2730, 8553103/6, -23749461029/870, 8615841276005/14322, -7709321041217/510, 2577687858367/6, -26315271553053477373/1919190, 2929993913841559/6, -261082718496449122051/13530, 1520097643918070802691/1806, -27833269579301024235023/690, 596451111593912163277961/282, -5609403368997817686249127547/46410, 495057205241079648212477525/66] }}} {{{id=206| B = matrix(5,5,a); show(B) ///

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrrr} \frac{1}{6} & -\frac{1}{30} & \frac{1}{42} & -\frac{1}{30} & \frac{5}{66} \\ -\frac{691}{2730} & \frac{7}{6} & -\frac{3617}{510} & \frac{43867}{798} & -\frac{174611}{330} \\ \frac{854513}{138} & -\frac{236364091}{2730} & \frac{8553103}{6} & -\frac{23749461029}{870} & \frac{8615841276005}{14322} \\ -\frac{7709321041217}{510} & \frac{2577687858367}{6} & -\frac{26315271553053477373}{1919190} & \frac{2929993913841559}{6} & -\frac{261082718496449122051}{13530} \\ \frac{1520097643918070802691}{1806} & -\frac{27833269579301024235023}{690} & \frac{596451111593912163277961}{282} & -\frac{5609403368997817686249127547}{46410} & \frac{495057205241079648212477525}{66} \end{array}\right)

}}} {{{id=215| maxima(B) /// matrix([1/6,-1/30,1/42,-1/30,5/66],[-691/2730,7/6,-3617/510,43867/798,-174611/330],[854513/138,-236364091/2730,8553103/6,-23749461029/870,8615841276005/14322],[-7709321041217/510,2577687858367/6,-26315271553053477373/1919190,2929993913841559/6,-261082718496449122051/13530],[1520097643918070802691/1806,-27833269579301024235023/690,596451111593912163277961/282,-5609403368997817686249127547/46410,495057205241079648212477525/66]) }}} {{{id=210| type(B) /// }}} {{{id=209| f = B.charpoly(); f /// x^5 - 495057205273309581358818895/66*x^4 + 61983227840206665836404733579108736902585440527440571508/46582559287006925*x^3 - 3329536450676795979922301816677295701621727182818226937358357764336/5750644893516577095405*x^2 + 6493333576199389777336154216471826424372955666712765638117933711360/16375645934871014776439*x - 3431762080971147893771985302569072595725008815917760783195308032000/73857505134826413583531 }}} {{{id=212| plot(f, 0,1e25) /// }}} {{{id=217| plot(f, 0,1e25) /// }}} {{{id=213| f.roots(RDF) /// [(0.150059334256, 1), (0.534801951741, 1), (435126.664288, 1), (1.77394190156e+14, 1), (7.50086674639e+24, 1)] }}} {{{id=218| RDF /// Real Double Field }}} {{{id=220| a = sqrt(2) + 5/3; a /// sqrt(2) + 5/3 }}} {{{id=221| a /// sqrt(2) + 5/3 }}} {{{id=222| parent(5/3) /// Rational Field }}} {{{id=223| parent(sqrt(2)) /// Symbolic Ring }}} {{{id=225| N(a, 100) /// 3.0808802290397617154683553909 }}} {{{id=224| v = RealIntervalField(300)(a); v /// 3.08088022903976171546835539087636474523633854204361473984334640465739914512877370551705421? }}} {{{id=226| v.lower() /// 3.08088022903976171546835539087636474523633854204361473984334640465739914512877370551705420 }}} {{{id=227| v.upper() /// 3.08088022903976171546835539087636474523633854204361473984334640465739914512877370551705421 }}} {{{id=219| RealField(300) /// Real Field with 300 bits of precision }}} {{{id=214| f.roots(RealField(300)) /// [(0.150059333481093391773686375578264240701638829767703759632823388736689908987599290621254000, 1), (0.534801953521090603789758819043126592253340136980208644293456816768698841917499312647873830, 1), (435126.664297256454269261184284331297953186551891568161740994260752652307965164882644080304, 1), (1.77394190155680591105539556683033543721657724243993243309540258438532706303343039717196551e14, 1), (7.50086674638790249666402380216015941110613395097305878065441149224416725615049334908804157e24, 1)] }}} {{{id=207| v = B.eigenvalues(); v /// [0.1500593334810934?, 0.5348019535210906?, 435126.6642972565?, 1.773941901556806?e14, 7.500866746387902?e24] }}} {{{id=205| alpha = v[0]; alpha /// 0.1500593334810934? }}} {{{id=208| alpha.minpoly() /// x^5 - 495057205273309581358818895/66*x^4 + 61983227840206665836404733579108736902585440527440571508/46582559287006925*x^3 - 3329536450676795979922301816677295701621727182818226937358357764336/5750644893516577095405*x^2 + 6493333576199389777336154216471826424372955666712765638117933711360/16375645934871014776439*x - 3431762080971147893771985302569072595725008815917760783195308032000/73857505134826413583531 }}} {{{id=204| /// }}}

Interactive Image Compression

(using numpy) 

{{{id=29| import pylab, numpy X = pylab.imread(DATA + 'focm.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=(8,4)) html("Compressed to %.1f%% of size using %s eigenvalues."%( 100*(2.0*i*n+i)/(n*n), i)) ///

Eigenvalues (quality)

}}} {{{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(10^7) /// 92027175502604546685596278166825605430729405281023979395328576351741298526232350197882291654710333933219876431112892669442996519201446933718057425885425510196566971369272243936886123704944390011846626724222935883880949646021554674211449712293631879438242092222979701858787035045131791561718499094276679781015502944193307504577212918898104161448934354538420643899518683659226259312517022343012768006249066347774384224200200491423135789948628712467862610060006610227873354093344771970346402912468019502617741296485750068965727678736574879683519236357061319134860914524427627076446580477740857594944050855144756641881148963046419111504530928013165254773178279374714115048498031936743061146399094602347281946619566715867818368113040887581799683872172944577575391666322871295451048112070492385908727524159239222366587691028630013147462129464573569940173628469758175515190001641345140899367093190859080267185611792170429465197857968117435299141079155705662249617336911859509285578584413441733816964914269258918743530174261522145886491433067881470395832647408578195566042566460494284912372612048505127243987254597660690323804252237148083121685300106473794690689801747504661946299472005981442948094926764563853171727831538610914056742150737497538427502124024964209080033657278768694168219946346309906686670220404292685540282106667661584147201155740243521818020629234011924167214100674819982674459329845176126920689181532303574746823885977198276651255478126323692431978852519785180336170541394768609366389114772406136683349412746708709224596935940110215929918473383014389065632764805267657590865513593978244244311680854217735536592823219001199569770431572718888108452817614295777255622715362408122265800848932276196267545354426556723347685735897551768373108977390874033016418618266068861412680046926417299574367992118065543824471964711585682146618651730116890773279227432137954488457335074000412015820488115033194548391813526724286716271750346470118187739995800431441608729698516332428256345862484406623639956273335472545741134762782834725659303362718667415197368798040219223371637134331256426629887430519590196027703231191899072357523708159719377437782971846471928181434133277347743728346062720786947891744911516248998265758156411629802345973249527007535170095624359891373084013345497111826508558570667664488643633183476426421938705419176750248730025527079093923132020478303851642054200888831310378483811088425548758508373324093873479730273437693679342081086702018520971877967092726990888474755041255077282275352308667980953624552474645295774554823817830271752015905175487656200680986962732503598211618667111264793583287147032328796244469851598884862190118775508811358918545701650113647140734902415801399931326365622784208807398682277421689531141855076998289838068746126554377321750652288128533274971246389513818051037817849704769088911818484659276794986806465634286299339344423718490667472287384981129496643999747556606122301402027411096367172771385437211572122448922846957076711812454645572143320128971630906382167345381586565940316748316082706877584540948496188398233399350019423833682423300814059493159013417591894979385650634324308841947277121816559279335938926911510683549043290690287102733571315222761846482615431786061813454633634415974179413942024706129986725734471552346167738613509470760758338637870579921007168514417341548481513953296373455058614174692678013759737246724696931125240457406888289154055030387593548942880549262383621259594080699698643245355453826567378500963781681659096276126857969078217677288980 Time: CPU 0.32 s, Wall: 0.50 s }}} {{{id=102| @interact def _(n=(25..10000)): plot(prime_pi, 0, n, gridlines='minor').show() ///

n

}}} {{{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| E.change_ring(GF(next_prime(10^50))).cardinality() # pari SEA /// 100000000000000000000000001917684156174529696959920 }}} {{{id=6| /// }}}

Graph Theory

{{{id=216| graphs.BuckyBall().plot().show(figsize=15) /// }}} {{{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=78| graph_editor(G) ///

SaveClose

}}} {{{id=74| graphs. /// }}} {{{id=113| /// }}}

Cython

• Smooth transition between Python and compiled C code.
• Make code that involves lots of manipulation of C-level data structures optimally fast (orthogonal to algorithm choice).
• Heavily used in scientific computing using Python.

{{{id=111| def python_sum2(n): s = int(0) for i in xrange(1, n+1): s += i*i return s /// }}} {{{id=123| python_sum2(3) /// 14 }}} {{{id=142| python_sum2(2*10^6) /// 2666668666667000000 }}} {{{id=110| timeit('python_sum2(2*10^6)') /// 5 loops, best of 3: 170 ms per loop }}} {{{id=120| %cython def cython_sum2(long n): cdef long i, s = 0 for i in range(1, n+1): s += i*i return s /// }}} {{{id=124| cython_sum2(3) /// 14 }}} {{{id=141| cython_sum2(2*10^6) /// 2666668666667000000 }}} {{{id=109| timeit('cython_sum2(2*10^6)') /// 125 loops, best of 3: 1.94 ms per loop }}} {{{id=122| 153/1.88 /// 81.3829787234043 }}} {{{id=140| /// }}} Of course, it is better to choose a different algorithm: {{{id=126| var('k, n') show(factor(sum(k^2, k, 1, n))) ///

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6} \, {\left(n + 1\right)} {\left(2 \, n + 1\right)} n

}}} {{{id=121| def sum2(n): return n*(2*n+1)*(n+1)/6 /// }}} {{{id=146| sum2(2*10^6) /// 2666668666667000000 }}} {{{id=136| timeit('sum2(2*10^6)') /// 625 loops, best of 3: 2.81 µs per loop }}}

Even then, Cython provides a speedup:

{{{id=143| %cython def c_sum2(long n): return n*(2*n+1)*(n+1)/6 /// }}} {{{id=197| c_sum2(3) /// 14 }}} {{{id=144| timeit('c_sum2(2*10^6)') /// 625 loops, best of 3: 653 ns per loop }}}

But at a cost!

{{{id=145| c_sum2(2*10^6) # WARNING: overflow -- it's just like C... /// -407788678951258603 }}} {{{id=147| n=2*10^6; n*(2*n+1)*(n+1) > 2^63 /// True }}} {{{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 2.08 s, Wall: 2.00 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| /// }}}

Question Break 2 (of 2)

????

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