An Introduction to Sage

William Stein, Professor of Mathematics, University of Washington

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Mission Statement

Create a viable free open source alternative to Magma, Maple, Mathematica, and Matlab.

Firefox <--> Internet Explorer, Opera

Open Office, Latex <--> Microsoft Office

Linux <--> Microsoft Windows

PostgreSQL, MySQL <--> Oracle, Microsoft SQLserver

GIMP <--> Photoshop

Sage <--> Magma, Maple, Mathematica, Matlab

History of Sage

I started Sage at Harvard in January 2005.

Sage-1.0 released February 2006 at Sage Days 1 (UC San Diego).

Nearly 30 Sage Days Workshops (!) at UCLA, UW, Cambridge, Bristol, Austin, France, San Diego, Seattle, MSRI, ..., Barcelona, ...

Sage won first prize in the Trophees du Libre (November 2007)

Funding from Microsoft, Univ of Washington, UC San Diego, NSF, DoD, Google, Sun, private donations, etc.

Hundreds of other people subsequently got involved in Sage's development, and the scope of the project has widened to cover all mathematical computation. There is interest from all areas of mathematics, physics, engineering, etc. (Developer mailing list has 1184 subscribers and about 50 messages/day... and sage-support has 1718 subscribers.)

What is Sage?

Sage = Python + Math
A unified self-contained distribution of open source free mathematical software.
Nearly a half million lines of new Python (and Cython) code that implements new capabilities and algorithms.
A "cloud" application like GMail or Google Docs: http://sagenb.org (currently about 30,000 users); Of course Sage also runs on your desktop and supercomputer.

Tour of the http://sagemath.org website

Google page search
Documentation
Live Chat: http://sagemath.org/help-irc.html
Downloading and Installing Sage
Developer map

Summary:
Sage is about building the car instead of reinventing the wheel

Sage uses Python, a mainstream programming language, instead of inventing a custom mathematics language
Use straightforward method to link programs together -- C library and pseudotty's, instead of XML servers/OpenMath. We implement all conversion routines, instead of expecting upstream to do it: we make them communicate with Sage, whether they want to or not.
Give copious credit to contributors and be very developer friendly (easily build from source).
Reuse, improve, and contribute to existing libraries and projects (e.g., Scipy, Numpy, R, ATLAS, CVXopt, GSL), instead of starting over and competing with them.
Make the GUI using a web browser: the world of java and javascript is available and Sage integrates with the web.

And now a demo...

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Very Big Numbers

Very big numbers: In less than a second, Sage exactly computes $(10^6)!$, which has over 5 million digits.

{{{id=12| time n = factorial(10^6) /// Time: CPU 0.68 s, Wall: 0.79 s }}} {{{id=14| n.ndigits() /// 5565709 }}} {{{id=20| time m = n * (n+1) /// Time: CPU 0.38 s, Wall: 0.39 s }}}

and a matrix with a big determinant:

{{{id=63| a = random_matrix(ZZ,200, x=2^128); a /// 200 x 200 dense matrix over Integer Ring (type 'print a.str()' to see all of the entries) }}} {{{id=64| a = random_matrix(ZZ,200, x=2^128); a /// 200 x 200 dense matrix over Integer Ring (type 'print a.str()' to see all of the entries) }}} {{{id=61| a[0,0] /// 50110420315670456383223454274407788191 }}} {{{id=67| time a.determinant() /// 11218105205230809746566316190198723341407927476765957799203199799212703595012684629333238328097186325704207831003229127678788755296132397468592341142460886190515509450222101477918147750547051743861165817737683676760178130730972784974992498200917894088217369828814426381521688535283126383787262165539698594143403476047047431785870841431339615899823418060606351060595348606580507982277525447817776456629742788523583811463136959111115491705163683530094679015309382866665731315521616670050657542027347823700597311223576219233101799188047803166368566346484642689366265149406543140717265621618149134812073698237363997318271201783691087129265728799818647928427690121945189129598175747254695679743510243159043438135187406689278710284075826660699729451760719538176327755629532401174627191322417176387908815565452056556065853085903279721517235744488355018097688236451490342335703719833553002032041040527723310092389070197093394769262864684323426695422197081740459445714114735549193400919169917066050555345309267128927803869233377774865301821745146086882512456793949710051994179606493276721768324076620700442112373116486512668830086481557594920909147506407177885537006647236022333825913105571797564132067148418354302852936982767159543041205177629871746713292358466722388015976917307283109743686639937667590676853581042627996236017545386232873479026793270905963262161309337673897798027036836663035257833812737315617990728234078234073667898882284894758914468536936037657776743249137606007621820781471553390128598377135204505346779179200573800205110632590122831948650044473843560709296192627890557705528815055345642742046508326499486537673210799666860208171933281829730266996519370558481425900852230916139167450167250694529555796931243510108115154206280470145167137406337317972127204937699640879285304093854511591269050029370930299731337801812579525661192772233522349653629370232146260210768259464031211172129132856749553749721041373250852304851950031822653866903774390544270118854687839050306503433784204145360149229374849248362487882759140506584691478169534425723699208701101709949212704306747050151551781741561566841735020649685019353818198408667386282791216983262332621588640484060724199188986857204529434792968649459368354238874081537890087303720022981698942290902502720625056975526130074457387275255804618348916869945642288681297739396782676708083604593127106033283699253880506833768957520139775388829132231775970652018474762047685741020477397064598308635631066155225444160633087987459306424267754628610362635796675584603295830364517189899979709625058903549693440228240846220992323608258502361577773506354022881793685659977176811363016962518236264428107264768416120144702680626472758916786551978716758256872075806240495877172100804645513042403808845678174836777082909943157159774472621396192397668884135542471439835938124172713633537608106701264778386863204761899866705962880793071648495429651005708607218495003914399368148614944391501324379708317174737119003622929693043656285984819884379821055606389566124987431410843693631248288603858746496696281112465700475117355515676200278363212508757842369551504300299513396776640978385735134087083919864433578206227715179063964150961416665970254232788930580034791337176494122843497349052838762846193304315803500730282856066253799830308497863056102411261371663242669647300814264449673852656793777608491847554101000766952392786094232148801826887523658347628268147860059622751178838747891907713841008977454119680045974014697538706531251492282807250095846769541957310495654011719061046611217885367794624044406869851757670867129266967441563119868953144170612077460039377512806963798446503494009323285791272162784174447771492728678150778580604251296371859970625272498800302753447839956185490560440331864626292584154766061293688806193537810653979452163009190456657940203550672249702896190641434352167543038912113598962114200009657897804337381576895634396723292082082246697056259211285124748049919801146513508999321755323096768230095487606896590780661172161212349285850841051800764370261467451915922843035227545481310739716429253129574475155538988454752204961910216368465824111660352224343069883730979723967867931404724023782000267723181803352362804793764165639092072764433570660448255563541939811391287327167176688071237664266219636981331082393679105852227932112190310614499997682147538448620478595536435618168906821658251363182269281798898723482840560269845528721682881098845284660573289709161287615494324695976868334097309391487107738489682379679031847505302902193482674469832144990668581654967828227583056627920769560889135070764507314995618199778157610923460763882713574352073696721448362720731879731336199666553906750116272274821167333377636596401281794787449491791490148098831472967579987886676850191787507586465584897586589612696945164121828683085502012096140261165489103935794029973458510972990205387319186249640786733348432956887812532713415095996433354907791271004005214594065845378474452236421962896353120457343640151763622660562656352401164478364833421597776416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Time: CPU 2.38 s, Wall: 2.50 s }}} {{{id=66| /// }}}

Calculus

Sage does Calculus:

{{{id=8| f(x) = sin(x)^2*cos(x)*exp(x) show(f) ///

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ e^{x} \sin\left(x\right)^{2} \cos\left(x\right)

}}} {{{id=9| plot(f, 0, 3) ///

}}} {{{id=5| show(integrate(f, x)) ///

\newcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ -\frac{3}{40} \, e^{x} \sin\left(3 \, x\right) + \frac{1}{8} \, e^{x} \sin\left(x\right) - \frac{1}{40} \, e^{x} \cos\left(3 \, x\right) + \frac{1}{8} \, e^{x} \cos\left(x\right)

}}} {{{id=68| /// }}} {{{id=65| /// }}}

Graph Theory

Sage can do graph theory:

{{{id=69| g = graphs.FlowerSnark(); g /// Flower Snark: Graph on 20 vertices }}} {{{id=24| graph_editor(g) ///

}}}

Sage contains many unique and deep algorithms:

{{{id=70| g.automorphism_group() /// Permutation Group with generators [(2,11)(3,12)(4,13)(5,8)(6,9)(7,10)(16,19)(17,18), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,20)(15,16,17,18,19), (1,14)(2,4)(5,7)(8,10)(11,13)] }}} {{{id=38| /// }}}

Statistics

Sage includes R, scipy.stats, and GSL (=Gnu Scientific Library).

{{{id=41| t = stats.TimeSeries(10^6).randomize('normal') plot(t.sums(), frame=True, figsize=[8,3]) ///

}}} {{{id=42| t.plot_histogram() ///

}}} {{{id=43| mean(t) /// -0.00033107089746577014 }}} {{{id=44| std(t) /// 1.0004268203066153 }}} {{{id=45| show(std([1,pi,e])) ///

\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{\frac{1}{18} \, {\left(\pi - 2 \, e + 1\right)}^{2} + \frac{1}{18} \, {\left(\pi + e - 2\right)}^{2} + \frac{1}{18} \, {\left(2 \, \pi - e - 1\right)}^{2}}

}}}

Generalized Hidden Markov Models Library (a complete new implementation in Sage, by me) -- useful for math biology, computer learning, etc.:

{{{id=46| m = hmm.DiscreteHiddenMarkovModel([[0.4,0.5,0.1],[0.1,0.8,0.1],[.2,.3,.5]], [[0.1,0.9],[0.5,0.5],[1,0]], [.2,.5,.3], emission_symbols=['a','b']); m /// Discrete Hidden Markov Model with 3 States and 2 Emissions Transition matrix: [0.4 0.5 0.1] [0.1 0.8 0.1] [0.2 0.3 0.5] Emission matrix: [0.1 0.9] [0.5 0.5] [1.0 0.0] Initial probabilities: [0.2000, 0.5000, 0.3000] Emission symbols: ['a', 'b'] }}} {{{id=72| m.baum_welch(['a','b','b','a','b','b','a','b','b']) /// (-4.6196788513299064e-05, 20) }}} {{{id=75| m /// Discrete Hidden Markov Model with 3 States and 2 Emissions Transition matrix: [ 2.30981274908e-05 0.999976901873 1.89806678835e-23] [7.14454696757e-221 1.98424978494e-232 1.0] [ 1.0 0.0 0.0] Emission matrix: [6.75306548194e-69 1.0] [ 0.0 1.0] [ 1.0 0.0] Initial probabilities: [0.0000, 0.0000, 1.0000] Emission symbols: ['a', 'b'] }}} {{{id=76| m.sample(10) /// ['a', 'b', 'b', 'a', 'b', 'b', 'a', 'b', 'b', 'a'] }}} {{{id=79| g = m.graph() /// }}} {{{id=78| show(plot(m.graph(),graph_border=1), figsize=3) ///

}}} {{{id=77| A = [[0.5,0.5],[0.5,0.5]] B = [[(0.9,(0.0,1.0)), (0.1,(1,10000))],[(1,(1,1)), (0,(0,0.1))]] m = hmm.GaussianMixtureHiddenMarkovModel(A, B, [1,0]); m /// Gaussian Mixture Hidden Markov Model with 2 States Transition matrix: [0.5 0.5] [0.5 0.5] Emission parameters: [0.9*N(0.0,1.0) + 0.1*N(1.0,10000.0), 1.0*N(1.0,1.0) + 0.0*N(0.0,0.1)] Initial probabilities: [1.0000, 0.0000] }}} {{{id=74| plot(m.sample(1000).sums()) ///

}}} {{{id=84| /// }}}

Fortran!

Example here: http://sagenb.org/home/pub/1708/

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Matrices

And numerical linear algebra:

{{{id=31| import pylab A_image = pylab.mean(pylab.imread(DATA + 'lonestar.png'), 2) @interact def svd_image(i = ("Eigenvalues (quality)",(20,(1..100))), display_axes = ("Display Axes", True)): u,s,v = pylab.linalg.svd(A_image) A = sum(s[j]*pylab.outer(u[0:,j], v[j,0:]) for j in range(i)) g = graphics_array([matrix_plot(A),matrix_plot(A_image)]) show(g, axes=display_axes, figsize=(8,3)) html('

Lonestar: compressed using %s eigenvalues

'%i) ///

Eigenvalues (quality)

Display Axes

}}} {{{id=30| /// }}}

FemHub -- a customized Sage

See http://femhub.org/

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3-Dimensional Plots

Sage can draw 3d plots:

{{{id=10| var('x,y') b = 2.2 (plot3d(sin(x^2-y^2),(x,-b,b),(y,-b,b), opacity=.9) + plot3d(0, (x,-b,b), (y,-b,b), color='red')) /// }}} {{{id=48| var('x,y,z') T = golden_ratio 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)) r = 4.78 implicit_plot3d(p, (x, -r, r), (y, -r, r), (z, -r, r), plot_points=50) /// }}} {{{id=47| /// }}}

Sage can plot Yoda:

{{{id=4| from scipy import io x = io.loadmat(DATA + 'yodapose.mat', struct_as_record=True) from sage.plot.plot3d.index_face_set import IndexFaceSet V = x['V']; F3 = x['F3']-1; F4 = x['F4']-1 Y = (IndexFaceSet(F3, V, color = Color('#00aa00')) + IndexFaceSet(F4, V, color = Color('#00aa00'))) Y = Y.rotateX(-1) Y.show(aspect_ratio = [1,1,1], frame = False, figsize = 4) /// }}} {{{id=3| /// }}}

And if you're really serious, Sage is Python, so you can also use:

Mayavi: Python interface to VTK
Chaco: interact 2d graphics system developed at Enthought (local Austin company)
VisIt: https://wci.llnl.gov/codes/visit/manuals.html (python bindings)
ParaView: http://www.cmake.org/Wiki/ParaView/Python_Scripting
VPython: http://vpython.org/
PyGame: http://www.pygame.org/news.html

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Key take-away points:

FOSS: 100% free open source software: good for clusters, sharing, research, collaboration
Python: very mainstream, scientific-computing friendly programming language
Cython: optimized Python to C/C++ compiler we develop with support for C/C++ datatypes
Interfaces: to control Matlab, Octave, Mathematica, etc., from Sage
Self contained: can have many copies of Sage at once, so no "dependency hell"
Peer reviewed: get your code refereed
Web based: notebook interface

Questions?

Now, take a break, then start with the tutorials...

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