20090210 -- Sun Sage Talk system:sage



SAGE

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

William Stein, Associate Professor, University of Washington


 

 

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History

 

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Sun's Interest in Sage

{{{id=160| 2 + 3 /// 5 }}} {{{id=80| /// }}}

Sage is 100% Free

Active user community; 964 members of the sage-support mailing list.


Free webapp -- sagenb.org -- has about 3000 users (and there are other servers at universities around the world..)


sagemath.org

 


Sage Devel Headquarters: Four 24-core Sun X4450's with 128GB RAM each + 1 Sun X4540 with 24TB disk

 

 

 

sagenb.org

{{{id=161| A. = QuaternionAlgebra(QQ, -1,-1) A /// Quaternion algebra with generators (i, j, k) over Rational Field }}} {{{id=162| i*j + k /// 2*k }}} {{{id=62| /// }}}

Some Advantages of Sage over the Ma*'s

  1. Python-- general purpose language at core of Sage; huge user base compared to Matlab, Mathematica, Maple and Magma
  2. Cython -- blazingly fast compiled code
  3. Integrated web server -- (twisted, etc.)
  4. Free and Open source
  5. Excellent Peer review of Code: "I do really and truly believe that the Sage refereeing model results in better code." -- John Cremona
{{{id=119| email('wstein@gmail.com', 'The calculation finished!') /// Child process 50883 is sending email to wstein@gmail.com... }}} {{{id=118| /// }}}

 

So What is Sage?

Reference Manual (over 3000 pages, all new)

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PART 2 -- Examples

The first example from the Python tutorial:

{{{id=22| the_world_is_flat = 1 if the_world_is_flat: print "Be careful not to fall off!" /// Be careful not to fall off! }}}

Symbolic expressions:

{{{id=23| x, y = var('x,y') type(x) /// }}} {{{id=19| a = 1 + sqrt(2) + pi + 2/3 + x^y a /// x^y + pi + sqrt(2) + 5/3 }}} {{{id=163| show(a) ///
{x}^{y} + \pi + \sqrt{ 2 } + \frac{5}{3}
}}} {{{id=20| show(expand(a^3)) ///
{x}^{{3 y}} + {{3 \pi} {x}^{{2 y}} } + {{3 \sqrt{ 2 }} {x}^{{2 y}} } + {5 {x}^{{2 y}} } + {{3 {\pi}^{2} } {x}^{y} } + {{{6 \sqrt{ 2 }} \pi} {x}^{y} } + {{10 \pi} {x}^{y} } + {{10 \sqrt{ 2 }} {x}^{y} } + \frac{{43 {x}^{y} }}{3} + {\pi}^{3} + {{3 \sqrt{ 2 }} {\pi}^{2} } + {5 {\pi}^{2} } + {{10 \sqrt{ 2 }} \pi} + \frac{{43 \pi}}{3} + \frac{{31 \sqrt{ 2 }}}{3} + \frac{395}{27}
}}}

Solve equations

{{{id=101| var('a,b,c,x') show(solve(x^3 + sqrt(17)*a*x + b == 0, x)[0]) ///
x = {\left( \frac{{-\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) {\left( \frac{\sqrt{ {27 {b}^{2} } + {{68 \sqrt{ 17 }} {a}^{3} } }}{{6 \sqrt{ 3 }}} - \frac{b}{2} \right)}^{\frac{1}{3}} } - \frac{{{\sqrt{ 17 } \left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right)} a}}{{3 {\left( \frac{\sqrt{ {27 {b}^{2} } + {{68 \sqrt{ 17 }} {a}^{3} } }}{{6 \sqrt{ 3 }}} - \frac{b}{2} \right)}^{\frac{1}{3}} }}
}}} {{{id=164| # pynac var('a,b,c,x',ns=1) /// (a, b, c, x) }}} {{{id=165| time f = expand((a+b+c)^20) /// Time: CPU 0.01 s, Wall: 0.15 s }}} {{{id=166| time f = expand((a+b+c)^20) /// Time: CPU 0.07 s, Wall: 0.17 s }}} {{{id=3| /// }}} {{{id=32| /// }}}

Example: A Huge Integer Determinant

{{{id=31| a = random_matrix(ZZ,200,x=-2^127,y=2^127) time d = a.determinant(); d /// -3206345835841150086532012665544966318382751531427225751731662808227201875273081172829921248048278905259337954771231526849892612295627112667032702643641680113636489925504741612628511886339279503720307980341971832438658666228859902602522435003314362891900243033913370220018212575553019273831166631343588048716695985009486692567253408047136948885990299315171555944432935726908013352849027185005121483615467128246747108315021270196769346616634956742917837600567006487412417725151019263971709588639664984012921326642928182131583594161684811001706543787319214556189240558790862516476527044183131785129678385887546508818405919637644425510228795876962583134333062240364156552384557506451068806629892111767095933731485419230445214168880305195675325657148775189145907836542346834982036263320324461399479825104667361728255698857644069151308518088864708816270614337424571100114218122574503521754372813022792585071218715938620853725651054818184815056826006871806035434909793477068551490520450250848680795465514946198914241134786098698328248557871669223236406526182975163610174500733006772810268380445285445542799980956562893809056304576045368708348349002848128080612894644202666117232852682996980743095505633992962848808746908455979404133300839508596664619427076496272812564189916746835759370438689115248184387660039996364504583239485655705070019647528133183009559027827812683104952043165319422682527464902587985639745321159115171994377452016509132861162378305497239324702626728517229975173571056377193891822580856936183148551924820718157875317301154420991012275606897863280594510678216626144463762666385272458309785894342041303339805888640029056610189462240650742679104514954719597787899089116801811043958966477301390380314714093118482735605955357181250980758417946065562959336741494385171121884505153666613698200701027964241470205052483436703073249282905707369474114879163890641116586510234499486207571508356355504350175184060322719600033257294318603312231488085464447151372179900364608526075305595132882279601171060946607034196066306997436438751036706837382227585771913931930718050200338581104030819287316013221281009355527480531227382281806575029042929237511863736211285693300668580204212865086408408424533105380355832502370672304096803948591259319891256887399890346576938795857247732827620245174415622424512789510073898487376705555057838264735048995499398758615222804075659611661514999020151793476738577965492812153100933358829319889104384726627057374922099849235201580507398345213801767955515763976587830173724221813595095656123941737541308010844562222296682723124616644878048369904630780583474030523016622010231721780485858154369925182714968249116932552327547454952429633720591481311201176649275462370561979382794922486041964157445761461739136982100372269404377787564253195938134783602448099909969643767065958801912291144001208326537486160756154603499691195781074993906266595438486687552869875346162917097482031859167051177444876880450573327361805330870896106742571543470898052663676194616978800434561206021079705500314463777901222456287819498590123859025549832322760829418373013683747708643687633742848210695095168425114072080017183168661422002060581520069567513947189079028827099003397715833738087568756085853212643370854057258786576894490947880445896467854021921586318678705840858541269236967111288191464942302598780086921526039909630988728893506466638760365347792991218102938171851566943380338190376168966984681934259720408470265649444812419400725067901075584104190046140320107777302894810629022433429388926975183053907955603857644715178669147292202587548909990806813279996072704823792121608499016066686195557074834803331722975533001291666843868512454916458397075353289281139256805211742037719868255744306738406026715024805035153125121049750455462485687414581718785783210639189487633266745998813015979159850453599778185565536565698464749671194160919941250862337761874050298546096691647842758740827582874005191129613516278110667071321471693664820958666589682325616330367309490813603008647460142880670561213238334195764651618855856492988206287247979703083867745161374268119000031079229279977876431006188823851298808903954304696702747923172480004845843958829343625728644937411754671220680517704301663194821777208717721788024967544156359128872361322083773457342631423157617030489743275820665217602965793786178434372729838874830643889301280866456111712165658176243085652958263975136054687468037943196669316429849701612094075550546083462880018873447865106646009172217196937061734734995156928877162857262720383390575973743250570089189537406573014574950041418877682384430776053210312356867966147682596262607896366115881775796976217057647105670334784461899668065950525746877628945060367492610311697132856015553572952831341855616647947328270090240166119308013217459518268456777576071594065606714453191534327004510346332194168732909793154013398797652983050535749996162527215200095372546287421243171914660941772112055907154786037826253658635211125605684954171662932443912293638683154107361381795441363311233909417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Time: CPU 2.87 s, Wall: 3.28 s }}}

We can also copy this matrix over to Maple and compute the same determinant there...

{{{id=167| a[0,0] /// 5437434447110237200823494268835742396 }}} {{{id=27| maple.with_package('LinearAlgebra') B = maple(a) t = maple.cputime() time c = B.Determinant() # new fast Determinant in maple (don't use "det"!) maple.cputime(t) /// 21.969999999999999 }}} {{{id=168| c == d /// True }}} {{{id=169| R. = GF(389)[] /// }}} {{{id=170| f = (x+y+z+1)^20 time g = f*(f+1) /// Time: CPU 0.11 s, Wall: 0.11 s }}} {{{id=26| len(str(g)) /// 216557 }}}

We can copy to Magma also.  This ability to copy objects around is unique to Sage.

{{{id=82| B = magma(a) t = magma.cputime() time e = B.Determinant() magma.cputime(t) /// Time: CPU 0.00 s, Wall: 11.36 s 10.69 }}} {{{id=112| /// }}}

IRC a few days ago...

22:41 < linuxgus> I can show people at work tomorrow 
that they don't have to abandon Matlab [if they switch
to Sage].  Can Sage convert Matlab data types to 
its own?  That would be awesome!
{{{id=25| /// }}}

Example: A Symbolic Expression

{{{id=50| x = var('x') f(x) = sin(3*x)*x+log(x) + 1/(x+1)^2 show(f) ///
x \ {\mapsto}\ {x \sin \left( {3 x} \right)} + \log \left( x \right) + \frac{1}{{\left( x + 1 \right)}^{2} }
}}}

Plotting functions has similar syntax to Mathematica:

{{{id=171| show(f.integrate(x)) ///
x \ {\mapsto}\ \frac{\sin \left( {3 x} \right) - {{3 x} \cos \left( {3 x} \right)}}{9} + {x \log \left( x \right)} - \frac{1}{x + 1} - x
}}} {{{id=109| plot(f,(0.01,2), thickness=4) + text("Mathematica-style plotting in Sage", (1,-2), rgbcolor='black') /// }}}


Sage also has 2d plotting that is almost identical to MATLAB:

{{{id=159| import pylab as p p.figure() t = p.arange(0.01, 2.0, 0.01) s = p.sin(2 * p.pi * t) s = p.array([float(f(x)) for x in t]) P = p.plot(t, s, linewidth=4) p.xlabel('time (s)'); p.ylabel('voltage (mV)') p.title('Matlab-style plotting in Sage') p.grid(True) p.savefig('sage.png') /// }}} {{{id=110| /// }}} {{{id=73| /// }}}


_fast_float_ yields super-fast evaluation of Sage symbolic expressions -- e.g., here it is 10 times faster than native Python!

{{{id=105| var('x') show(f) ///
x \ {\mapsto}\ {x \sin \left( {3 x} \right)} + \log \left( x \right) + \frac{1}{{\left( x + 1 \right)}^{2} }
}}} {{{id=172| f(x,y,z) = x^3 * sin(x^2) + cos(x*y) - 1/(x+y+z) /// }}} {{{id=74| g = f._fast_float_(x,y,z) timeit('g(4.5r,3.2r,5.7r)') /// 625 loops, best of 3: 715 ns per loop }}} {{{id=75| %python # %python, so no preparsing so uses pure python import math def g(x): return math.sin(3*x)*x + log(x) + 1/(1+x)**2 /// }}} {{{id=77| timeit('g(4.5r)') /// 625 loops, best of 3: 6.88 µs per loop }}} {{{id=48| /// }}} {{{id=76| /// }}}

Example: Compare Answers with Maple

{{{id=83| var('x') f = sin(3*x)*x+log(x) + 1/(x+1)^2 show(integrate(f)) ///
\frac{\sin \left( {3 x} \right) - {{3 x} \cos \left( {3 x} \right)}}{9} + {x \log \left( x \right)} - \frac{1}{x + 1} - x
}}}

The command maple(...) fires up Maple (if you have it!), and creates a reference to a live object:

{{{id=0| m = maple(f); m /// sin(3*x)*x+ln(x)+1/(x+1)^2 }}} {{{id=89| type(m) /// }}} {{{id=90| m.parent() /// Maple }}} {{{id=91| m.parent().pid() /// 24038 }}} {{{id=92| os.system('ps ax |grep 24038') /// 24038 s007 Ss+ 0:00.01 /bin/sh /Users/wstein/bin/maple -t 24233 s015 S+ 0:00.00 sh -c ps ax |grep 24038 24235 s015 R+ 0:00.00 grep 24038 0 }}}

Use Maple objects via a Pythonic notation:

{{{id=88| show(m.integrate('x')) ///
1/9\,\sin \left( 3\,x \right) -1/3\,\cos \left( 3\,x \right) x+\ln \left( x \right) x-x- \left( x+1 \right) ^{-1}
}}} {{{id=42| mathematica(f).Integrate(x) /// -x - (1 + x)^(-1) - (x*Cos[3*x])/3 + x*Log[x] + Sin[3*x]/9 }}} {{{id=43| /// }}}

Example: Interactive Image Compression

This illustrates pylab (matplotlib + numpy), Sage plotting, html output, and @interact.

{{{id=97| # first just play import pylab A = pylab.imread(DATA + 'seattle.png') graphics_array([matrix_plot(A^(-1)), matrix_plot(1-A[0:,0:,2])]).show(figsize=[10,4]) /// }}} {{{id=174| A[0,0,] /// array([ 0.37254903, 0.42745098, 0.80784315, 1. ], dtype=float32) }}} {{{id=44| import pylab A_image = pylab.mean(pylab.imread(DATA + 'seattle.png'), 2) @interact def svd_image(i=(20,(1..100)), display_axes=True, m=matrix(ZZ,2,2)): 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('

Compressed using %s eigenvalues

'%i) /// }}} {{{id=94| /// }}} {{{id=55| /// }}}

Example: 3d Plots

{{{id=84| var('x,y') plot3d(sin(x*y) - cos(x^5), (x,-2,2), (y,-2,2)) /// }}} {{{id=124| show(sphere((0,0,0)) + sphere((0,1,0), color='red', opacity=0.5) + icosahedron((1,1,0), color='orange'), figsize=10) /// }}} {{{id=116| L = [] @interact def random_list(number_of_points=(10..50), dots=True): n = normalvariate global L if len(L) != number_of_points: L = [(n(0,1), n(0,1), n(0,1)) for i in range(number_of_points)] G = list_plot3d(L,interpolation_type='nn', texture=Color('#ff7500'),num_points=120) if dots: G += point3d(L) G.show() /// }}} {{{id=54| /// }}}

3d plotting (using jmol) is fast even though it does not use Java3d or OpenGL or require any special signed code or drivers. 

{{{id=53| # Yoda! -- over 50,000 triangles. from scipy import io X = io.loadmat(DATA + 'yodapose.mat') 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='green') + IndexFaceSet(F4,V,color='green') Y = Y.rotateX(-1) Y.show(aspect_ratio=[1,1,1], frame=False, figsize=4) html('"Use the source, Luke..."') /// "Use the source, Luke..." }}} {{{id=145| /// }}}

Real World Example: Super Fast Code (using Cython)

 

to	sage-support 
date	Sat, Jan 31, 2009 at 11:15 AM
Hi,

I received first a MemoryError, and later on Sage reported:

uitkomst1=[]
uitkomst2=[]
eind=int((10^9+2)/(2*sqrt(3)))
print eind
for y in srange(1,eind):
 test1=is_square(3*y^2+1,True)
 test2=is_square(48*y^2+1,True)
 if test1[0] and test1[1]%3==2: uitkomst1.append((y,(2*test1[1]-1)/3))
 if test2[0] and test2[1]%3==1: uitkomst2.append((y,(2*test2[1]+1)/3))
print uitkomst1
een=sum([3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9])
print uitkomst2
twee=sum([3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9])
print een+twee

If you replace 10^9 with 10^6, the above listing works properly.

Maybe I made a mistake?

Rolandb

I rewrote Roland's code slightly so it wouldn't waste memory by constructing big lists... but the result was slow.

{{{id=150| def f_python(n): uitkomst1=[] uitkomst2=[] eind=int((n+2)/(2*sqrt(3))) print eind for y in (1..eind): test1=is_square(3*y^2+1,True) test2=is_square(48*y^2+1,True) if test1[0] and test1[1]%3==2: uitkomst1.append((y,(2*test1[1]-1)/3)) if test2[0] and test2[1]%3==1: uitkomst2.append((y,(2*test2[1]+1)/3)) print uitkomst1 een=sum(3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9) print uitkomst2 twee=sum(3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9) print een+twee /// }}} {{{id=7| time f_python(10^5) /// 28868 [(1, 1), (15, 17), (209, 241), (2911, 3361)] [(1, 5), (14, 65), (195, 901), (2716, 12545)] 51408 Time: CPU 0.74 s, Wall: 0.86 s }}} {{{id=8| time f_python(10^6) /// 288675 [(1, 1), (15, 17), (209, 241), (2911, 3361), (40545, 46817)] [(1, 5), (14, 65), (195, 901), (2716, 12545), (37829, 174725)] 716034 Time: CPU 7.14 s, Wall: 7.65 s }}}

While waiting to see if f_python(10^6) would finish, I decided to try the Cython compiler.  I declared a few data types, put %cython at the top of the cell, and wham, it got over 200 times faster.

{{{id=1| %cython from sage.all import is_square cdef extern from "math.h": long double sqrtl(long double) def f(n): uitkomst1=[] uitkomst2=[] cdef long long eind=int((n+2)/(2*sqrt(3))) cdef long long y, t print eind for y in range(1,eind): t = sqrtl( (3*y*y + 1)) if t * t == 3*y*y + 1: uitkomst1.append((y, (2*t-1)/3)) t = sqrtl( (48*y*y + 1)) if t * t == 48*y*y + 1: uitkomst2.append((y, (2*t+1)/3)) print uitkomst1 een=sum([3*x-1 for (y,x) in uitkomst1 if 3*x-1<10^9]) print uitkomst2 twee=sum([3*x+1 for (y,x) in uitkomst2 if 3*x+1<10^9]) print een+twee /// }}} {{{id=45| time f(10^5) /// 28868 [(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L)] [(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L)] 2 Time: CPU 0.00 s, Wall: 0.00 s }}} {{{id=4| time f(10^6) /// 288675 [(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L), (40545L, 46817L), (151316L, 174724L)] [(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L), (37829L, 174725L)] 2 Time: CPU 0.03 s, Wall: 0.03 s }}} {{{id=5| time f(10^9) /// 288675135 [(1L, 1L), (4L, 4L), (15L, 17L), (56L, 64L), (209L, 241L), (780L, 900L), (2911L, 3361L), (10864L, 12544L), (40545L, 46817L), (151316L, 174724L), (564719L, 652081L), (2107560L, 2433600L), (7865521L, 9082321L), (29354524L, 33895684L), (109552575L, 126500417L)] [(1L, 5L), (14L, 65L), (195L, 901L), (2716L, 12545L), (37829L, 174725L), (526890L, 2433601L), (7338631L, 33895685L), (102213944L, 472105985L)] 2 Time: CPU 25.60 s, Wall: 26.50 s }}} {{{id=6| 7.14/0.03 /// 238.000000000000 }}}

This is not a contrived example.  This is a real world example that came up a this weekend.  For C-style computations, Sage (via Cython) is as fast as C.

{{{id=151| /// }}} {{{id=148| /// }}} {{{id=149| /// }}} {{{id=144| /// }}}

Example: Do some number theory

{{{id=128| /// }}}

Plotting an elliptic curve over a finite field

{{{id=127| p = 389 E = EllipticCurve(GF(p).random_element()) E.plot() /// }}}

and finding its group structure...

{{{id=137| E.abelian_group() /// (Multiplicative Abelian Group isomorphic to C180 x C2, ((262 : 190 : 1), (228 : 0 : 1))) }}}

Creating another random elliptic curve over a bigger finite field and compute its cardinality in seconds:

{{{id=129| p = next_prime(10^60) E = EllipticCurve(GF(p).random_element()) show(E) ///
y^2 = x^3 + 425356051223648747023108902736172620897278673931104973974164x + 920733503445411966314417165378315096439996780491982901461664
}}} {{{id=130| time E.cardinality() /// 1000000000000000000000000000000753054185359723724596110573310 Time: CPU 0.00 s, Wall: 5.66 s }}}

Some weight 3 modular forms on $\Gamma_1(12)$:

{{{id=138| show(ModularForms(Gamma1(12),3,prec=20).basis()) ///
\begin{array}{l}[q - 6q^{4} - 3q^{5} + 6q^{6} + 5q^{7} + 12q^{8} - 9q^{9} - 6q^{11} - 6q^{12} + 14q^{13} - 24q^{14} + 3q^{15} + 15q^{17} - 31q^{19} + O(q^{20}),\\ q^{2} - 4q^{4} - q^{5} + 3q^{6} + 5q^{7} + 4q^{8} - 6q^{9} - 2q^{10} - 6q^{11} + 12q^{13} - 12q^{14} + 3q^{15} + 8q^{16} + 5q^{17} - 3q^{18} - 31q^{19} + O(q^{20}),\\ q^{3} - 2q^{4} - q^{5} + 2q^{6} + q^{7} + 4q^{8} - 6q^{9} - 2q^{11} - 2q^{12} + 12q^{13} - 8q^{14} + q^{15} + 5q^{17} - 19q^{19} + O(q^{20}),\\ 1 + 528q^{10} - 960q^{11} + 572q^{12} - 1176q^{14} + 3200q^{15} - 420q^{16} - 4224q^{17} - 216q^{18} + 2496q^{19} + O(q^{20}),\\ q + \frac{32429}{72}q^{10} - \frac{7945}{9}q^{11} + \frac{96131}{216}q^{12} + 170q^{13} - \frac{70211}{72}q^{14} + \frac{67282}{27}q^{15} - \frac{23447}{72}q^{16} - \frac{30008}{9}q^{17} - \frac{637}{4}q^{18} + \frac{17615}{9}q^{19} + O(q^{20}),\\ q^{2} + \frac{2458}{9}q^{10} - \frac{4960}{9}q^{11} + \frac{8722}{27}q^{12} - \frac{5626}{9}q^{14} + \frac{49600}{27}q^{15} - \frac{2305}{9}q^{16} - \frac{21824}{9}q^{17} - 133q^{18} + \frac{12896}{9}q^{19} + O(q^{20}),\\ q^{3} + \frac{373}{3}q^{10} - \frac{760}{3}q^{11} + \frac{1435}{9}q^{12} - \frac{931}{3}q^{14} + \frac{7834}{9}q^{15} - \frac{355}{3}q^{16} - \frac{3344}{3}q^{17} - 70q^{18} + \frac{1976}{3}q^{19} + O(q^{20}),\\ q^{4} + \frac{88}{3}q^{10} - \frac{160}{3}q^{11} + \frac{331}{9}q^{12} - \frac{196}{3}q^{14} + \frac{1600}{9}q^{15} - \frac{31}{3}q^{16} - \frac{704}{3}q^{17} - 12q^{18} + \frac{416}{3}q^{19} + O(q^{20}),\\ q^{5} - \frac{1027}{72}q^{10} + \frac{350}{9}q^{11} - \frac{5005}{216}q^{12} + \frac{2989}{72}q^{14} - \frac{3032}{27}q^{15} + \frac{1225}{72}q^{16} + \frac{1450}{9}q^{17} + \frac{35}{4}q^{18} - \frac{793}{9}q^{19} + O(q^{20}),\\ q^{6} - \frac{70}{3}q^{10} + \frac{160}{3}q^{11} - \frac{286}{9}q^{12} + \frac{196}{3}q^{14} - \frac{1600}{9}q^{15} + \frac{79}{3}q^{16} + \frac{704}{3}q^{17} + 22q^{18} - \frac{416}{3}q^{19} + O(q^{20}),\\ q^{7} - \frac{1315}{72}q^{10} + \frac{350}{9}q^{11} - \frac{5005}{216}q^{12} + \frac{3277}{72}q^{14} - \frac{3275}{27}q^{15} + \frac{1225}{72}q^{16} + \frac{1441}{9}q^{17} + \frac{35}{4}q^{18} - \frac{784}{9}q^{19} + O(q^{20}),\\ q^{8} - \frac{88}{9}q^{10} + \frac{160}{9}q^{11} - \frac{250}{27}q^{12} + \frac{196}{9}q^{14} - \frac{1600}{27}q^{15} + \frac{70}{9}q^{16} + \frac{704}{9}q^{17} + 4q^{18} - \frac{416}{9}q^{19} + O(q^{20}),\\ q^{9} - \frac{31}{8}q^{10} + 5q^{11} - \frac{49}{24}q^{12} + \frac{49}{8}q^{14} - \frac{50}{3}q^{15} + \frac{13}{8}q^{16} + 22q^{17} + \frac{7}{4}q^{18} - 13q^{19} + O(q^{20})]\end{array}
}}} {{{id=133| /// }}} {{{id=61| /// }}}

Questions?


{{{id=175| a = random_matrix(RDF,500) /// }}} {{{id=176| matrix_plot(a) /// }}} {{{id=177| time v = a.eigenvalues() /// Time: CPU 0.77 s, Wall: 0.87 s }}} {{{id=60| import scipy.special /// }}} {{{id=179| import scipy.optimize /// }}} {{{id=180| /// }}} {{{id=59| /// }}} {{{id=52| /// }}} {{{id=107| /// }}} {{{id=106| /// }}} {{{id=108| /// }}}