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


 

BTW: I really, REALLY like sage.  I'm just surprised i haven't heard
of it before 5 days ago.

 -- Clinton Bowen on the sage-support mailing list Mon, Feb 9, 2009 at 11:03 PM.

 

{{{id=2| /// }}}

 

History

 

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Sun's Interest in Sage

  1. Sage will very soon (weeks) fully support Solaris.  This porting work was fully funded by the Department of Defense, and will continue.



  2. Sage is the only project whose goal is to create a viable open source alternative to all of Mathematica, Maple, Matlab and Magma.  It is the analogue of Star Office or Firefox, as alternatives to Microsoft Office or Internet Explorer.

 

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

Sage is 100% Free and Open Source

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

 

Each day about 1500-2000 different real human visitors each day.


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| /// }}}

Many Advantages of Sage over the Ma*'s

  1. Python-- a general purpose language at core of Sage; huge user base compared to Matlab, Mathematica, Maple and Magma

  2. Cython -- write blazingly fast compiled code in Sage

  3. Free and Open source

  4. 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| /// }}}

 

Sage Is...

{{{id=96| /// }}}

 

Examples

Symbolic expressions:

{{{id=23| x, y = var('x,y') type(x) /// }}} {{{id=19| a = 1 + sqrt(2) + pi + 2/3 + x^y /// }}} {{{id=163| show(a) ///
{x}^{y} + \pi + \sqrt{ 2 } + \frac{5}{3}
}}} {{{id=20| show(expand(a^2)) ///
{x}^{{2 y}} + {{2 \pi} {x}^{y} } + {{2 \sqrt{ 2 }} {x}^{y} } + \frac{{10 {x}^{y} }}{3} + {\pi}^{2} + {{2 \sqrt{ 2 }} \pi} + \frac{{10 \pi}}{3} + \frac{{10 \sqrt{ 2 }}}{3} + \frac{43}{9}
}}}

Solve equations

{{{id=101| var('a,b,c,x') show(solve(x^2 + sqrt(17)*a*x + b == 0, x)) ///
\begin{array}{l}[x = \frac{-\left( \sqrt{ {17 {a}^{2} } - {4 b} } \right) - {\sqrt{ 17 } a}}{2},\\ x = \frac{\sqrt{ {17 {a}^{2} } - {4 b} } - {\sqrt{ 17 } a}}{2}]\end{array}
}}} {{{id=181| var('a,b,c,x') show(solve(a*x^3 + b*x + c == 0, x)[0]) ///
x = {\left( \frac{{-\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) {\left( \frac{\sqrt{ \frac{{{27 a} {c}^{2} } + {4 {b}^{3} }}{a} }}{{{6 \sqrt{ 3 }} a}} - \frac{c}{{2 a}} \right)}^{\frac{1}{3}} } - \frac{{\left( \frac{{\sqrt{ 3 } i}}{2} - \frac{1}{2} \right) b}}{{{3 a} {\left( \frac{\sqrt{ \frac{{{27 a} {c}^{2} } + {4 {b}^{3} }}{a} }}{{{6 \sqrt{ 3 }} a}} - \frac{c}{{2 a}} \right)}^{\frac{1}{3}} }}
}}} {{{id=164| A = random_matrix(QQ, 500); v = random_matrix(QQ,500,1) time x = A \ v /// Time: CPU 1.69 s, Wall: 2.20 s }}} {{{id=3| len(str(x[0])) /// 1486 }}} {{{id=32| /// }}}

Example: A Huge Integer Determinant

{{{id=31| a = random_matrix(ZZ,200,x=-2^127,y=2^127) time d = a.determinant() len(str(d)) /// Time: CPU 3.14 s, Wall: 4.25 s 7786 }}}

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

{{{id=167| a[0,0] /// -10495856349494057617178227541504239545 }}} {{{id=27| maple.with_package('LinearAlgebra') B = maple(a) t = maple.cputime() time c = B.Determinant() maple.cputime(t) /// 21.969999999999999 }}} {{{id=168| c == d /// True }}}

This ability to easily move objects between math software 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=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: 709 ns per loop }}} {{{id=75| %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: 6.89 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): 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( 4*x*exp(-x^2-y^2), (x,-2,2), (y,-2,2) ) /// }}} {{{id=124| ( sphere((0,0,0), opacity=0.7) + sphere((0,1,0), color='red', opacity=0.5) + icosahedron((1,1,0), color='green') ) /// }}} {{{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=182| 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 0.72 s, Wall: 0.77 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 }}}

3d Plots                                                              

{{{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| /// }}}

Numerical Matrix Algebra

{{{id=184| a = random_matrix(RDF, 500) # entries in [-1,1] G = graphics_array([matrix_plot(a^i) for i in [-3,-2,-1,1,2,3]], 2, 3) show(G, axes=False) /// }}} {{{id=148| @interact def f(i=slider([-10..-1]+[1..10])): show(matrix_plot(a^i),axes=False) ///
i 
}}} {{{id=149| import matplotlib.cm; matplotlib.cm.datad.keys() /// ['Spectral', 'summer', 'RdBu', 'gist_earth', 'Set1', 'Set2', 'Set3', 'Dark2', 'hot', 'PuOr_r', 'PuBuGn_r', 'RdPu', 'gist_ncar_r', 'gist_yarg_r', 'Dark2_r', 'YlGnBu', 'RdYlBu', 'hot_r', 'gist_rainbow_r', 'gist_stern', 'cool_r', 'cool', 'gray', 'copper_r', 'Greens_r', 'GnBu', 'gist_ncar', 'spring_r', 'gist_rainbow', 'RdYlBu_r', 'gist_heat_r', 'OrRd_r', 'bone', 'gist_stern_r', 'RdYlGn', 'Pastel2_r', 'spring', 'Accent', 'YlOrRd_r', 'Set2_r', 'PuBu', 'RdGy_r', 'spectral', 'flag_r', 'jet_r', 'RdPu_r', 'gist_yarg', 'BuGn', 'Paired_r', 'hsv_r', 'YlOrRd', 'Greens', 'PRGn', 'gist_heat', 'spectral_r', 'Paired', 'hsv', 'Oranges_r', 'prism_r', 'Pastel2', 'Pastel1_r', 'Pastel1', 'gray_r', 'PuRd_r', 'Spectral_r', 'BuGn_r', 'YlGnBu_r', 'copper', 'gist_earth_r', 'Set3_r', 'OrRd', 'PuBu_r', 'winter_r', 'jet', 'bone_r', 'BuPu', 'Oranges', 'RdYlGn_r', 'PiYG', 'YlGn', 'binary_r', 'gist_gray_r', 'BuPu_r', 'gist_gray', 'flag', 'RdBu_r', 'BrBG', 'Reds', 'summer_r', 'GnBu_r', 'BrBG_r', 'Reds_r', 'RdGy', 'PuRd', 'Accent_r', 'Blues', 'Greys', 'autumn', 'PRGn_r', 'Greys_r', 'pink', 'binary', 'winter', 'pink_r', 'prism', 'YlOrBr', 'Purples_r', 'PiYG_r', 'YlGn_r', 'Blues_r', 'YlOrBr_r', 'Purples', 'autumn_r', 'Set1_r', 'PuOr', 'PuBuGn'] }}} {{{id=144| /// }}}

Cython: Sage's Compiler

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=128| /// }}}

Plotting an elliptic curve over a finite field

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

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

{{{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| /// }}}