Sage Days 14 Talk
system:sage

<hr />
<h2 style="text-align: center;"><span style="color: #333399;"><strong><br /><br /></strong></span></h2>
<h1 style="text-align: center;"><span style="font-size: xx-large;"><span style="color: #000000;"><strong>Sage </strong></span></span></h1>
<h1 style="text-align: center;"><span style="color: #800000;"><strong>Creating a viable free open source alternative to <br />Magma, Maple, Mathematica and Matlab</strong></span></h1>
<address style="text-align: center;"><span style="color: #800000;"><strong><span style="color: #000000;">William Stein, Associate Professor, University of Washington</span><br /></strong></span></address>
<p><span style="color: #800000;"><strong><br /></strong></span></p>
<p style="text-align: center;"><img src="sage_banner.png" alt="" width="637" height="491" /></p>
<p>&nbsp;</p>

{{{id=215|

///
}}}

{{{id=2|

///
}}}

<p>&nbsp;</p>
<h1 style="text-align: center;"><span style="color: #800000;"><strong>History</strong></span></h1>
<p><span style="background-color: #ffffff;"><span style="font-size: x-large;"> 
<ul>
<li> <em>I started Sage</em> at <span style="color: #800000;">Harvard</span> in <strong><em>January 2005</em></strong>. </li>
<br />
<li> No existing math software <strong><em>good enough</em></strong>.</li>
<br />
<li>I got <em><strong>very annoyed</strong></em> that my students and colleagues had to pay a <em><strong>ridiculous amount</strong></em> to use the code I wrote in Magma for <em><strong>Arithmetic Geometry (modular forms, elliptic curves, etc.)</strong></em>.</li>
<br />
<li> Sage-1.0 released <strong><em>February 2006</em></strong> at Sage Days 1 (San Diego). </li>
<br />
<li> <strong><em>Sage Days Workshops</em></strong> 1, 2, ..., 13, at UCLA, UW, Cambridge, Bristol, Austin, France, San Diego, Seattle, Georgia, etc. </li>
<br />
<li>Sage <strong><em>won first prize</em></strong> in Trophees du Libre (November 2007)</li>
<br />
<li> Funding from <em><strong><span style="background-color: #ffffff;"><span style="font-size: x-large;">Microsoft, </span></span>UW, NSF, DoD, Google, Sun,</strong></em> private donations, etc. </li>
</ul>
</span></span></p>
<p style="text-align: center;">
<table style="height: 223px;" border="0" width="850" align="center">
<tbody>
<tr>
<td><img src="sun_logo.png" alt="" width="245" height="110" /> <img src="dod.png" alt="" width="110" height="110" /><img src="nsf.png" alt="" width="110" height="110" /><img src="uw.png" alt="" width="110" height="110" /><img src="msr.png" alt="" width="141" height="110" /></td>
</tr>
</tbody>
</table>
</p>
<p>&nbsp;</p>

{{{id=117|

///
}}}

{{{id=214|

///
}}}

<p style="text-align: center;">&nbsp;</p>
<hr style="text-align: center;" />
<h2 style="text-align: center;"><span style="background-color: #ffffff;"><span style="color: #800000;">sagemath.org</span></span><br /></h2>
<p>&nbsp;</p>
<p style="text-align: center;"><img src="sage_webpage.png" alt="" width="749" height="722" /></p>
<p style="text-align: center;"><span style="font-size: large;">Each day about 1500-2000 different real human visitors each day.</span></p>
<p style="text-align: center;">&nbsp;</p>
<p style="text-align: center;">&nbsp;</p>
<hr />
<p style="text-align: center;">&nbsp;</p>
<p style="text-align: center;"><span style="font-size: large;"><br /></span></p>
<p style="text-align: center;"><img src="screms.jpg" alt="" width="461" height="613" /></p>
<p style="text-align: center;"><em><strong><br /></strong></em></p>
<p style="text-align: center;"><span style="font-size: large;"><span style="font-family: trebuchet ms,geneva;"><em><strong>Sage Devel Headquarters: </strong></em>Four 24-core Sun X4450's with 128GB RAM each + 1 Sun X4540 with 24TB disk<br /></span></span></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<hr />
<p style="text-align: center;">&nbsp;</p>
<h2 style="text-align: center;"><strong><span style="color: #800000;">sagenb.org</span></strong></h2>
<p style="text-align: center;"><img src="sagenb.png" alt="" width="968" height="563" /></p>

{{{id=62|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Some Advantages of Sage...<br /></span></h1>
<ul>
</ul>
<ol>
<li><span style="background-color: #ffffff;"><span style="font-size: x-large;">Python-- <em><strong>a general purpose</strong></em> language at core of Sage; huge </span><span style="font-size: x-large;">user base compared to Matlab, Mathematica, Maple and Magma<br /></span></span></li>
<br />
<li><span style="background-color: #ffffff;"><span style="font-size: x-large;">Cython -- <em><strong>write fast</strong></em> <em><strong>compiled</strong></em> code in Sage<br /></span></span></li>
<br />
<li><span style="font-size: x-large;"><span style="background-color: #ffcc00;"><span style="background-color: #ffffff;"><span style="background-color: #ffffff;"><em><strong>Free</strong></em> and <em><strong>Open source</strong></em></span></span></span></span></li>
<br />
<li><span style="font-size: x-large;"><span style="background-color: #ffcc00;"><span style="background-color: #ffffff;"><span style="background-color: #ffffff;"><em><strong>Peer review</strong></em> of Code: "<em>I do really and truly believe that the Sage refereeing model results in better code.</em>" -- John Cremona<br /></span></span></span></span></li>
</ol> 
<ul>
</ul>

{{{id=119|

///
}}}

{{{id=118|

///
}}}

<p>&nbsp;</p>
<hr />
<h1 style="text-align: center;"><span style="color: #800000;">Sage Is...</span><br /></h1>
<p><span style="background-color: #ffffff;"><span style="font-size: x-large;"> 
<ul>
<li>Over <strong><em>300,000 lines</em></strong> of new Python/Cython code</li>
<br />
<li><em><strong>Distribution</strong></em> of mathematical software; easy to build from source (over 5 million lines of code).</li>
<br />
<li>About <em><strong>150 developers</strong></em>: <a href="http://sagemath.org/development-map.html" target="_new">developer map</a></li>
<br />
<li><em><strong>Interfaces</strong></em> to most math software, including Magma, Macaulay 2, Singular, PHCpack, Polymake. </li>
<br />
<li>Exact and numerical<strong> </strong><strong><em>linear algebra</em>,</strong> optimization, <em><strong>statistics</strong></em> (numpy, scipy, R, and gsl all included), group theory, combinatorics, cryptography. </li>
<strong><br /></strong>
<li><strong> Number Theory:</strong> elliptic curves, modular forms, <em>L</em>-functions -- this is <em>my research area</em></li>
<br />
<li> 2d and 3d <em><strong>plotting</strong> </em></li>
</ul>
</span></span></p>
<hr />

{{{id=96|

///
}}}

<p>&nbsp;</p>
<h1 style="text-align: center;"><span style="color: #800000;">Examples: Calculus</span><br /></h1>
<p><strong>Symbolic expressions:</strong></p>

{{{id=23|
x, y = var('x,y')
type(x)
///

<class 'sage.calculus.calculus.SymbolicVariable'>
}}}

{{{id=197|
reset('e')
///
}}}

{{{id=19|
a = 1 + sqrt(7)^2 + pi^e + 2/3 + x^y
///
}}}

{{{id=163|
show(a)
///

<html><div class="math">{x}^{y}  + {\pi}^{e}  + \frac{26}{3}</div></html>
}}}

{{{id=253|
show(expand(a^2))
///

<html><div class="math">{x}^{{2 y}}  + {{2 {\pi}^{e} } {x}^{y} } + \frac{{52 {x}^{y} }}{3} + {\pi}^{{2 e}}  + \frac{{52 {\pi}^{e} }}{3} + \frac{676}{9}</div></html>
}}}

{{{id=254|
show(integrate(sin(x^2)))
///

<html><div class="math">\frac{{\sqrt{ \pi } \left( {\left( {\sqrt{ 2 } i} + \sqrt{ 2 } \right) \text{erf} \left( \frac{{\left( {\sqrt{ 2 } i} + \sqrt{ 2 } \right) x}}{2} \right)} + {\left( {\sqrt{ 2 } i} - \sqrt{ 2 } \right) \text{erf} \left( \frac{{\left( {\sqrt{ 2 } i} - \sqrt{ 2 } \right) x}}{2} \right)} \right)}}{8}</div></html>
}}}

{{{id=252|

///
}}}

<p><strong><br /></strong></p>

<h1 style="text-align: center;"><span style="color: #800000;">Example: Some Rings<br /></span></h1>

{{{id=207|
QQ[sqrt(2)]
///
Number Field in sqrt2 with defining polynomial x^2 - 2
}}}

{{{id=201|
RDF
///
Real Double Field
}}}

{{{id=202|
R = RealIntervalField(100); R
///
Real Interval Field with 100 bits of precision
}}}

{{{id=203|
R((1.01030,1.010332))
///
1.0103?
}}}

{{{id=164|
A = random_matrix(QQ, 500); v = random_matrix(QQ,500,1)
time x = A \ v
///
Time: CPU 1.56 s, Wall: 1.54 s
}}}

{{{id=3|
len(str(x[0]))
///
1484
}}}

{{{id=32|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: A Huge Integer Determinant<br /></span></h1>

{{{id=31|
a = random_matrix(ZZ,200,200,x=-2^127,y=2^127)
time d = a.determinant()
len(str(d))
///

Time: CPU 3.26 s, Wall: 3.33 s
7785
}}}

<p><span style="font-size: large;">We can also <strong><em>copy this matrix</em></strong><em> </em>over to Maple and compute the same determinant there...</span></p>

{{{id=167|
a[0,0]
///

-57511480170851074417469190457182575086
}}}

{{{id=27|
maple.with_package('LinearAlgebra')
B = maple(a)
t = maple.cputime()
time c = B.Determinant()
maple.cputime(t)
///

Time: CPU 0.00 s, Wall: 23.42 s
21.962
}}}

{{{id=168|
c == d
///

True
}}}

<p><span style="font-size: large;">This ability to easily move objects between math software is <strong><em>unique to Sage</em></strong>.</span></p>

{{{id=82|
B = magma(a)
t = magma.cputime()
time e = B.Determinant()
magma.cputime(t)
///

Time: CPU 0.00 s, Wall: 11.01 s
10.75
}}}

{{{id=208|
e == d
///

True
}}}

{{{id=261|
import scipy.stats
///
}}}

{{{id=262|

///
}}}

{{{id=260|
[ZZ.random_element(-5,5) for _ in [1..1000]]
///

[-3, -3, 3, -3, 2, 1, 4, -1, -2, -5, -4, -3, 3, -1, -1, 0, -1, 0, 2, -2, 2, 1, 4, 4, 3, 4, 0, -1, -2, -1, -2, 0, 1, 3, -4, 4, 0, -3, -3, 3, -4, -1, 1, 0, 3, 0, -1, -3, 2, 4, -5, -3, -5, -2, -3, -1, 4, 2, 0, -1, -4, 0, -4, -5, 0, 1, 2, 2, 1, 1, -5, 3, 1, -3, -1, 2, 0, -3, 4, 1, 3, 2, -1, -1, 2, -2, -2, 2, 2, 4, -5, 2, -1, -3, 2, 3, -5, 3, -4, -2, -2, -3, 0, -5, -3, 0, 4, 4, 0, -5, -3, -3, 3, -5, 4, -3, -5, 1, 4, -3, 1, -1, 2, 2, 2, -1, 3, 3, -4, 1, -5, 3, 0, -1, -1, -4, -3, 3, 1, -5, -5, -2, -2, 1, -4, -2, 4, -3, 1, -3, 1, 2, -3, -4, -4, 0, -5, -4, -1, 2, -2, 2, 2, 4, 3, -3, -5, 0, -1, 2, 1, -4, 1, 4, 4, -5, 4, 3, -1, -5, -3, -5, 2, 4, 3, -4, 1, -3, -3, -4, 3, -2, -4, 4, 2, 1, 0, -1, 1, 4, -5, -5, -2, -3, 1, -5, -5, -3, -2, 1, -1, 4, 4, -5, -4, -5, 4, -4, 1, 4, 4, -4, -5, 2, -1, -3, -1, 4, 0, 0, -4, -5, 0, 1, 0, -5, 4, 0, -1, 2, 1, -1, -5, -2, 2, -5, 1, 0, -4, 3, 0, -5, 4, 3, -2, 4, 4, 0, -3, 1, 1, 1, -2, 0, -3, -4, -3, -4, -5, 1, 0, 2, 3, 3, 3, -2, 3, 0, -2, 2, 3, 4, -5, 1, 3, -5, 1, -4, -2, 4, -3, 4, -1, 1, 1, 3, -2, 1, 4, 3, 2, 4, -5, -2, -3, 4, 1, -4, 3, 3, 4, -5, -4, 0, -4, -4, 1, 4, -2, 0, 0, -4, -4, 0, -5, -2, -3, -2, -1, -5, 2, -5, 4, -5, -4, 0, -3, 0, 3, 4, 0, -2, -2, 2, 4, -4, -3, 3, 2, 3, 0, -4, 1, -3, -1, -3, -4, -1, -5, 0, -3, 0, -5, 2, -3, -4, -3, 3, -1, -3, 0, 0, -4, -1, -2, -2, 2, -4, 1, -5, 1, -2, 0, 2, -5, 2, -5, -3, -3, -2, 2, 4, 3, 3, -4, 0, 2, -3, -1, 1, -3, 0, 1, 4, -2, 3, 2, 2, 4, -3, 2, 3, 0, 1, -3, -4, -3, 0, -4, 0, -3, -1, -3, -5, -2, 4, 4, 3, -3, 0, 3, 2, -2, -3, 2, -4, -2, -5, -5, -2, -3, -5, -4, -2, 0, 2, -5, 1, -3, -4, -4, -2, -5, 3, 1, 3, 2, 4, 3, 3, 2, 2, -5, 1, 4, -5, -1, 4, -2, -2, -3, 3, -3, 1, -5, 3, 0, 2, 1, -1, 3, 4, 4, -5, 3, -3, -3, 3, -2, -4, -2, 1, -3, -3, -4, 4, -4, 4, -1, -5, 0, 3, -5, 3, 3, -3, -2, -5, -3, -3, 3, 1, -4, -1, 0, -2, -4, -2, -2, 1, 1, -2, -1, -2, -3, -5, 4, 2, -2, 0, 3, 2, -5, -2, -4, 0, -3, -3, 4, 0, -3, -1, -4, -4, -1, -2, -2, 2, 2, -3, -1, -4, 2, 1, 2, 4, 3, -3, -2, 2, -5, -3, 4, 1, 4, 2, 4, 4, 4, 2, -2, 1, -2, -1, 3, -4, 3, -4, -2, -4, 1, -5, -2, -5, -5, 1, -4, 0, 4, -5, 0, -3, 4, -3, 0, -4, -2, -3, -1, -4, 0, 2, -4, -3, -3, -4, 1, 2, 3, 0, 2, -4, -1, 2, 2, 4, -1, -4, -2, -2, 1, 3, 1, -3, -4, 1, -2, 0, -5, 2, 3, -1, -3, -1, 2, 3, 1, -2, -4, -1, 0, -5, -4, 4, 4, 2, -2, -4, -5, 4, -3, -1, 4, -4, 4, 3, 1, 1, -4, 4, -1, 4, 1, -4, 4, -2, 3, 3, 2, -1, -1, 4, 4, 2, 0, -1, 2, -3, 2, 2, -2, -2, 3, 0, 2, 2, 0, 4, -3, 2, 4, -4, 3, 3, 0, 0, 1, -5, -1, -3, 0, 3, 2, -1, 3, -4, -3, -5, -4, -4, 4, -2, 3, 3, 0, -3, -4, 0, 2, -1, 0, 0, 4, 3, 4, 2, 2, -1, -2, -4, 4, 2, -3, -2, -5, -1, -4, 2, 2, 1, -2, -1, -3, 3, 0, -1, -1, 2, 4, 4, 3, 4, -1, -1, 3, 4, 0, 1, 3, 1, 0, -3, 1, -3, 1, 0, -1, -3, -3, 1, -4, -2, -1, 0, 3, 3, -5, 4, -1, -4, 0, 2, 0, -4, 1, -5, 1, -2, 4, 4, 3, -2, 1, 4, 2, 2, -4, 0, 2, -5, 3, 1, 2, -4, -3, 4, 2, 4, 4, 0, 4, 4, -5, -1, -4, -5, -2, 2, -1, -4, -4, 0, 1, 1, -2, 4, -3, 0, 4, 4, 2, -3, -3, -3, -5, 3, -4, -1, -2, -1, 0, -1, -1, -2, 1, 2, -2, -4, 3, 1, -5, 2, -3, 4, -4, -2, 0, 2, -4, 1, -4, 3, 3, -3, 2, -5, -1, 1, -2, -2, -3, 4, 4, 3, -4, -1, 0, 2, 4, 2, -2, 4, -5, -4, 4, -2, -3, -1, 3, 0, 3, -2, -3, -5, 2, -4, 4, 3, -3, -2, -2, -5, -4, 2, -5, -2, 4, -2, -5, 0, 2, 4, 1, -5, -5, 0, -1, 0, -1, 0, 2, 0, 4, -4, 2, 0, -3, -4, -3, -1, -2, 1, -1, 3, -1, 3, 1, 0, 0, 3, 3, 4, -4, -2, -4, 2, 3, 4, 4, 2, -4, -3, 2, -2, 4, -2, -3, -3, -2, 0, 0, -3, -3, 4, -3, -5, 0, -2, -5, -3, 4, 1, -5, 2, 3, -5, -3, -4, -2, -5, -3, 0, -2, -3, 4, 3, -5, 0, 1, -5, -1, -4, 2, -1, -2, -3, -3, -2, -2, -5]
}}}

{{{id=259|
QQ.random_element??
///
}}}

{{{id=25|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example:&nbsp; Plotting a Symbolic Expression</span><br /></h1>

{{{id=50|
x = var('x')
f(x) = sin(3*x)*x+log(x) + 1/(x+1)^2
show(f)
///
<html><div class="math">x \ {\mapsto}\ {x \sin \left( {3 x} \right)} + \log \left( x \right) + \frac{1}{{\left( x + 1 \right)}^{2} }</div></html>
}}}

<p><span style="color: #333300;"><strong>Plotting functions has similar syntax to <span style="background-color: #ffffff;">Mathematica:</span></strong></span></p>

{{{id=109|
plot(f,(0.01,2), thickness=4) + text("Mathematica-style plotting in Sage", (1,-2), rgbcolor='black')
///
}}}

<p><span style="font-size: large;"><span style="color: #333300;"><strong><br /></strong></span></span></p>
<p><span style="color: #333300;"><strong><span style="font-size: large;">NOTE: Sage also has 2d plotting that is almos<span style="background-color: #ffffff;">t</span><em><span style="background-color: #ffffff;"> identical to MATLAB's 2d plotting</span></em>:</span><br /></strong></span></p>

{{{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|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Fast Evaluation</span><br /></h1>

{{{id=172|
f(x,y,z) = sin(3*x)*x + log(x*y*z)^3 + 1/(1+y)^2
///
}}}

{{{id=263|
timeit('f(4.5r,3.2r,5.7r)')
///

625 loops, best of 3: 398 µs per loop
}}}

{{{id=74|
g = f._fast_float_(x,y,z)
timeit('g(4.5r,3.2r,5.7r)')
///

625 loops, best of 3: 726 ns per loop
}}}

{{{id=75|
%python   
import math
def g(x,y,z): return math.sin(3*x)*x + log(x*y*z)**3 + 1/(1+y)**2
///
}}}

{{{id=77|
timeit('g(4.5r,3.2r,5.7r)')
///

625 loops, best of 3: 7.18 µs per loop
}}}

{{{id=213|
7.15/.669
///
10.6875934230194
}}}

{{{id=48|

///
}}}

{{{id=76|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Interface with Maple and Mathematica<br /></span></h1>

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

<p>The command <span style="background-color: #ffffff;">maple(...) fires up Maple (if you have it!), and creates a reference to a live object:</span></p>

{{{id=0|
m = maple(f); m
///
sin(3*x)*x+ln(x)+1/(x+1)^2
}}}

{{{id=89|
type(m)
///
<class 'sage.interfaces.maple.MapleElement'>
}}}

{{{id=90|
m.parent()
///
Maple
}}}

{{{id=91|
m.parent().pid()
///
67798
}}}

{{{id=92|
os.system('ps ax |grep 24038')
///
67818 s001  S+     0:00.00 sh -c ps ax |grep 24038
67820 s001  R+     0:00.00 grep 24038
0
}}}

<p>Use Maple objects via a <strong><em><span style="background-color: #ffffff;">Pythonic notation</span></em></strong>:</p>

{{{id=88|
show(m.integrate('x'))
///
<html><div class="math">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}</div></html>
}}}

{{{id=42|
mathematica(f).Integrate(x)
///
-x - (1 + x)^(-1) - (x*Cos[3*x])/3 + x*Log[x] + Sin[3*x]/9
}}}

{{{id=43|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Interactive Image Compression</span></h1>
<p>This illustrates pylab (matplotlib + numpy), Sage plotting, html output, and @interact.</p>

{{{id=97|
# first just play
import pylab
A = pylab.imread(DATA + 'simons.png')
graphics_array([matrix_plot(A), matrix_plot(A[0:,0:,1]*A[0:,0:,1])]).show(figsize=[10,4])
///
}}}

{{{id=174|
A[0,0,]
///
array([ 0.03529412,  0.04705882,  0.01960784,  1.        ], dtype=float32)
}}}

{{{id=44|
import pylab
A_image = pylab.mean(pylab.imread(DATA + 'simons.png'), 2)
@interact
def svd_image(i=(20,(1..100)), j=[1..30], display_axes=True, m=matrix(QQ,2,2), color=Color('red'), f=sin(x^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('<h2>Compressed using %s eigenvalues</h2>'%i)
///

<html><!--notruncate--><div padding=6 id='div-interact-44'> <table width=800px height=20px bgcolor='#c5c5c5'
                 cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color="black">i&nbsp;</font></td><td><table><tr><td>
    	<div id='slider-i-44' class='ui-slider ui-slider-3' style='margin:0px;'><span class='ui-slider-handle'></span></div>
    	</td><td><font color='black' id='slider-i-44-lbl'></font></td></tr></table><script>(function(){ var values = ["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100"]; setTimeout(function() {
    $('#slider-i-44').slider({
    	stepping: 1, min: 0, max: 99, startValue: 19,
    	change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-i-44-lbl').text(values[position]); interact(44, "sage.server.notebook.interact.update(44, \"i\", 22, sage.server.notebook.interact.standard_b64decode(\""+encode64(position)+"\"), globals());sage.server.notebook.interact.recompute(44)"); },
    	slide: function(e,ui) { if(values!=null) $('#slider-i-44-lbl').text(values[ui.value]); }
    });
    if(values != null) $('#slider-i-44-lbl').text(values[$('#slider-i-44').slider('value')]);
    }, 1); })();</script></td></tr>
<tr><td align=right><font color="black">j&nbsp;</font></td><td><select onchange='interact(44, "sage.server.notebook.interact.update(44, \"j\", 23, sage.server.notebook.interact.standard_b64decode(\""+encode64(this.options[this.selectedIndex].value)+"\"), globals());sage.server.notebook.interact.recompute(44)");'><option value='0' selected>1</option>
<option value='1' >2</option>
<option value='2' >3</option>
<option value='3' >4</option>
<option value='4' >5</option>
<option value='5' >6</option>
<option value='6' >7</option>
<option value='7' >8</option>
<option value='8' >9</option>
<option value='9' >10</option>
<option value='10' >11</option>
<option value='11' >12</option>
<option value='12' >13</option>
<option value='13' >14</option>
<option value='14' >15</option>
<option value='15' >16</option>
<option value='16' >17</option>
<option value='17' >18</option>
<option value='18' >19</option>
<option value='19' >20</option>
<option value='20' >21</option>
<option value='21' >22</option>
<option value='22' >23</option>
<option value='23' >24</option>
<option value='24' >25</option>
<option value='25' >26</option>
<option value='26' >27</option>
<option value='27' >28</option>
<option value='28' >29</option>
<option value='29' >30</option>
</select></td></tr>
<tr><td align=right><font color="black">display_axes&nbsp;</font></td><td><input type='checkbox' checked width=200px onchange='interact(44, "sage.server.notebook.interact.update(44, \"display_axes\", 24, sage.server.notebook.interact.standard_b64decode(\""+encode64(this.checked)+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td></tr>
<tr><td align=right><font color="black">m&nbsp;</font></td><td><table><tr><td><input type='text' value='0' size='4' onchange='interact(44, "sage.server.notebook.interact.update(44, \"m\", 25, sage.server.notebook.interact.standard_b64decode(\""+encode64( "[["+jQuery(this).parents("table").eq(0).find("tr").map(function(){return jQuery(this).find("input").map(function() {return jQuery(this).val();}).get().join(",");}).get().join("],[")+"]]" )+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td><td><input type='text' value='0' size='4' onchange='interact(44, "sage.server.notebook.interact.update(44, \"m\", 25, sage.server.notebook.interact.standard_b64decode(\""+encode64( "[["+jQuery(this).parents("table").eq(0).find("tr").map(function(){return jQuery(this).find("input").map(function() {return jQuery(this).val();}).get().join(",");}).get().join("],[")+"]]" )+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td></tr><tr><td><input type='text' value='0' size='4' onchange='interact(44, "sage.server.notebook.interact.update(44, \"m\", 25, sage.server.notebook.interact.standard_b64decode(\""+encode64( "[["+jQuery(this).parents("table").eq(0).find("tr").map(function(){return jQuery(this).find("input").map(function() {return jQuery(this).val();}).get().join(",");}).get().join("],[")+"]]" )+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td><td><input type='text' value='0' size='4' onchange='interact(44, "sage.server.notebook.interact.update(44, \"m\", 25, sage.server.notebook.interact.standard_b64decode(\""+encode64( "[["+jQuery(this).parents("table").eq(0).find("tr").map(function(){return jQuery(this).find("input").map(function() {return jQuery(this).val();}).get().join(",");}).get().join("],[")+"]]" )+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td></tr></table></td></tr>
<tr><td align=right><font color="black">color&nbsp;</font></td><td><table><tr><td><div id='color-selector-color-44-picker'></div></td><td>
<input type='text' id='color-selector-color-44' name='color' onchange='interact(44, "sage.server.notebook.interact.update(44, \"color\", 26, sage.server.notebook.interact.standard_b64decode(\""+encode64(this.value)+"\"), globals());sage.server.notebook.interact.recompute(44)");$.farbtastic("#color-selector-color-44-picker").setColor(this.value);' value='#ff0000'/></td></tr></table><script>setTimeout(function() {
          $('#color-selector-color-44-picker').farbtastic('#color-selector-color-44');
          $.farbtastic('#color-selector-color-44-picker').linkTo(function(color) {
              var t = get_element('color-selector-color-44');
              if(color!=t.value) {
                  t.value = color;
                  t.style.backgroundColor = color; 
                  interact(44, "sage.server.notebook.interact.update(44, \"color\", 26, sage.server.notebook.interact.standard_b64decode(\""+encode64(color)+"\"), globals());sage.server.notebook.interact.recompute(44)");
              }
              return;
            })
       }, 1);</script></td></tr>
<tr><td align=right><font color="black">f&nbsp;</font></td><td><input type='text' value="sin(x^2)" size=80 onchange='interact(44, "sage.server.notebook.interact.update(44, \"f\", 27, sage.server.notebook.interact.standard_b64decode(\""+encode64(this.value)+"\"), globals());sage.server.notebook.interact.recompute(44)")'></input></td></tr>
</table><div id='cell-interact-44'><?__SAGE__START>
        <table border=0 bgcolor='#white' width=100% height=100%>
        <tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>
        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>
        </table><?__SAGE__END></div></td>
                 </tr></table></div>
                 </html>
}}}

{{{id=94|

///
}}}

{{{id=220|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Implicit 2d Plot</span></h1>

{{{id=223|
var("x y")
@interact
def foo(f = y^3 + x*y - x^2 - x, n=(10..500)):
    implicit_plot(f, (-2,3), (-3,3), plot_points=n, cmap='hsv').show(aspect_ratio=1)
///

<html><!--notruncate--><div padding=6 id='div-interact-223'> <table width=800px height=20px bgcolor='#c5c5c5'
                 cellpadding=15><tr><td bgcolor='#f9f9f9' valign=top align=left><table><tr><td align=right><font color="black">f&nbsp;</font></td><td><input type='text' value="y^3 + x*y - x^2 - x" size=80 onchange='interact(223, "sage.server.notebook.interact.update(223, \"f\", 35, sage.server.notebook.interact.standard_b64decode(\""+encode64(this.value)+"\"), globals());sage.server.notebook.interact.recompute(223)")'></input></td></tr>
<tr><td align=right><font color="black">n&nbsp;</font></td><td><table><tr><td>
    	<div id='slider-n-223' class='ui-slider ui-slider-3' style='margin:0px;'><span class='ui-slider-handle'></span></div>
    	</td><td><font color='black' id='slider-n-223-lbl'></font></td></tr></table><script>(function(){ var values = ["10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50","51","52","53","54","55","56","57","58","59","60","61","62","63","64","65","66","67","68","69","70","71","72","73","74","75","76","77","78","79","80","81","82","83","84","85","86","87","88","89","90","91","92","93","94","95","96","97","98","99","100","101","102","103","104","105","106","107","108","109","110","111","112","113","114","115","116","117","118","119","120","121","122","123","124","125","126","127","128","129","130","131","132","133","134","135","136","137","138","139","140","141","142","143","144","145","146","147","148","149","150","151","152","153","154","155","156","157","158","159","160","161","162","163","164","165","166","167","168","169","170","171","172","173","174","175","176","177","178","179","180","181","182","183","184","185","186","187","188","189","190","191","192","193","194","195","196","197","198","199","200","201","202","203","204","205","206","207","208","209","210","211","212","213","214","215","216","217","218","219","220","221","222","223","224","225","226","227","228","229","230","231","232","233","234","235","236","237","238","239","240","241","242","243","244","245","246","247","248","249","250","251","252","253","254","255","256","257","258","259","260","261","262","263","264","265","266","267","268","269","270","271","272","273","274","275","276","277","278","279","280","281","282","283","284","285","286","287","288","289","290","291","292","293","294","295","296","297","298","299","300","301","302","303","304","305","306","307","308","309","310","311","312","313","314","315","316","317","318","319","320","321","322","323","324","325","326","327","328","329","330","331","332","333","334","335","336","337","338","339","340","341","342","343","344","345","346","347","348","349","350","351","352","353","354","355","356","357","358","359","360","361","362","363","364","365","366","367","368","369","370","371","372","373","374","375","376","377","378","379","380","381","382","383","384","385","386","387","388","389","390","391","392","393","394","395","396","397","398","399","400","401","402","403","404","405","406","407","408","409","410","411","412","413","414","415","416","417","418","419","420","421","422","423","424","425","426","427","428","429","430","431","432","433","434","435","436","437","438","439","440","441","442","443","444","445","446","447","448","449","450","451","452","453","454","455","456","457","458","459","460","461","462","463","464","465","466","467","468","469","470","471","472","473","474","475","476","477","478","479","480","481","482","483","484","485","486","487","488","489","490","491","492","493","494","495","496","497","498","499","500"]; setTimeout(function() {
    $('#slider-n-223').slider({
    	stepping: 1, min: 0, max: 490, startValue: 0,
    	change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-n-223-lbl').text(values[position]); interact(223, "sage.server.notebook.interact.update(223, \"n\", 36, sage.server.notebook.interact.standard_b64decode(\""+encode64(position)+"\"), globals());sage.server.notebook.interact.recompute(223)"); },
    	slide: function(e,ui) { if(values!=null) $('#slider-n-223-lbl').text(values[ui.value]); }
    });
    if(values != null) $('#slider-n-223-lbl').text(values[$('#slider-n-223').slider('value')]);
    }, 1); })();</script></td></tr>
</table><div id='cell-interact-223'><?__SAGE__START>
        <table border=0 bgcolor='#white' width=100% height=100%>
        <tr><td bgcolor=white align=left valign=top><pre><?__SAGE__TEXT></pre></td></tr>
        <tr><td  align=left valign=top><?__SAGE__HTML></td></tr>
        </table><?__SAGE__END></div></td>
                 </tr></table></div>
                 </html>
}}}

{{{id=55|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: 3d Plots</span></h1>

{{{id=84|
var('x,y')
P = plot3d(sqrt(x^2-y^2+3), (x,-3,3), (y,-2,2), color='blue', opacity=0.7)
P + sphere((0,0,0),2,color='red', aspect_ratio=[1,1,1], opacity=0.6)
///
}}}

{{{id=124|
A = ( sphere((0,0,0), opacity=0.7) + sphere((0,1,0), color='red', opacity=0.5) 
        + icosahedron((1,1,0), color='green') )
        
A.show(viewer='tachyon')
///
}}}

{{{id=224|
# implicit_plot3d -- not yet released 
# (see http://trac.sagemath.org/sage_trac/ticket/5249)!!
var('x,y,z')
T = RDF(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 = 4.77
implicit_plot3d(p, (-r, r), (-r, r), (-r, r), plot_points=40) 
///
}}}

<p>3d plotting (using <a href="http://jmol.sourceforge.net/" target="_new">jmol</a>) is fast even though it does<strong> not </strong>use Java3d or OpenGL or require any special signed code or drivers. </p>

{{{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..."')
///

<html><font color='black'>"Use the source, Luke..."</font></html>
}}}

{{{id=145|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Write Fast Code (using Cython)</span></h1>
<p>&nbsp;</p>
<pre>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&lt;10^9])
print uitkomst2
twee=sum([3*x+1 for (y,x) in uitkomst2 if 3*x+1&lt;10^9])
print een+twee

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

Maybe I made a mistake?

Rolandb
</pre>
<p>I rewrote Roland's code slightly so it wouldn't waste memory by constructing big lists... but the result was slow.</p>

{{{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
}}}

<p>Rewrite using <em><strong>CYTHON</strong></em></p>

{{{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
    for y in range(1,eind):   
        t = <long long>sqrtl(<long long> (3*y*y + 1))
        if t * t == 3*y*y + 1:
            uitkomst1.append((y, (2*t-1)/3))
        t = <long long>sqrtl(<long long> (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=4|
time f(10^6)
///

[(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
}}}

<p>A speedup by a factor of 238!!</p>

{{{id=255|
7.14/0.03
///

238.000000000000
}}}

{{{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|

///
}}}

{{{id=242|

///
}}}

{{{id=243|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Multiply Polynomials</span></h1>

{{{id=245|
R.<x,y,z> = GF(997)[]; R
///

Multivariate Polynomial Ring in x, y, z over Finite Field of size 997
}}}

{{{id=247|
show((x+y+z+3)^2)
///

<html><div class="math">x^{2} + 2 x y + y^{2} + 2 x z + 2 y z + z^{2} + 6 x + 6 y + 6 z + 9</div></html>
}}}

{{{id=246|
f = (x+y+z+1)^20
time g = f*(f+1)
///

Time: CPU 0.11 s, Wall: 0.12 s
}}}

{{{id=248|
%magma
R<x,y,z> := PolynomialRing(GF(997),3);
f := (x+y+z+1)^20;
time g := f*(f+1);
///



Time: 0.280
}}}

{{{id=249|
mf = magma(f)
time g = mf*(mf+1)
///

Time: CPU 0.00 s, Wall: 0.30 s
}}}

{{{id=250|

///
}}}

{{{id=151|

///
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Compute a Groebner Bases</span></h1>

{{{id=228|
P.<a,b,c> = PolynomialRing(QQ,3, order='lex'); P
///

Multivariate Polynomial Ring in a, b, c over Rational Field
}}}

{{{id=258|
I = sage.rings.ideal.Katsura(P,3); show(I.gens())
///

<html><div class="math">\left(a + 2 b + 2 c - 1, 
 a^{2} - a + 2 b^{2} + 2 c^{2}, 
 2 a b + 2 b c - b\right)</div></html>
}}}

{{{id=227|
I.groebner_basis()
///

[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}

<p><strong>NOTE</strong>: Sage includes Singular and is very tightly integrated with it.</p>

{{{id=229|
sage.rings.ideal.Katsura(P,3).groebner_basis('singular:std')
///

[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}

{{{id=230|
sage.rings.ideal.Katsura(P,3).groebner_basis('singular:slimgb')
///

[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}

<p><strong>NOTE: </strong>You must have Macaulay 2 installed for this to work...</p>

{{{id=231|
sage.rings.ideal.Katsura(P,3).groebner_basis('macaulay2')
///

[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}

{{{id=232|
sage.rings.ideal.Katsura(P,3).groebner_basis('magma')
///

[a - 60*c^3 + 158/7*c^2 + 8/7*c - 1, b + 30*c^3 - 79/7*c^2 + 3/7*c, c^4 - 10/21*c^3 + 1/84*c^2 + 1/84*c]
}}}

<h1 style="text-align: center;"><span style="color: #800000;">Example: Compute a Groebner Fan</span></h1>

{{{id=133|
P.<a,b,c> = PolynomialRing(QQ,3, order='lex'); P
I = sage.rings.ideal.Katsura(P,3)
///
}}}

{{{id=61|
F = I.groebner_fan(); F
///

Groebner fan of the ideal:
Ideal (a + 2*b + 2*c - 1, a^2 - a + 2*b^2 + 2*c^2, 2*a*b + 2*b*c - b) of Multivariate Polynomial Ring in a, b, c over Rational Field
}}}

{{{id=233|
F.weight_vectors()
///

[(4, 4, 1), (3, 2, 1), (5, 2, 3), (2, 1, 3), (2, 3, 5), (2, 3, 1), (2, 5, 3), (1, 4, 4)]
}}}

{{{id=108|
F.render().show(axes=False)
///
}}}

{{{id=239|

///
}}}

{{{id=238|

///
}}}

{{{id=237|

///
}}}

{{{id=236|

///
}}}

{{{id=234|

///
}}}