/
2010-06-11-SageDays20.5-demo-Franco.html
372 lines (365 loc) · 24.5 KB
/
2010-06-11-SageDays20.5-demo-Franco.html
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
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
jsMath
SageThe Sage Notebook Searching for Sage server...
Version 4.4.3
* admin
* Toggle
* Home
* Published
* Log
* Settings
* Help
* Report a Problem
* Sign out
Sage Days 20.5: Demo
last edited on June 11, 2010 12:40 AM by admin
SaveSave & quitDiscard & quit
Typeset
PrintPrint Worksheet Edit Text Undo Share Publish
> >> Exit
<< <
View plain text
<h1 style="text-align: center;">Sage Days 20.5 : Demo</h1>
<p> </p>
<h2>The Sage notebook: an interface to your software!</h2>
sage: 2+3
<h2>Interface with Magma (you need to have Magma installed)</h2>
<p>We use Sage to construct a random $150\times150$ matrix over the integers with entries of size up to $2^{128}$.</p>
sage: a = random_matrix(ZZ,150,x=-2^128,y=2^128)
<p>A random entry of the matrix:</p>
sage: a[72,17]
<p>We use Magma to compute the determinant of the matrix:</p>
sage: b = magma(a)
sage: magma.eval('time e := Determinant(%s);'%b.name())
<p>We can also do this with Sage, and it's faster!</p>
sage: time d = a.determinant()
<p>And we get the same answer:</p>
sage: d == magma('e')
<p>Here's an example of Sage and Magma computing the <a title="Wikipedia: Hermite normal form" href="http://en.wikipedia.org/wiki/Hermite_normal_form" target="_blank">Hermite normal form</a> of a random matrix of size $100\times100$ over the integers. </p>
sage: a = random_matrix(ZZ,100,x=-2^64,y=2^64)
sage: time h = a.hermite_form()
sage: b = magma(a)
sage: magma.eval('time e := HermiteForm(%s);'%b.name())
sage: h == magma('e')
<h2>Interface with Maple (you need to have Maple installed)</h2>
<p>Below we use Maple and Sage to compute the number of <a title="Wikipedia: Partition (number theory)" href="http://en.wikipedia.org/wiki/Partition_(number_theory)" target="_blank">partitions</a> of $27020$.</p>
sage: time maple('combinat[numbpart](27020)')
sage: time number_of_partitions(27020)
<p>Not only is Sage faster, it also returns the correct answer: <a href="http://www.research.att.com/~njas/sequences/A110375">http://www.research.att.com/~njas/sequences/A110375</a>.</p>
<h2>Embed Java Applets into the Sage Notebook</h2>
sage: load DATA+'EuclideanK44Mech.sage'
sage: html(jsp_string)
<h2>Beautiful documentation</h2>
<p>Type <strong>plot(</strong> and then press the <strong>TAB</strong> key.</p>
sage: plot(
<p>Click on <a title="documentation" href="../../../help" target="_blank">Help</a> at the top of this page.</p>
<p>There is also <a title="live documentation" href="../../../doc/live/reference/index.html" target="_blank">Live Documentation</a> in which you can evaluate the examples.</p>
<h2>Interact with your mathematics by creating interactive widgets</h2>
<h2>interact: curves of pursuit</h2>
sage: npi = RDF(pi)
sage: def rot(t):
....: from math import cos,sin
....: return matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
....:
sage: def pursuit(n,x0,y0,lamb,steps = 100, threshold = .01):
....: paths = [[[x0,y0]]]
....: for i in range(1,n):
....: rx,ry = list(rot(2*npi*i/n)*vector([x0,y0]))
....: paths.append([[rx,ry]])
....: oldpath = [x[-1] for x in paths]
....: for q in range(steps):
....: diffs = [[oldpath[(j+1)%n][0]-oldpath[j][0],oldpath[(j+1)%n][1]-oldpath[j][1]] for j in range(n)]
....: npath = [[oldpath[j][0]+lamb*diffs[j][0],oldpath[j][1]+lamb*diffs[j][1]] for j in range(n)]
....: for j in range(n):
....: paths[j].append(npath[j])
....: oldpath = npath
....: return paths
....:
sage: html('<h3>Curves of Pursuit</h3>')
sage: @interact
sage: def curves_of_pursuit(n = slider([2..20],default = 5, label="# of points"),steps = slider([floor(1.4^i) for i in range(2,18)],default = 10, label="# of steps"), stepsize = slider(srange(.01,1,.01),default = .2, label="stepsize"), colorize = selector(['BW','Line color', 'Filled'],default = 'BW')):
....: outpaths = pursuit(n,0,1,stepsize, steps = steps)
....: mcolor = (0,0,0)
....: outer = line([q[0] for q in outpaths]+[outpaths[0][0]], rgbcolor = mcolor)
....: polys = Graphics()
....: if colorize=='Line color':
....: colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))]
....: elif colorize == 'BW':
....: colors = [(0,0,0) for j in range(len(outpaths[0]))]
....: else:
....: colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))]
....: polys = sum([polygon([outpaths[(i+1)%n][j+1],outpaths[(i+1)%n][j], outpaths[i][j+1]], rgbcolor = colors[j]) for i in range(n) for j in range(len(outpaths[0])-1)])
....: #polys = polys[0]
....: colors = [(0,0,0) for j in range(len(outpaths[0]))]
....: nested = sum([line([q[j] for q in outpaths]+[outpaths[0][j]], rgbcolor = colors[j]) for j in range(len(outpaths[0]))])
....: lpaths = [line(x, rgbcolor = mcolor) for x in outpaths]
....: show(sum(lpaths)+nested+polys, axes = False, figsize = [5,5], xmin = -1, xmax = 1, ymin = -1, ymax =1)
<html><font color='black'><h3>Curves of Pursuit</h3></font></html>
<html><!--notruncate--><div padding=6 id="div-interact-36"> <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"># of points </font></td><td><table><tr><td>
<div id="slider-n-36" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
</td><td><font color="black" id="slider-n-36-lbl"></font></td></tr></table><script>(function(){ var values = ["2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20"]; setTimeout(function() {
$('#slider-n-36').slider({
step: 1, min: 0, max: 18, value: 3,
change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-n-36-lbl').text(values[position]); interact(36, '_interact_.update(\'36\', \'n\', 11, _interact_.standard_b64decode(\''+encode64(position)+'\'), globals()); _interact_.recompute(\'36\');'); },
slide: function(e,ui) { if(values!=null) $('#slider-n-36-lbl').text(values[ui.value]); }
});
if(values != null) $('#slider-n-36-lbl').text(values[$('#slider-n-36').slider('value')]);
}, 1); })();</script></td></tr>
<tr><td align=right><font color="black"># of steps </font></td><td><table><tr><td>
<div id="slider-steps-36" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
</td><td><font color="black" id="slider-steps-36-lbl"></font></td></tr></table><script>(function(){ var values = ["1","2","3","5","7","10","14","20","28","40","56","79","111","155","217","304"]; setTimeout(function() {
$('#slider-steps-36').slider({
step: 1, min: 0, max: 15, value: 5,
change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-steps-36-lbl').text(values[position]); interact(36, '_interact_.update(\'36\', \'steps\', 12, _interact_.standard_b64decode(\''+encode64(position)+'\'), globals()); _interact_.recompute(\'36\');'); },
slide: function(e,ui) { if(values!=null) $('#slider-steps-36-lbl').text(values[ui.value]); }
});
if(values != null) $('#slider-steps-36-lbl').text(values[$('#slider-steps-36').slider('value')]);
}, 1); })();</script></td></tr>
<tr><td align=right><font color="black">stepsize </font></td><td><table><tr><td>
<div id="slider-stepsize-36" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
</td><td><font color="black" id="slider-stepsize-36-lbl"></font></td></tr></table><script>(function(){ var values = ["0.0100000000000000","0.0200000000000000","0.0300000000000000","0.0400000000000000","0.0500000000000000","0.0600000000000000","0.0700000000000000","0.0800000000000000","0.0900000000000000","0.100000000000000","0.110000000000000","0.120000000000000","0.130000000000000","0.140000000000000","0.150000000000000","0.160000000000000","0.170000000000000","0.180000000000000","0.190000000000000","0.200000000000000","0.210000000000000","0.220000000000000","0.230000000000000","0.240000000000000","0.250000000000000","0.260000000000000","0.270000000000000","0.280000000000000","0.290000000000000","0.300000000000000","0.310000000000000","0.320000000000000","0.330000000000000","0.340000000000000","0.350000000000000","0.360000000000000","0.370000000000000","0.380000000000000","0.390000000000000","0.400000000000000","0.410000000000000","0.420000000000000","0.430000000000000","0.440000000000000","0.450000000000000","0.460000000000000","0.470000000000000","0.480000000000000","0.490000000000000","0.500000000000000","0.510000000000000","0.520000000000000","0.530000000000000","0.540000000000000","0.550000000000000","0.560000000000000","0.570000000000000","0.580000000000000","0.590000000000000","0.600000000000000","0.610000000000000","0.620000000000000","0.630000000000000","0.640000000000000","0.650000000000000","0.660000000000000","0.670000000000000","0.680000000000000","0.690000000000000","0.700000000000000","0.710000000000000","0.720000000000000","0.730000000000000","0.740000000000000","0.750000000000000","0.760000000000000","0.770000000000000","0.780000000000000","0.790000000000000","0.800000000000000","0.810000000000000","0.820000000000001","0.830000000000001","0.840000000000001","0.850000000000001","0.860000000000001","0.870000000000001","0.880000000000001","0.890000000000001","0.900000000000001","0.910000000000001","0.920000000000001","0.930000000000001","0.940000000000001","0.950000000000001","0.960000000000001","0.970000000000001","0.980000000000001","0.990000000000001"]; setTimeout(function() {
$('#slider-stepsize-36').slider({
step: 1, min: 0, max: 98, value: 19,
change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-stepsize-36-lbl').text(values[position]); interact(36, '_interact_.update(\'36\', \'stepsize\', 13, _interact_.standard_b64decode(\''+encode64(position)+'\'), globals()); _interact_.recompute(\'36\');'); },
slide: function(e,ui) { if(values!=null) $('#slider-stepsize-36-lbl').text(values[ui.value]); }
});
if(values != null) $('#slider-stepsize-36-lbl').text(values[$('#slider-stepsize-36').slider('value')]);
}, 1); })();</script></td></tr>
<tr><td align=right><font color="black">colorize </font></td><td><select onchange="interact(36, '_interact_.update(\'36\', \'colorize\', 14, _interact_.standard_b64decode(\''+encode64(this.options[this.selectedIndex].value)+'\'), globals()); _interact_.recompute(\'36\');');"><option value="0" selected>BW</option>
<option value="1" >Line color</option>
<option value="2" >Filled</option>
</select></td></tr>
</table><div id="cell-interact-36"><?__SAGE__START>
<table border=0 bgcolor="white" width=100% height=100%>
<tr><td bgcolor="white" align=left valign=top><pre>
<font color='black'><img src=/home/admin/41/cells/36/sage0.png?12></font>
</pre></td></tr>
<tr><td align=left valign=top></td></tr>
</table><?__SAGE__END></div></td>
</tr></table></div>
</html>
<h2>interact: Taylor polynomials</h2>
sage: var('x')
sage: @interact
sage: def g(f=sin(x)*e^(-x), c=0, n=(1..30), xinterval=range_slider(-5, 5, 1, default=(-1,4), label="x-interval"), yinterval=range_slider(-50, 50, 1, default=(-1,1), label="y-interval")):
....: x0 = c
....: degree = n
....: xmin,xmax = xinterval
....: ymin,ymax = yinterval
....: p = plot(f, xmin, xmax, thickness=4)
....: dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
....: ft = f.taylor(x,x0,degree)
....: pt = plot(ft, xmin, xmax, color='red', thickness=2, fill=f)
....: show(dot + p + pt, ymin=ymin, ymax=ymax, xmin=xmin, xmax=xmax)
....: html('$f(x)\;=\;%s$'%latex(f))
....: html('$P_{%s}(x)\;=\;%s+R_{%s}(x)$'%(degree,latex(ft),degree))
<html><!--notruncate--><div padding=6 id="div-interact-38"> <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 </font></td><td><input type="text" value="e^(-x)*sin(x)" size=80 onchange="interact(38, '_interact_.update(\'38\', \'f\', 1, _interact_.standard_b64decode(\''+encode64(this.value)+'\'), globals()); _interact_.recompute(\'38\');')"></input></td></tr>
<tr><td align=right><font color="black">c </font></td><td><input type="text" value="0" size=80 onchange="interact(38, '_interact_.update(\'38\', \'c\', 2, _interact_.standard_b64decode(\''+encode64(this.value)+'\'), globals()); _interact_.recompute(\'38\');')"></input></td></tr>
<tr><td align=right><font color="black">n </font></td><td><table><tr><td>
<div id="slider-n-38" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
</td><td><font color="black" id="slider-n-38-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"]; setTimeout(function() {
$('#slider-n-38').slider({
step: 1, min: 0, max: 29, value: 0,
change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-n-38-lbl').text(values[position]); interact(38, '_interact_.update(\'38\', \'n\', 3, _interact_.standard_b64decode(\''+encode64(position)+'\'), globals()); _interact_.recompute(\'38\');'); },
slide: function(e,ui) { if(values!=null) $('#slider-n-38-lbl').text(values[ui.value]); }
});
if(values != null) $('#slider-n-38-lbl').text(values[$('#slider-n-38').slider('value')]);
}, 1); })();</script></td></tr>
<tr><td align=right><font color="black">x-interval </font></td><td><table><tr><td>
<div id="slider-xinterval-38" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;"></div>
</td></tr><tr><td><font color="black" id="slider-xinterval-38-lbl"></font></td></tr></table><script>(function()
{
var values = ["-5","-4","-3","-2","-1","0","1","2","3","4","5"];
var pos = [4, 9];
var sel = '#slider-xinterval-38';
var updatePos = function()
{
pos[0]=$(sel).slider('values', 0);
pos[1]=$(sel).slider('values', 1);
if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');
};
setTimeout(function()
{
$(sel).slider(
{
range: true,
step: 1,
min: 0,
max: 10,
values: [4, 9],
change: function(e,ui){ updatePos(); interact(38, '_interact_.update(\'38\', \'xinterval\', 4, _interact_.standard_b64decode(\''+encode64(pos[0]+' '+pos[1])+'\'), globals()); _interact_.recompute(\'38\');'); },
slide: updatePos
});
updatePos();
}, 1);
})();</script></td></tr>
<tr><td align=right><font color="black">y-interval </font></td><td><table><tr><td>
<div id="slider-yinterval-38" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;"></div>
</td></tr><tr><td><font color="black" id="slider-yinterval-38-lbl"></font></td></tr></table><script>(function()
{
var values = ["-50","-49","-48","-47","-46","-45","-44","-43","-42","-41","-40","-39","-38","-37","-36","-35","-34","-33","-32","-31","-30","-29","-28","-27","-26","-25","-24","-23","-22","-21","-20","-19","-18","-17","-16","-15","-14","-13","-12","-11","-10","-9","-8","-7","-6","-5","-4","-3","-2","-1","0","1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50"];
var pos = [49, 51];
var sel = '#slider-yinterval-38';
var updatePos = function()
{
pos[0]=$(sel).slider('values', 0);
pos[1]=$(sel).slider('values', 1);
if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');
};
setTimeout(function()
{
$(sel).slider(
{
range: true,
step: 1,
min: 0,
max: 100,
values: [49, 51],
change: function(e,ui){ updatePos(); interact(38, '_interact_.update(\'38\', \'yinterval\', 5, _interact_.standard_b64decode(\''+encode64(pos[0]+' '+pos[1])+'\'), globals()); _interact_.recompute(\'38\');'); },
slide: updatePos
});
updatePos();
}, 1);
})();</script></td></tr>
</table><div id="cell-interact-38"><?__SAGE__START>
<table border=0 bgcolor="white" width=100% height=100%>
<tr><td bgcolor="white" align=left valign=top><pre>
<font color='black'><img src=/home/admin/41/cells/38/sage0.png?7></font>
<font color='black'><span class="math">f(x)\;=\;e^{\left(-x\right)} \sin\left(x\right)</span></font>
<font color='black'><span class="math">P_{1}(x)\;=\;x+R_{1}(x)</span></font>
</pre></td></tr>
<tr><td align=left valign=top></td></tr>
</table><?__SAGE__END></div></td>
</tr></table></div>
</html>
<h2>interact: Taylor polynomials</h2>
sage: var('x')
sage: @interact
sage: def _(f=arctan(x), c=0, n=(0..25), xinterval=range_slider(-5, 5, 1, default=(-1,1), label="x-interval"), yinterval=range_slider(-50, 50, 1, default=(-1,1), label="y-interval")):
....: x0 = c
....: degree = n
....: xmin,xmax = xinterval
....: ymin,ymax = yinterval
....: p = plot(f, xmin, xmax, thickness=4)
....: dot = point((x0,f(x=x0)),pointsize=80,rgbcolor=(1,0,0))
....: ft = f.taylor(x,x0,degree)
....: pt = plot(ft, xmin, xmax, color='red', thickness=2, fill=f)
....: show(dot + p + pt, ymin = ymin, ymax = ymax, xmin=xmin, xmax=xmax)
....: html('$f(x)\;=\;%s$'%latex(f))
....: html('$P_{%s}(x)\;=\;%s+R_{%s}(x)$'%(degree,latex(ft),degree))
<html><!--notruncate--><div padding=6 id="div-interact-35"> <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 </font></td><td><input type="text" value="arctan(x)" size=80 onchange="interact(35, '_interact_.update(\'35\', \'f\', 6, _interact_.standard_b64decode(\''+encode64(this.value)+'\'), globals()); _interact_.recompute(\'35\');')"></input></td></tr>
<tr><td align=right><font color="black">c </font></td><td><input type="text" value="0" size=80 onchange="interact(35, '_interact_.update(\'35\', \'c\', 7, _interact_.standard_b64decode(\''+encode64(this.value)+'\'), globals()); _interact_.recompute(\'35\');')"></input></td></tr>
<tr><td align=right><font color="black">n </font></td><td><table><tr><td>
<div id="slider-n-35" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
</td><td><font color="black" id="slider-n-35-lbl"></font></td></tr></table><script>(function(){ var values = ["0","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"]; setTimeout(function() {
$('#slider-n-35').slider({
step: 1, min: 0, max: 25, value: 0,
change: function (e,ui) { var position = ui.value; if(values!=null) $('#slider-n-35-lbl').text(values[position]); interact(35, '_interact_.update(\'35\', \'n\', 8, _interact_.standard_b64decode(\''+encode64(position)+'\'), globals()); _interact_.recompute(\'35\');'); },
slide: function(e,ui) { if(values!=null) $('#slider-n-35-lbl').text(values[ui.value]); }
});
if(values != null) $('#slider-n-35-lbl').text(values[$('#slider-n-35').slider('value')]);
}, 1); })();</script></td></tr>
<tr><td align=right><font color="black">x-interval </font></td><td><table><tr><td>
<div id="slider-xinterval-35" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;"></div>
</td></tr><tr><td><font color="black" id="slider-xinterval-35-lbl"></font></td></tr></table><script>(function()
{
var values = ["-5","-4","-3","-2","-1","0","1","2","3","4","5"];
var pos = [4, 6];
var sel = '#slider-xinterval-35';
var updatePos = function()
{
pos[0]=$(sel).slider('values', 0);
pos[1]=$(sel).slider('values', 1);
if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');
};
setTimeout(function()
{
$(sel).slider(
{
range: true,
step: 1,
min: 0,
max: 10,
values: [4, 6],
change: function(e,ui){ updatePos(); interact(35, '_interact_.update(\'35\', \'xinterval\', 9, _interact_.standard_b64decode(\''+encode64(pos[0]+' '+pos[1])+'\'), globals()); _interact_.recompute(\'35\');'); },
slide: updatePos
});
updatePos();
}, 1);
})();</script></td></tr>
<tr><td align=right><font color="black">y-interval </font></td><td><table><tr><td>
<div id="slider-yinterval-35" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 20.0em;"></div>
</td></tr><tr><td><font color="black" id="slider-yinterval-35-lbl"></font></td></tr></table><script>(function()
{
var values = ["-50","-49","-48","-47","-46","-45","-44","-43","-42","-41","-40","-39","-38","-37","-36","-35","-34","-33","-32","-31","-30","-29","-28","-27","-26","-25","-24","-23","-22","-21","-20","-19","-18","-17","-16","-15","-14","-13","-12","-11","-10","-9","-8","-7","-6","-5","-4","-3","-2","-1","0","1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20","21","22","23","24","25","26","27","28","29","30","31","32","33","34","35","36","37","38","39","40","41","42","43","44","45","46","47","48","49","50"];
var pos = [49, 51];
var sel = '#slider-yinterval-35';
var updatePos = function()
{
pos[0]=$(sel).slider('values', 0);
pos[1]=$(sel).slider('values', 1);
if(values!=null) $(sel+'-lbl').text('('+values[pos[0]]+', '+values[pos[1]]+')');
};
setTimeout(function()
{
$(sel).slider(
{
range: true,
step: 1,
min: 0,
max: 100,
values: [49, 51],
change: function(e,ui){ updatePos(); interact(35, '_interact_.update(\'35\', \'yinterval\', 10, _interact_.standard_b64decode(\''+encode64(pos[0]+' '+pos[1])+'\'), globals()); _interact_.recompute(\'35\');'); },
slide: updatePos
});
updatePos();
}, 1);
})();</script></td></tr>
</table><div id="cell-interact-35"><?__SAGE__START>
<table border=0 bgcolor="white" width=100% height=100%>
<tr><td bgcolor="white" align=left valign=top><pre>
<font color='black'><img src=/home/admin/41/cells/35/sage0.png?11></font>
<font color='black'><span class="math">f(x)\;=\;\sin\left(x\right)</span></font>
<font color='black'><span class="math">P_{15}(x)\;=\;-\frac{1}{1307674368000} \, x^{15} + \frac{1}{6227020800} \, x^{13} - \frac{1}{39916800} \, x^{11} + \frac{1}{362880} \, x^{9} - \frac{1}{5040} \, x^{7} + \frac{1}{120} \, x^{5} - \frac{1}{6} \, x^{3} + x+R_{15}(x)</span></font>
</pre></td></tr>
<tr><td align=left valign=top></td></tr>
</table><?__SAGE__END></div></td>
</tr></table></div>
</html>
<h2>Cython: Python $\longmapsto$ C</h2>
<p>Here is a function that computes $\sum_{k=0}^N k$ in pure Python.</p>
sage: def mysum(N):
....: s = int(0)
....: for k in range(1,N):
....: s += k
....: return s
sage: time mysum(10^7)
<p>Let us compare this with the Cython version.</p>
sage: %cython
sage: def mysum_cython(N):
....: cdef long long s = 0
....: cdef int k
....: for k in range(1,N):
....: s += k
....: return s
sage: time mysum_cython(10^7)
<h2>Drawing graphs</h2>
sage: HS = graphs.HyperStarGraph(6,3)
sage: HS.plot()
<html><font color='black'><img src='cell://sage0.png'></font></html>
sage: HS.plot3d()
sage: graph_editor(HS)
<h2>Use the source!</h2>
sage: import scipy
sage: from scipy import io
sage: x=io.loadmat('%s/yodapose.mat'%DATA) # you can change it to yodapose/yodapose_low to use the high/low res version
sage: from sage.plot.plot3d.index_face_set import IndexFaceSet
sage: V=x['V']
sage: F3=x['F3']-1
sage: F4=x['F4']-1
sage: yoda=IndexFaceSet(F3,V,color='green')+IndexFaceSet(F4,V,color='green')
sage: yoda.show(figsize=5, frame=False)
/opt/sage/local/lib/python2.6/site-packages/scipy/io/matlab/mio.py:84: FutureWarning: Using struct_as_record default value (False) This will change to True in future versions
return MatFile5Reader(byte_stream, **kwargs)