Sage Days 20.5 : Demo
The Sage notebook: an interface to your software!
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2+3
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Interface with Magma (you need to have Magma installed)
We use Sage to construct a random $150\times150$ matrix over the integers with entries of size up to $2^{128}$.
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a = random_matrix(ZZ,150,x=-2^128,y=2^128)
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A random entry of the matrix:
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a[72,17]
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We use Magma to compute the determinant of the matrix:
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b = magma(a)
magma.eval('time e := Determinant(%s);'%b.name())
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We can also do this with Sage, and it's faster!
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time d = a.determinant()
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And we get the same answer:
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d == magma('e')
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Here's an example of Sage and Magma computing the Hermite normal form of a random matrix of size $100\times100$ over the integers.
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a = random_matrix(ZZ,100,x=-2^64,y=2^64)
time h = a.hermite_form()
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b = magma(a)
magma.eval('time e := HermiteForm(%s);'%b.name())
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h == magma('e')
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Interface with Maple (you need to have Maple installed)
Below we use Maple and Sage to compute the number of partitions of $27020$.
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time maple('combinat[numbpart](27020)')
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time number_of_partitions(27020)
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Not only is Sage faster, it also returns the correct answer: http://www.research.att.com/~njas/sequences/A110375.
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Embed Java Applets into the Sage Notebook
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load DATA+'EuclideanK44Mech.sage'
html(jsp_string)
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Beautiful documentation
Type plot( and then press the TAB key.
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plot(
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Click on Help at the top of this page.
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There is also Live Documentation in which you can evaluate the examples.
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Interact with your mathematics by creating interactive widgets
interact: curves of pursuit
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npi = RDF(pi)
def rot(t):
from math import cos,sin
return matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
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
html('Curves of Pursuit
')
@interact
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)
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interact: Taylor polynomials
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var('x')
@interact
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))
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interact: Taylor polynomials
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var('x')
@interact
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))
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Cython: Python $\longmapsto$ C
Here is a function that computes $\sum_{k=0}^N k$ in pure Python.
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def mysum(N):
s = int(0)
for k in range(1,N):
s += k
return s
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time mysum(10^7)
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Let us compare this with the Cython version.
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%cython
def mysum_cython(N):
cdef long long s = 0
cdef int k
for k in range(1,N):
s += k
return s
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time mysum_cython(10^7)
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Drawing graphs
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HS = graphs.HyperStarGraph(6,3)
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HS.plot()
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HS.plot3d()
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graph_editor(HS)
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Use the source!
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import scipy
from scipy import io
x=io.loadmat('%s/yodapose.mat'%DATA) # you can change it to yodapose/yodapose_low to use the high/low res version
from sage.plot.plot3d.index_face_set import IndexFaceSet
V=x['V']
F3=x['F3']-1
F4=x['F4']-1
yoda=IndexFaceSet(F3,V,color='green')+IndexFaceSet(F4,V,color='green')
yoda.show(figsize=5, frame=False)
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