# Sage Interactions

Post code that demonstrates the use of the interact command in Sage here. It should be easy for people to just scroll through and paste examples out of here into their own sage notebooks.

We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out.

## Miscellaneous

### Evaluate a bit of code in a given system

by William Stein (there is no way yet to make the text box big):

@interact
def _(system=selector([('sage0', 'Sage'), ('gp', 'PARI'), ('magma', 'Magma')]), code='2+2'):
print globals()[system].eval(code)

attachment:evalsys.png

## Graph Theory

### Automorphism Groups of some Graphs

by William Stein (I spent less than five minutes on this):

@interact
def _(graph=['CycleGraph', 'CubeGraph', 'RandomGNP'],
n=selector([1..10],nrows=1), p=selector([10,20,..,100],nrows=1)):
print graph
if graph == 'CycleGraph':
print "n (=%s): number of vertices"%n
G = graphs.CycleGraph(n)
elif graph == 'CubeGraph':
if n > 8:
print "n reduced to 8"
n = 8
print "n (=%s): dimension"%n
G = graphs.CubeGraph(n)
elif graph == 'RandomGNP':
print "n (=%s) vertices"%n
print "p (=%s%%) probability"%p
G = graphs.RandomGNP(n, p/100.0)

print G.automorphism_group()
show(plot(G))

attachment:autograph.png

## Calculus

### A contour map and 3d plot of two inverse distance functions

by William Stein

@interact
def _(q1=(-1,(-3,3)), q2=(-2,(-3,3)),
cmap=['autumn', 'bone', 'cool', 'copper', 'gray', 'hot', 'hsv',
'jet', 'pink', 'prism', 'spring', 'summer', 'winter']):
x,y = var('x,y')
f = q1/sqrt((x+1)^2 + y^2) + q2/sqrt((x-1)^2+(y+0.5)^2)
C = contour_plot(f, (-2,2), (-2,2), plot_points=30, contours=15, cmap=cmap)
show(C, figsize=3, aspect_ratio=1)
show(plot3d(f, (x,-2,2), (y,-2,2)), figsize=5, viewer='tachyon')     

attachment:mountains.png

### A simple tangent line grapher

by Marshall Hampton

html('<h2>Tangent line grapher</h2>')
@interact
def tangent_line(f = input_box(default=sin(x)), xbegin = slider(0,10,1/10,0), xend = slider(0,10,1/10,10), x0 = slider(0, 1, 1/100, 1/2)):
prange = [xbegin, xend]
x0i = xbegin + x0*(xend-xbegin)
var('x')
df = diff(f)
tanf = f(x0i) + df(x0i)*(x-x0i)
fplot = plot(f, prange[0], prange[1])
print 'Tangent line is y = ' + tanf._repr_()
tanplot = plot(tanf, prange[0], prange[1], rgbcolor = (1,0,0))
fmax = f.find_maximum_on_interval(prange[0], prange[1])[0]
fmin = f.find_minimum_on_interval(prange[0], prange[1])[0]
show(fplot + tanplot, xmin = prange[0], xmax = prange[1], ymax = fmax, ymin = fmin)

attachment:tangents.png

## Differential Equations

### Euler's Method in one variable

by Marshall Hampton. This needs some polishing but its usable as is.

def tab_list(y, headers = None):
'''
Converts a list into an html table with borders.
'''
s = '<table border = 1>'
if headers:
for q in headers:
s = s + '<th>' + str(q) + '</th>'
for x in y:
s = s + '<tr>'
for q in x:
s = s + '<td>' + str(q) + '</td>'
s = s + '</tr>'
s = s + '</table>'
return s
var('x y')
@interact
def euler_method(y_exact_in = input_box('-cos(x)+1.0', type = str, label = 'Exact solution = '), y_prime_in = input_box('sin(x)', type = str, label = "y' = "), start = input_box(0.0, label = 'x starting value: '), stop = input_box(6.0, label = 'x stopping value: '), startval = input_box(0.0, label = 'y starting value: '), nsteps = slider([2^m for m in range(0,10)], default = 10, label = 'Number of steps: '), show_steps = slider([2^m for m in range(0,10)], default = 8, label = 'Number of steps shown in table: ')):
y_exact = lambda x: eval(y_exact_in)
y_prime = lambda x,y: eval(y_prime_in)
stepsize = float((stop-start)/nsteps)
steps_shown = max(nsteps,show_steps)
sol = [startval]
xvals = [start]
for step in range(nsteps):
sol.append(sol[-1] + stepsize*y_prime(xvals[-1],sol[-1]))
xvals.append(xvals[-1] + stepsize)
sol_max = max(sol + [find_maximum_on_interval(y_exact,start,stop)[0]])
sol_min = min(sol + [find_minimum_on_interval(y_exact,start,stop)[0]])
show(plot(y_exact(x),start,stop,rgbcolor=(1,0,0))+line([[xvals[index],sol[index]] for index in range(len(sol))]),xmin=start,xmax = stop, ymax = sol_max, ymin = sol_min)
if nsteps < steps_shown:
table_range = range(len(sol))
else:
table_range = range(0,floor(steps_shown/2)) + range(len(sol)-floor(steps_shown/2),len(sol))
html(tab_list([[i,xvals[i],sol[i]] for i in table_range], headers = ['step','x','y']))

attachment:eulermethod.png

## Linear Algebra

### Numerical instability of the classical Gram-Schmidt algorithm

by Marshall Hampton (tested by William Stein, who thinks this is really nice!)

def GS_classic(a_list):
'''
Given a list of vectors or a matrix, returns the QR factorization using the classical (and numerically unstable) Gram-Schmidt algorithm.
'''
if type(a_list) != list:
cols = a_list.cols()
a_list = [x for x in cols]
indices = range(len(a_list))
q = []
r = [[0 for i in indices] for j in indices]
v = [a_list[i].copy() for i in indices]
for i in indices:
for j in range(0,i):
r[j][i] = q[j].inner_product(a_list[i])
v[i] = v[i] - r[j][i]*q[j]
r[i][i] = (v[i]*v[i])^(1/2)
q.append(v[i]/r[i][i])
q = matrix([q[i] for i in indices]).transpose()
return q, matrix(r)
def GS_modern(a_list):
'''
Given a list of vectors or a matrix, returns the QR factorization using the 'modern' Gram-Schmidt algorithm.
'''
if type(a_list) != list:
cols = a_list.cols()
a_list = [x for x in cols]
indices = range(len(a_list))
q = []
r = [[0 for i in indices] for j in indices]
v = [a_list[i].copy() for i in indices]
for i in indices:
r[i][i] = v[i].norm(2)
q.append(v[i]/r[i][i])
for j in range(i+1, len(indices)):
r[i][j] = q[i].inner_product(v[j])
v[j] = v[j] - r[i][j]*q[i]
q = matrix([q[i] for i in indices]).transpose()
return q, matrix(r)
html('<h2>Numerical instability of the classical Gram-Schmidt algorithm</h2>')
@interact
def gstest(precision = slider(range(3,53), default = 10), a1 = input_box([1,1/1000,1/1000]), a2 = input_box([1,1/1000,0]), a3 = input_box([1,0,1/1000])):
myR = RealField(precision)
displayR = RealField(5)
html('precision in bits: ' + str(precision) + '<br>')
A = matrix([a1,a2,a3])
A = [vector(myR,x) for x in A]
qn, rn = GS_classic(A)
qb, rb = GS_modern(A)
html('Classical Gram-Schmidt:')
show(matrix(displayR,qn))
html('Stable Gram-Schmidt:')
show(matrix(displayR,qb))

attachment:GramSchmidt.png

### Linear transformations

by Jason Grout

@interact
def linear_transformation(theta=slider(0, 2*pi, .1), r=slider(0.1, 2, .1, default=1)):
A=matrix([[1,-1],[-1,1/2]])
v=vector([r*cos(theta), r*sin(theta)])
w = A*v
circles = sum([circle((0,0), radius=i, rgbcolor=(0,0,0)) for i in [1..2]])
print jsmath("v = %s,\; %s v=%s"%(v.n(4),latex(A),w.n(4)))
show(v.plot(rgbcolor=(1,0,0))+w.plot(rgbcolor=(0,0,1))+circles,aspect_ratio=1)

attachment:Linear-Transformations.png

## Number Theory

### Continued Fraction Plotter

by William Stein

@interact
def _(number=e, ymax=selector([None,5,20,..,400],nrows=2), clr=Color('purple'), prec=[500,1000,..,5000]):
c = list(continued_fraction(RealField(prec)(number))); print c
show(line([(i,z) for i, z in enumerate(c)],rgbcolor=clr),ymax=ymax,figsize=[10,2])

attachment:contfracplot.png

### Illustrating the prime number thoerem

by William Stein

@interact
def _(N=(100,(2..2000))):
html("<font color='red'>$\pi(x)$</font> and <font color='blue'>$x/(\log(x)-1)$</font> for $x < %s$"%N)
show(plot(prime_pi, 0, N, rgbcolor='red') + plot(x/(log(x)-1), 5, N, rgbcolor='blue'))

attachment:primes.png

### Computing Generalized Bernoulli Numbers

by William Stein (Sage-2.10.3)

@interact
def _(m=selector([1..15],nrows=2), n=(7,(3..10))):
G = DirichletGroup(m)
s = "<h3>First n=%s Bernoulli numbers attached to characters with modulus m=%s</h3>"%(n,m)
s += '<table border=1>'
s += '<tr bgcolor="#edcc9c"><td align=center>$\\chi$</td><td>Conductor</td>' + \
''.join('<td>$B_{%s,\chi}$</td>'%k for k in [1..n]) + '</tr>'
for eps in G.list():
v = ''.join(['<td align=center bgcolor="#efe5cd">$%s$</td>'%latex(eps.bernoulli(k)) for k in [1..n]])
s += '<tr><td bgcolor="#edcc9c">%s</td><td bgcolor="#efe5cd" align=center>%s</td>%s</tr>\n'%(
eps, eps.conductor(), v)
s += '</table>'
html(s)

attachment:bernoulli.png

### Computing modular forms

by William Stein

j = 0
@interact
def _(N=[1..100], k=selector([2,4,..,12],nrows=1), prec=(3..40),
group=[(Gamma0, 'Gamma0'), (Gamma1, 'Gamma1')]):
M = CuspForms(group(N),k)
print j; global j; j += 1
print M; print '\n'*3
print "Computing basis...\n\n"
if M.dimension() == 0:
print "Space has dimension 0"
else:
prec = max(prec, M.dimension()+1)
for f in M.basis():
view(f.q_expansion(prec))
print "\n\n\nDone computing basis."

attachment:modformbasis.png

### Computing the cuspidal subgroup

by William Stein

html('<h1>Cuspidal Subgroups of Modular Jacobians J0(N)</h1>')
@interact
def _(N=selector([1..8*13], ncols=8, width=10, default=10)):
A = J0(N)
print A.cuspidal_subgroup()

attachment:cuspgroup.png

### A Charpoly and Hecke Operator Graph

by William Stein

# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots
@interact
def f(N = prime_range(11,400),
p = selector(prime_range(2,12),nrows=1),
three_d = ("Three Dimensional", False)):
S = SupersingularModule(N)
T = S.hecke_matrix(p)
G = Graph(T, multiedges=True, loops=not three_d)
html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p))
show(T.charpoly().factor())
if three_d:
show(G.plot3d(), aspect_ratio=[1,1,1])
else:
show(G.plot(),figsize=7)

attachment:heckegraph.png

### Demonstrating the Diffie-Hellman Key Exchange Protocol

by Timothy Clemans (refereed by William Stein)

@interact
def diffie_hellman(button=selector(["New example"],label='',buttons=True),
bits=("Number of bits of prime", (8,12,..512))):
maxp = 2^bits
p = random_prime(maxp)
k = GF(p)
g = k.multiplicative_generator()
a = ZZ.random_element(10, maxp)
b = ZZ.random_element(10, maxp)

print """
<html>
<style>
.gamodp {
background:yellow
}
.gbmodp {
background:orange
}
.dhsame {
color:green;
font-weight:bold
}
</style>
<h2>%s-Bit Diffie-Hellman Key Exchange</h2>
<ol style="color:#000;font:12px Arial, Helvetica, sans-serif">
<li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li>
<li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li>
<li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li>
<li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
<li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li>
</ol></html>
""" % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p,
(g^ b)^a, g^a, b, p, (g^a)^b)

attachment:dh.png

### Plotting an elliptic curve over a finite field

E = EllipticCurve('37a')
@interact
def _(p=slider(prime_range(1000), default=389)):
show(E)
print "p = %s"%p
show(E.change_ring(GF(p)).plot(),xmin=0,ymin=0)

attachment:ellffplot.png

## Bioinformatics

### Web app: protein browser

by Marshall Hampton (tested by William Stein)

import urllib2 as U
@interact
def protein_browser(GenBank_ID = input_box('165940577', type = str), file_type = selector([(1,'fasta'),(2,'GenPept')])):
if file_type == 2:
gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&id='
else:
gen_str = 'http://www.ncbi.nlm.nih.gov/entrez/viewer.fcgi?db=protein&sendto=t&dopt=fasta&id='
f = U.urlopen(gen_str + GenBank_ID)
g = f.read()
f.close()
html(g)

attachment:biobrowse.png

### Coalescent simulator

by Marshall Hampton

def next_gen(x, selection=1.0):
'''Creates the next generation from the previous; also returns parent-child indexing list'''
next_x = []
for ind in range(len(x)):
if random() < (1 + selection)/len(x):
rind = 0
else:
rind = int(round(random()*(len(x)-1)+1/2))
next_x.append((x[rind],rind))
next_x.sort()
return [[x[0] for x in next_x],[x[1] for x in next_x]]
def coal_plot(some_data):
'''Creates a graphics object from coalescent data'''
gens = some_data[0]
inds = some_data[1]
gen_lines = line([[0,0]])
pts = Graphics()
ngens = len(gens)
gen_size = len(gens[0])
for x in range(gen_size):
pts += point((x,ngens-1), hue = gens[0][x]/float(gen_size*1.1))
p_frame = line([[-.5,-.5],[-.5,ngens-.5], [gen_size-.5,ngens-.5], [gen_size-.5,-.5], [-.5,-.5]])
for g in range(1,ngens):
for x in range(gen_size):
old_x = inds[g-1][x]
gen_lines += line([[x,ngens-g-1],[old_x,ngens-g]], hue = gens[g-1][old_x]/float(gen_size*1.1))
pts += point((x,ngens-g-1), hue = gens[g][x]/float(gen_size*1.1))
return pts+gen_lines+p_frame
@interact
def coalescents(pop_size = slider(2,100,1,15,'Population size'), selection = slider(-1,1,.1,0, 'Selection for first taxon'), s = selector(['Again!'], label='Refresh', buttons=True)):
print 'Population size: ' + str(pop_size)
print 'Selection coefficient for first taxon: ' + str(d_field(selection))
start = [i for i in range(pop_size)]
gens = [start]
inds = []
while gens[-1][0] != gens[-1][-1]:
g_index = len(gens) - 1
n_gen = next_gen(gens[g_index], selection = selection)
gens.append(n_gen[0])
inds.append(n_gen[1])
coal_data1 = [gens,inds]
print 'Generations until coalescence: ' + str(len(gens))
show(coal_plot(coal_data1), axes = False, figsize = [8,4.0*len(gens)/pop_size], ymax = len(gens)-1)

attachment:coalescent.png

## Miscellaneous Graphics

### Interactive rotatable raytracing with Tachyon3d

C = cube(color=['red', 'green', 'blue'], aspect_ratio=[1,1,1],
viewer='tachyon') + sphere((1,0,0),0.2)
@interact
def example(theta=(0,2*pi), phi=(0,2*pi), zoom=(1,(1,4))):
show(C.rotate((0,0,1), theta).rotate((0,1,0),phi), zoom=zoom)

attachment:tachyonrotate.png

### Somewhat Silly Egg Painter

by Marshall Hampton (refereed by William Stein)

var('s,t')
g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s))
def P(color, rng):
return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1)
colorlist = ['red','blue','red','blue']
@interact
def _(band_number = selector(range(1,5)), current_color = Color('red')):
html('<h1 align=center>Egg Painter</h1>')
colorlist[band_number-1] = current_color
egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)])
show(egg)

attachment:eggpaint.png