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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. Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].
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We'll likely restructure and reorganize this once we have some nontrivial content and get a sense of how it is laid out.

[[TableOfContents]]		  * [:interact/graph_theory:Graph Theory]
 * [:interact/calculus:Calculus]
 * [:interact/diffeq:Differential Equations]
 * [:interact/linear_algebra:Linear Algebra]
 * [:interact/algebra:Algebra]
 * [:interact/number_theory:Number Theory]
 * [:interact/web:Web Applications]
 * [:interact/bio:Bioinformatics]
 * [:interact/graphics:Drawing Graphics]
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== Profile a snippet of code ==
{{{
html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'),
      do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
    if do_preparse: cmd = preparse(cmd)
    print "<html>" # trick to avoid word wrap
    if cprof:
        cProfile.run(cmd)
    else:
        profile.run(cmd)
    print "</html>"
}}}
attachment:profile.png
Line 14: Line 38:
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=== A Random Walk ===
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== 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

A square matrix defines a linear transformation which rotates and/or scales vectors. In the interact command below, the red vector represents the original vector (v) and the blue vector represents the image w under the linear transformation. You can change the angle and length of v by changing theta and r.

{{{
@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 ===
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# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots html('<h1>A Random Walk</h1>')
vv = []; nn = 0
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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])		 def foo(pts = checkbox(True, "Show points"),
        refresh = checkbox(False, "New random walk every time"),
        steps = (50,(10..500))):
    # We cache the walk in the global variable vv, so that
    # checking or unchecking the points checkbox doesn't change
    # the random walk.
    html("<h2>%s steps</h2>"%steps)
    global vv
    if refresh or len(vv) == 0:
        s = 0; v = [(0,0)]
        for i in range(steps):
             s += random() - 0.5
             v.append((i, s))
        vv = v
    elif len(vv) != steps:
        # Add or subtract some points
        s = vv[-1][1]; j = len(vv)
        for i in range(steps - len(vv)):
            s += random() - 0.5
            vv.append((i+j,s))
        v = vv[:steps]
Line 308: Line 76:
        show(G.plot(),figsize=7)         v = vv
    L = line(v, rgbcolor='#4a8de2')
    if pts: L += points(v, pointsize=10, rgbcolor='red')
    show(L, xmin=0, figsize=[8,3])
Line 310: Line 81:
attachment:randomwalk.png
Line 311: Line 83:
attachment:heckegraph.png

=== Demonstrating the Diffie-Hellman Key Exchange Protocol ===
by Timothy Clemans (refereed by William Stein)		 === 3D Random Walk ===
Line 317: Line 86:
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)		 def rwalk3d(n=(50,1000), frame=True):
    pnt = [0,0,0]
    v = [copy(pnt)]
    for i in range(n):
        pnt[0] += random()-0.5
        pnt[1] += random()-0.5
        pnt[2] += random()-0.5
        v.append(copy(pnt))
    show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)
Line 351: Line 96:

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


[[Anchor(eggpaint)]]

=== 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		 attachment:randomwalk3d.png

Sage Interactions

Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].

  • [:interact/graph_theory:Graph Theory]
  • [:interact/calculus:Calculus]
  • [:interact/diffeq:Differential Equations]
  • [:interact/linear_algebra:Linear Algebra]
  • [:interact/algebra:Algebra]
  • [:interact/number_theory:Number Theory]
  • [:interact/web:Web Applications]
  • [:interact/bio:Bioinformatics]
  • [:interact/graphics:Drawing Graphics]

Miscellaneous

Profile a snippet of code

html('<h2>Profile the given input</h2>')
import cProfile; import profile
@interact
def _(cmd = ("Statement", '2 + 2'), 
      do_preparse=("Preparse?", True), cprof =("cProfile?", False)):
    if do_preparse: cmd = preparse(cmd)
    print "<html>"  # trick to avoid word wrap
    if cprof:
        cProfile.run(cmd)
    else:
        profile.run(cmd)
    print "</html>"

attachment:profile.png

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

A Random Walk

by William Stein 

html('<h1>A Random Walk</h1>')
vv = []; nn = 0
@interact
def foo(pts = checkbox(True, "Show points"), 
        refresh = checkbox(False, "New random walk every time"),
        steps = (50,(10..500))):
    # We cache the walk in the global variable vv, so that
    # checking or unchecking the points checkbox doesn't change
    # the random walk. 
    html("<h2>%s steps</h2>"%steps)
    global vv
    if refresh or len(vv) == 0:
        s = 0; v = [(0,0)]
        for i in range(steps): 
             s += random() - 0.5
             v.append((i, s)) 
        vv = v
    elif len(vv) != steps:
        # Add or subtract some points
        s = vv[-1][1]; j = len(vv)
        for i in range(steps - len(vv)):
            s += random() - 0.5
            vv.append((i+j,s))
        v = vv[:steps]
    else:
        v = vv
    L = line(v, rgbcolor='#4a8de2')
    if pts: L += points(v, pointsize=10, rgbcolor='red')
    show(L, xmin=0, figsize=[8,3])

attachment:randomwalk.png

3D Random Walk

@interact
def rwalk3d(n=(50,1000), frame=True):
    pnt = [0,0,0]
    v = [copy(pnt)]
    for i in range(n):
        pnt[0] += random()-0.5
        pnt[1] += random()-0.5
        pnt[2] += random()-0.5
        v.append(copy(pnt))
    show(line3d(v,color='black'),aspect_ratio=[1,1,1],frame=frame)

attachment:randomwalk3d.png

interact (last edited 2021-08-23 15:58:42 by anewton)