Differences between revisions 81 and 93 (spanning 12 versions)

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. If you have suggestions on how to improve interact, add them [:interactSuggestions: here] or email [email protected].

TableOfContents

Interact Examples

[:interact/graph_theory:]

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

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

Function tool

Enter symbolic functions f, g, and a, a range, then click the appropriate button to compute and plot some combination of f, g, and a along with f and g. This is inspired by the Matlab funtool GUI.

x = var('x')

@interact
def _(f=sin(x), g=cos(x), xrange=input_box((0,1)), yrange='auto', a=1,
      action=selector(['f', 'df/dx', 'int f', 'num f', 'den f', '1/f', 'finv',
                       'f+a', 'f-a', 'f*a', 'f/a', 'f^a', 'f(x+a)', 'f(x*a)', 
                       'f+g', 'f-g', 'f*g', 'f/g', 'f(g)'], 
             width=15, nrows=5, label="h = "),
      do_plot = ("Draw Plots", True)):

    try:
        f = SR(f); g = SR(g); a = SR(a)
    except TypeError, msg:
        print msg[-200:]
        print "Unable to make sense of f,g, or a as symbolic expressions."
        return 
    if not (isinstance(xrange, tuple) and len(xrange) == 2):
          xrange = (0,1)
    h = 0; lbl = ''
    if action == 'f':
        h = f
        lbl = 'f'
    elif action == 'df/dx':
        h = f.derivative(x)
        lbl = '\\frac{df}{dx}'
    elif action == 'int f':
        h = f.integrate(x)
        lbl = '\\int f dx'
    elif action == 'num f':
        h = f.numerator()
        lbl = '\\text{numer(f)}'
    elif action == 'den f':
        h = f.denominator()
        lbl = '\\text{denom(f)}'
    elif action == '1/f':
        h = 1/f
        lbl = '\\frac{1}{f}'
    elif action == 'finv':
        h = solve(f == var('y'), x)[0].rhs()
        lbl = 'f^{-1}(y)'
    elif action == 'f+a':
        h = f+a
        lbl = 'f + a'
    elif action == 'f-a':
        h = f-a
        lbl = 'f - a'
    elif action == 'f*a':
        h = f*a
        lbl = 'f \\times a'
    elif action == 'f/a':
        h = f/a
        lbl = '\\frac{f}{a}'
    elif action == 'f^a':
        h = f^a
        lbl = 'f^a'
    elif action == 'f^a':
        h = f^a
        lbl = 'f^a'
    elif action == 'f(x+a)':
        h = f(x+a)
        lbl = 'f(x+a)'
    elif action == 'f(x*a)':
        h = f(x*a)
        lbl = 'f(xa)'
    elif action == 'f+g':
        h = f+g
        lbl = 'f + g'
    elif action == 'f-g':
        h = f-g
        lbl = 'f - g'
    elif action == 'f*g':
        h = f*g
        lbl = 'f \\times g'
    elif action == 'f/g':
        h = f/g
        lbl = '\\frac{f}{g}'
    elif action == 'f(g)':
        h = f(g)
        lbl = 'f(g)'
      
      
    html('<center><font color="red">$f = %s$</font></center>'%latex(f))
    html('<center><font color="green">$g = %s$</font></center>'%latex(g))
    html('<center><font color="blue"><b>$h = %s = %s$</b></font></center>'%(lbl, latex(h)))
    if do_plot:
        P = plot(f, xrange, color='red', thickness=2) +  \
            plot(g, xrange, color='green', thickness=2) + \
            plot(h, xrange, color='blue', thickness=2)
        if yrange == 'auto':
            show(P, xmin=xrange[0], xmax=xrange[1])
        else:
            yrange = sage_eval(yrange)
            show(P, xmin=xrange[0], xmax=xrange[1], ymin=yrange[0], ymax=yrange[1])

attachment:funtool.png

Newton-Raphson Root Finding

by Neal Holtz

This allows user to display the Newton-Raphson procedure one step at a time. It uses the heuristic that, if any of the values of the controls change, then the procedure should be re-started, else it should be continued.

# ideas from 'A simple tangent line grapher' by Marshall Hampton
# http://wiki.sagemath.org/interact

State = Data = None   # globals to allow incremental additions to graphics

@interact
def newtraph(f = input_box(default=8*sin(x)*exp(-x)-1, label='f(x)'), 
             xmin = input_box(default=0), 
             xmax = input_box(default=4*pi), 
             x0 = input_box(default=3, label='x0'),
             show_calcs = ("Show Calcs",True),
             step = ['Next','Reset'] ):
    global State, Data
    prange = [xmin,xmax]
    state = [f,xmin,xmax,x0,show_calcs]
    if (state != State) or (step == 'Reset'):   # when any of the controls change
        X = [RR(x0)]                     # restart the plot
        df = diff(f)
        Fplot = plot(f, prange[0], prange[1])
        Data = [X, df, Fplot]
        State = state

    X, df, Fplot = Data
    i = len(X) - 1              # compute and append the next x value
    xi = X[i]
    fi = RR(f(xi))
    fpi = RR(df(xi))
    xip1 = xi - fi/fpi
    X.append(xip1)

    msg = xip1s = None          # now check x value for reasonableness
    is_inf = False
    if abs(xip1) > 10E6*(xmax-xmin):
        is_inf = True
        show_calcs = True
        msg = 'Derivative is 0!'
        xip1s = latex(xip1.sign()*infinity)
        X.pop()
    elif not ((xmin - 0.5*(xmax-xmin)) <= xip1 <= (xmax + 0.5*(xmax-xmin))):
        show_calcs = True
        msg = 'x value out of range; probable divergence!'
    if xip1s is None:
        xip1s = '%.4g' % (xip1,)

    def Disp( s, color="blue" ):
        if show_calcs:
            html( """<font color="%s">$ %s $</font>""" % (color,s,) )
    Disp( """f(x) = %s""" % (latex(f),)  +
          """~~~~f'(x) = %s""" % (latex(df),) )
    Disp( """i = %d""" % (i,)  +
          """~~~~x_{%d} = %.4g""" % (i,xi)  +
          """~~~~f(x_{%d}) = %.4g""" % (i,fi)  +
          """~~~~f'(x_{%d}) = %.4g""" % (i,fpi) )
    if msg:
        html( """<font color="red"><b>%s</b></font>""" % (msg,) )
        c = "red"
    else:
        c = "blue"
    Disp( r"""x_{%d} = %.4g - ({%.4g})/({%.4g}) = %s""" % (i+1,xi,fi,fpi,xip1s), color=c )

    Fplot += line( [(xi,0),(xi,fi)], linestyle=':', rgbcolor=(1,0,0) ) # vert dotted line
    Fplot += points( [(xi,0),(xi,fi)], rgbcolor=(1,0,0) )
    labi = text( '\nx%d\n' % (i,), (xi,0), rgbcolor=(1,0,0),
                 vertical_alignment="bottom" if fi < 0 else "top" )
    if is_inf:
        xl = xi - 0.05*(xmax-xmin)
        xr = xi + 0.05*(xmax-xmin)
        yl = yr = fi
    else:
        xl = min(xi,xip1) - 0.02*(xmax-xmin)
        xr = max(xi,xip1) + 0.02*(xmax-xmin)
        yl = -(xip1-xl)*fpi
        yr = (xr-xip1)*fpi
        Fplot += points( [(xip1,0)], rgbcolor=(0,0,1) )       # new x value
        labi += text( '\nx%d\n' % (i+1,), (xip1,0), rgbcolor=(1,0,0),
                 vertical_alignment="bottom" if fi < 0 else "top" )
    Fplot += line( [(xl,yl),(xr,yr)], rgbcolor=(1,0,0) )  # tangent

    show( Fplot+labi, xmin = prange[0], xmax = prange[1] )
    Data = [X, df, Fplot]

attachment:newtraph.png

Coordinate Transformations

by Jason Grout

var('u v')
from sage.ext.fast_eval import fast_float
@interact
def trans(x=input_box(u^2-v^2, label="x=",type=SR), \
    y=input_box(u*v, label="y=",type=SR), \
    t_val=slider(0,10,0.2,6, label="Length of curves"), \
    u_percent=slider(0,1,0.05,label="<font color='red'>u</font>", default=.7),
    v_percent=slider(0,1,0.05,label="<font color='blue'>v</font>", default=.7),
    u_range=input_box(range(-5,5,1), label="u lines"),
    v_range=input_box(range(-5,5,1), label="v lines")):
    thickness=4
    u_val = min(u_range)+(max(u_range)-min(u_range))*u_percent
    v_val = min(v_range)+(max(v_range)-min(v_range))*v_percent
    t_min = -t_val
    t_max = t_val

    g1=sum([parametric_plot((SR(u.subs(u=i))._fast_float_('v'),v.subs(u=i)._fast_float_('v')), t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])
    g2=sum([parametric_plot((u.subs(v=i)._fast_float_('u'),SR(v.subs(v=i))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])
    vline_straight=parametric_plot((SR(u.subs(v=v_val))._fast_float_('u'),SR(v.subs(v=v_val))._fast_float_('u')), t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
    uline_straight=parametric_plot((SR(u.subs(u=u_val))._fast_float_('v'),SR(v.subs(u=u_val))._fast_float_('v')), t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)

    (g1+g2+vline_straight+uline_straight).save("uv_coord.png",aspect_ratio=1, figsize=[5,5], axes_labels=['$u$','$v$'])


    g3=sum([parametric_plot((x.subs(u=i)._fast_float_('v'),y.subs(u=i)._fast_float_('v')),  t_min,t_max, rgbcolor=(1,0,0)) for i in u_range])
    g4=sum([parametric_plot((x.subs(v=i)._fast_float_('u'),y.subs(v=i)._fast_float_('u')),  t_min,t_max, rgbcolor=(0,0,1)) for i in v_range])
    vline=parametric_plot((SR(x.subs(v=v_val))._fast_float_('u'),SR(y.subs(v=v_val))._fast_float_('u')),  t_min,t_max, rgbcolor=(0,0,1), linestyle='-',thickness=thickness)
    uline=parametric_plot((SR(x.subs(u=u_val))._fast_float_('v'),SR(y.subs(u=u_val))._fast_float_('v')),  t_min,t_max,rgbcolor=(1,0,0), linestyle='-',thickness=thickness)
    (g3+g4+vline+uline).save("xy_coord.png", aspect_ratio=1, figsize=[5,5], axes_labels=['$x$','$y$'])


    print jsmath("x=%s, \: y=%s"%(latex(x), latex(y)))
    print "<html><table><tr><td><img src='cell://uv_coord.png'/></td><td><img src='cell://xy_coord.png'/></td></tr></table></html>"

attachment:coordinate-transform-1.png attachment:coordinate-transform-2.png

Taylor Series

by Harald Schilly

var('x')
x0  = 0
f   = sin(x)*e^(-x)
p   = plot(f,-1,5, thickness=2)
dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):
    ft = f.taylor(x,x0,order)
    pt = plot(ft,-1, 5, color='green', thickness=2)
    html('$f(x)\;=\;%s$'%latex(f))
    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
    show(dot + p + pt, ymin = -.5, ymax = 1)

attachment:taylor_series_animated.gif

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

Vector Fields and Euler's Method

by Mike Hansen (tested and updated by William Stein)

x,y = var('x,y')
@interact
def _(f = input_box(default=y), g=input_box(default=-x*y+x^3-x), 
      xmin=input_box(default=-1), xmax=input_box(default=1), 
      ymin=input_box(default=-1), ymax=input_box(default=1), 
      start_x=input_box(default=0.5), start_y=input_box(default=0.5),  
      step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
    old_f = f
    f = f.function(x,y)
    old_g = g
    g = g.function(x,y)
    steps = int(steps)

    points = [ (start_x, start_y) ]
    for i in range(steps):
        xx, yy = points[-1]
        try:
            points.append( (xx+step_size*f(xx,yy), yy+step_size*g(xx,yy)) )
        except (ValueError, ArithmeticError, TypeError):
            break

    starting_point = point(points[0], pointsize=50)
    solution = line(points)
    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )

    result = vector_field + starting_point + solution
    
    html(r"<h2>$ \frac{dx}{dt} = %s$  $ \frac{dy}{dt} = %s$</h2>"%(latex(old_f),latex(old_g)))
    print "Step size: %s"%step_size
    print "Steps: %s"%steps
    result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)

attachment:euler.png

Vector Field with Runga-Kutta-Fehlberg

by Harald Schilly

# Solve ODEs using sophisticated Methods like Runga-Kutta-Fehlberg
# by Harald Schilly, April 2008
# (jacobian doesn't work, please help ...)
var('x y')
@interact
def _(fin = input_box(default=y+exp(x/10)-1/3*((x-1/2)^2+y^3)*x-x*y^3), gin=input_box(default=x^3-x+1/100*exp(y*x^2+x*y^2)-0.7*x), 
      xmin=input_box(default=-1), xmax=input_box(default=1.8), 
      ymin=input_box(default=-1.3), ymax=input_box(default=1.5), 
      x_start=(-1,(-2,2)), y_start=(0,(-2,2)), error=(0.5,(0,1)), 
      t_length=(23,(0, 100)) , num_of_points = (1500,(5,2000)),
      algorithm = selector([
         ("rkf45" , "runga-kutta-felhberg (4,5)"),
         ("rk2" , "embedded runga-kutta (2,3)"),
         ("rk4" , "4th order classical runga-kutta"),
         ("rk8pd" , 'runga-kutta prince-dormand (8,9)'),
         ("rk2imp" , "implicit 2nd order runga-kutta at gaussian points"),
         ("rk4imp" , "implicit 4th order runga-kutta at gaussian points"),
         ("bsimp" , "implicit burlisch-stoer (requires jacobian)"),
         ("gear1" , "M=1 implicit gear"),
         ("gear2" , "M=2 implicit gear")
      ])):
    f(x,y)=fin
    g(x,y)=gin    
    
    ff = f._fast_float_(*f.args())
    gg = g._fast_float_(*g.args())
    
    #solve
    path = []      
    err = error
    xerr = 0
    for yerr in [-err, 0, +err]:
      T=ode_solver()
      T.algorithm=algorithm
      T.function = lambda t, yp: [ff(yp[0],yp[1]), gg(yp[0],yp[1])]  
      T.jacobian = lambda t, yp: [[diff(fun,dval)(yp[0],yp[1]) for dval in [x,y]] for fun in [f,g]]
      T.ode_solve(y_0=[x_start + xerr, y_start + yerr],t_span=[0,t_length],num_points=num_of_points)
      path.append(line([p[1] for p in T.solution]))

    #plot
    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )
    starting_point = point([x_start, y_start], pointsize=50)
    show(vector_field + starting_point + sum(path), aspect_ratio=1, figsize=[8,9])

attachment:ode_runga_kutta.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

Singular value decomposition

by Marshall Hampton

import scipy.linalg as lin
var('t')
def rotell(sig,umat,t,offset=0):
    temp = matrix(umat)*matrix(2,1,[sig[0]*cos(t),sig[1]*sin(t)])
    return [offset+temp[0][0],temp[1][0]]
@interact
def svd_vis(a11=slider(-1,1,.05,1),a12=slider(-1,1,.05,1),a21=slider(-1,1,.05,0),a22=slider(-1,1,.05,1),ofs= selector(['Off','On'],label='offset image from domain')):
    rf_low = RealField(12)
    my_mat = matrix(rf_low,2,2,[a11,a12,a21,a22])
    u,s,vh = lin.svd(my_mat.numpy())
    if ofs == 'On': 
        offset = 3
        fsize = 6
        colors = [(1,0,0),(0,0,1),(1,0,0),(0,0,1)]
    else: 
        offset = 0
        fsize = 5
        colors = [(1,0,0),(0,0,1),(.7,.2,0),(0,.3,.7)]
    vvects = sum([arrow([0,0],matrix(vh).row(i),rgbcolor = colors[i]) for i in (0,1)])    
    uvects = Graphics()
    for i in (0,1):
        if s[i] != 0: uvects += arrow([offset,0],vector([offset,0])+matrix(s*u).column(i),rgbcolor = colors[i+2])
    html('<h3>Singular value decomposition: image of the unit circle and the singular vectors</h3>')
    print jsmath("A = %s  = %s %s %s"%(latex(my_mat), latex(matrix(rf_low,u.tolist())), latex(matrix(rf_low,2,2,[s[0],0,0,s[1]])), latex(matrix(rf_low,vh.tolist())))) 
    image_ell = parametric_plot(rotell(s,u,t, offset),0,2*pi)
    graph_stuff=circle((0,0),1)+image_ell+vvects+uvects
    graph_stuff.set_aspect_ratio(1)
    show(graph_stuff,frame = False,axes=False,figsize=[fsize,fsize])

attachment:svd1.png

Discrete Fourier Transform

by Marshall Hampton

import scipy.fftpack as Fourier
@interact
def discrete_fourier(f = input_box(default=sum([sin(k*x) for k in range(1,5,2)])), scale = slider(.1,20,.1,5)):
    var('x')
    pbegin = -float(pi)*scale
    pend = float(pi)*scale
    html("<h3>Function plot and its discrete Fourier transform</h3>")
    show(plot(f, pbegin, pend, plot_points = 512), figsize = [4,3])
    f_vals = [f(ind) for ind in srange(pbegin, pend,(pend-pbegin)/512.0)]
    my_fft = Fourier.fft(f_vals)
    show(list_plot([abs(x) for x in my_fft], plotjoined=True), figsize = [4,3])

attachment:dfft1.png

Algebra

Groebner fan of an ideal

by Marshall Hampton; (needs sage-2.11 or higher, with gfan-0.3 interface)

@interact
def gfan_browse(p1 = input_box('x^3+y^2',type = str, label='polynomial 1: '), p2 = input_box('y^3+z^2',type = str, label='polynomial 2: '), p3 = input_box('z^3+x^2',type = str, label='polynomial 3: ')):
    R.<x,y,z> = PolynomialRing(QQ,3)
    i1 = ideal(R(p1),R(p2),R(p3))
    gf1 = i1.groebner_fan()
    testr = gf1.render()    
    html('Groebner fan of the ideal generated by: ' + str(p1) + ', ' + str(p2) + ', ' + str(p3))
    show(testr, axes = False, figsize=[8,8*(3^(.5))/2])

attachment:gfan_interact.png

Number Theory

Factor Trees

by William Stein

import random
def ftree(rows, v, i, F):
    if len(v) > 0: # add a row to g at the ith level.
        rows.append(v)
    w = []
    for i in range(len(v)):
        k, _, _ = v[i]
        if k is None or is_prime(k):
            w.append((None,None,None))
        else:
            d = random.choice(divisors(k)[1:-1])
            w.append((d,k,i))
            e = k//d
            if e == 1:
                w.append((None,None))
            else:
                w.append((e,k,i))
    if len(w) > len(v):
        ftree(rows, w, i+1, F)
def draw_ftree(rows,font):
    g = Graphics()
    for i in range(len(rows)):
        cur = rows[i]
        for j in range(len(cur)):
            e, f, k = cur[j]
            if not e is None:
                if is_prime(e):
                     c = (1,0,0)
                else:
                     c = (0,0,.4)
                g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c)
                if not k is None and not f is None:
                    g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], 
                    alpha=0.5)
    return g

@interact
def factor_tree(n=100, font=(10, (8..20)), redraw=['Redraw']):
    n = Integer(n)
    rows = []
    v = [(n,None,0)]
    ftree(rows, v, 0, factor(n))
    show(draw_ftree(rows, font), axes=False)

attachment:factortree.png

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

Fundamental Domains of SL_2(ZZ)

by Robert Miller

L = [[-0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000, -1, -1)]
R = [[0.5, 2.0^(x/100.0) - 1 + sqrt(3.0)/2] for x in xrange(1000)]
xes = [x/1000.0 for x in xrange(-500,501,1)]
M = [[x,abs(sqrt(x^2-1))] for x in xes]
fundamental_domain = L+M+R
fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
@interact
def _(gen = selector(['t+1', 't-1', '-1/t'], nrows=1)):
    global fundamental_domain
    if gen == 't+1':
        fundamental_domain = [[x+1,y] for x,y in fundamental_domain]
    elif gen == 't-1':
        fundamental_domain = [[x-1,y] for x,y in fundamental_domain]
    elif gen == '-1/t':
        new_dom = []
        for x,y in fundamental_domain:
            sq_mod = x^2 + y^2
            new_dom.append([(-1)*x/sq_mod, y/sq_mod])
        fundamental_domain = new_dom
    P = polygon(fundamental_domain)
    P.ymax(1.2); P.ymin(-0.1)
    P.show()

attachment:fund_domain.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)
    if bits>100:
        g = k(2)
    else:
        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

Web applications

Stock Market data, fetched from Yahoo and Google

by William Stein

import urllib

class Day:
    def __init__(self, date, open, high, low, close, volume):
        self.date = date
        self.open=float(open); self.high=float(high); self.low=float(low); self.close=float(close)
        self.volume=int(volume)
    def __repr__(self):
        return '%10s %4.2f %4.2f %4.2f %4.2f %10d'%(self.date, self.open, self.high, 
                   self.low, self.close, self.volume)

class Stock:
    def __init__(self, symbol):
        self.symbol = symbol.upper()

    def __repr__(self):
        return "%s (%s)"%(self.symbol, self.yahoo()['price'])
    
    def yahoo(self):
        url = 'http://finance.yahoo.com/d/quotes.csv?s=%s&f=%s' % (self.symbol, 'l1c1va2xj1b4j4dyekjm3m4rr5p5p6s7')
        values = urllib.urlopen(url).read().strip().strip('"').split(',')
        data = {}
        data['price'] = values[0]
        data['change'] = values[1]
        data['volume'] = values[2]
        data['avg_daily_volume'] = values[3]
        data['stock_exchange'] = values[4]
        data['market_cap'] = values[5]
        data['book_value'] = values[6]
        data['ebitda'] = values[7]
        data['dividend_per_share'] = values[8]
        data['dividend_yield'] = values[9]
        data['earnings_per_share'] = values[10]
        data['52_week_high'] = values[11]
        data['52_week_low'] = values[12]
        data['50day_moving_avg'] = values[13]
        data['200day_moving_avg'] = values[14]
        data['price_earnings_ratio'] = values[15]
        data['price_earnings_growth_ratio'] = values[16]
        data['price_sales_ratio'] = values[17]
        data['price_book_ratio'] = values[18]
        data['short_ratio'] = values[19]
        return data

    def historical(self):
        try:
            return self.__historical
        except AttributeError:
            pass
        symbol = self.symbol
        def get_data(exchange):
             name = get_remote_file('http://finance.google.com/finance/historical?q=%s:%s&output=csv'%(exchange, symbol.upper()), 
                       verbose=False)
             return open(name).read()
        R = get_data('NASDAQ')
        if "Bad Request" in R:
             R = get_data("NYSE")
        R = R.splitlines()
        headings = R[0].split(',')
        self.__historical = []
        try:
            for x in reversed(R[1:]):
                date, opn, high, low, close, volume = x.split(',')
                self.__historical.append(Day(date, opn,high,low,close,volume))
        except ValueError:
             pass
        self.__historical = Sequence(self.__historical,cr=True,universe=lambda x:x)
        return self.__historical

    def plot_average(self, spline_samples=10):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        avg = list(enumerate([(z.high+z.low)/2 for z in d]))
        P = line(avg) + points(avg, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, d[-1].low), horizontal_alignment='right', rgbcolor='black')
        if spline_samples > 0:
            k = 250//spline_samples
            spl = spline([avg[i*k] for i in range(len(d)//k)] + [avg[-1]])
            P += plot(spl, (0,len(d)+30), color=(0.7,0.7,0.7))
        P.xmax(260)
        return P

    def plot_diff(self):
        d = self.historical()
        if len(d) == 0:
            return text('no historical data at Google Finance about %s'%self.symbol, (0,3))
        diff = [] 
        for i in range(1, len(d)):
             z1 = d[i]; z0 = d[i-1]
             diff.append((i, (z1.high+z1.low)/2 - (z0.high + z0.low)/2))
        P = line(diff,thickness=0.5) + points(diff, rgbcolor='black', pointsize=4) + \
                 text(self.symbol, (len(d)*1.05, 0), horizontal_alignment='right', rgbcolor='black')
        P.xmax(260)
        return P

symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort()
stocks = dict([(s,Stock(s)) for s in symbols])

@interact
def data(symbol = symbols, other_symbol='', spline_samples=(8,[0..15])):
     if other_symbol != '':
         symbol = other_symbol
     S = Stock(symbol)
     html('<h1 align=center><font color="darkred">%s</font></h1>'%S)
     S.plot_average(spline_samples).save('avg.png', figsize=[10,2])
     S.plot_diff().save('diff.png', figsize=[10,2])

     Y = S.yahoo()
     k = Y.keys(); k.sort()
     html('Price during last 52 weeks:<br>Grey line is a spline through %s points (do not take seriously!):<br> <img src="cell://avg.png">'%spline_samples)
     html('Difference from previous day:<br> <img src="cell://diff.png">')
     html('<table align=center>' + '\n'.join('<tr><td>%s</td><td>%s</td></tr>'%(k[i], Y[k[i]]) for i in range(len(k))) + '</table>')

attachment:stocks.png

CO2 data plot, fetched from NOAA

by Marshall Hampton

While support for R is rapidly improving, scipy.stats has a lot of useful stuff too. This only scratches the surface.

import urllib2 as U
import scipy.stats as Stat
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
    if a_line.find('Creation:') != -1:
        cdate = a_line
    if a_line[0] != '#':
        temp = a_line.replace('\n','').split(' ')
        temp = [float(q) for q in temp if q != '']
        datalines.append(temp)
trdf = RealField(16)
@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
    htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
    htmls2 = '<h4>'+cdate+'</h4>'
    sel_data = [[q[2],q[4]] for q in datalines if start_date < q[2] < end_date]
    c_max = max([q[1] for q in sel_data])
    c_min = min([q[1] for q in sel_data])
    slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
    html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>')
    var('x,y')
    show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)

attachment:co2c.png

Pie Chart from the Google Chart API

by Harald Schilly

# Google Chart API: http://code.google.com/apis/chart
import urllib2 as inet
from pylab import imshow
@interact
def gChart(title="Google Chart API plots Pie Charts!", color1=Color('purple'), color2=Color('black'), color3=Color('yellow'), val1=slider(0,1,.05,.5), val2=slider(0,1,.05,.3), val3=slider(0,1,.05,0.1), label=("Maths Physics Chemistry")):
    url = "http://chart.apis.google.com/chart?cht=p3&chs=600x300"
    url += '&chtt=%s&chts=000000,25'%title.replace(" ","+")
    url += '&chco=%s'%(','.join([color1.html_color()[1:],color2.html_color()[1:],color3.html_color()[1:]]))
    url += '&chl=%s'%label.replace(" ","|")
    url += '&chd=t:%s'%(','.join(map(str,[val1,val2,val3])))
    print url
    html('<div style="border:3px dashed;text-align:center;padding:50px 0 50px 0"><img src="%s"></div>'%url)

attachment:interact_with_google_chart_api.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
d_field = RealField(10)
@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

Catalog of 3D Parametric Plots

var('u,v')
plots = ['Two Interlinked Tori', 'Star of David', 'Double Heart',
         'Heart', 'Green bowtie', "Boy's Surface", "Maeder's Owl",
         'Cross cap']
plots.sort()

@interact
def _(example=selector(plots, buttons=True, nrows=2),
      tachyon=("Raytrace", False), frame = ('Frame', False),
      opacity=(1,(0.1,1))):
    url = ''
    if example == 'Two Interlinked Tori':
        f1 = (4+(3+cos(v))*sin(u), 4+(3+cos(v))*cos(u), 4+sin(v))
        f2 = (8+(3+cos(v))*cos(u), 3+sin(v), 4+(3+cos(v))*sin(u))
        p1 = parametric_plot3d(f1, (u,0,2*pi), (v,0,2*pi), color="red", opacity=opacity)
        p2 = parametric_plot3d(f2, (u,0,2*pi), (v,0,2*pi), color="blue",opacity=opacity)
        P = p1 + p2
    elif example == 'Star of David':
        f_x = cos(u)*cos(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_y = cos(u)*sin(v)*(abs(cos(3*v/4))^500 + abs(sin(3*v/4))^500)^(-1/260)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        f_z = sin(u)*(abs(cos(4*u/4))^200 + abs(sin(4*u/4))^200)^(-1/200)
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, 0, 2*pi),opacity=opacity)
    elif example == 'Double Heart':
        f_x = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) + abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*sin(v)
        f_y = ( abs(v) - abs(u) - abs(tanh((1/sqrt(2))*u)/(1/sqrt(2))) - abs(tanh((1/sqrt(2))*v)/(1/sqrt(2))) )*cos(v)
        f_z = sin(u)*(abs(cos(4*u/4))^1 + abs(sin(4*u/4))^1)^(-1/1)
        P = parametric_plot3d([f_x, f_y, f_z], (u, 0, pi), (v, -pi, pi),opacity=opacity)
    elif example == 'Heart':
        f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u))
        f_z = v
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red",opacity=opacity)
    elif example == 'Green bowtie':
        f_x = sin(u) / (sqrt(2) + sin(v))
        f_y = sin(u) / (sqrt(2) + cos(v))
        f_z = cos(u) / (1 + sqrt(2))
        P = parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -pi, pi), frame=False, color="green",opacity=opacity)
    elif example == "Boy's Surface":
        url = "http://en.wikipedia.org/wiki/Boy's_surface"
        fx = 2/3* (cos(u)* cos(2*v) + sqrt(2)* sin(u)* cos(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fy = 2/3* (cos(u)* sin(2*v) - sqrt(2)* sin(u)* sin(v))* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        fz = sqrt(2)* cos(u)* cos(u) / (sqrt(2) - sin(2*u)* sin(3*v))
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, pi), plot_points = [90,90], frame=False, color="orange",opacity=opacity) 
    elif example == "Maeder's Owl":
        fx = v *cos(u) - 0.5* v^2 * cos(2* u)
        fy = -v *sin(u) - 0.5* v^2 * sin(2* u)
        fz = 4 *v^1.5 * cos(3 *u / 2) / 3
        P = parametric_plot3d([fx, fy, fz], (u, -2*pi, 2*pi), (v, 0, 1),plot_points = [90,90], frame=False, color="purple",opacity=opacity)
    elif example =='Cross cap':
        url = 'http://en.wikipedia.org/wiki/Cross-cap'
        fx = (1+cos(v))*cos(u)
        fy = (1+cos(v))*sin(u)
        fz = -tanh((2/3)*(u-pi))*sin(v)
        P = parametric_plot3d([fx, fy, fz], (u, 0, 2*pi), (v, 0, 2*pi), frame=False, color="red",opacity=opacity)
    else:
        print "Bug selecting plot?"
        return


    html('<h2>%s</h2>'%example)
    if url:
        html('<h3><a target="_new" href="%s">%s</a></h3>'%(url,url))
    show(P, viewer='tachyon' if tachyon else 'jmol', frame=frame)

attachment:parametricplot3d.png

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

Interactive 3d plotting

var('x,y')
@interact
def example(clr=Color('orange'), f=4*x*exp(-x^2-y^2), xrange='(-2, 2)', yrange='(-2,2)', 
    zrot=(0,pi), xrot=(0,pi), zoom=(1,(1/2,3)), square_aspect=('Square Frame', False),
    tachyon=('Ray Tracer', True)):
    xmin, xmax = sage_eval(xrange); ymin, ymax = sage_eval(yrange)
    P = plot3d(f, (x, xmin, xmax), (y, ymin, ymax), color=clr)
    html('<h1>Plot of $f(x,y) = %s$</h1>'%latex(f))
    aspect_ratio = [1,1,1] if square_aspect else [1,1,1/2]
    show(P.rotate((0,0,1), -zrot).rotate((1,0,0),xrot), 
         viewer='tachyon' if tachyon else 'jmol', 
         figsize=6, zoom=zoom, frame=False,
         frame_aspect_ratio=aspect_ratio)

attachment:tachyonplot3d.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

-  ⇤ ← Revision 81 as of 2008-04-14 09:20:09 → 
  Size: 43246
  Editor: MikeHansen
  Comment:
+   ← Revision 93 as of 2008-05-07 13:08:35 → ⇥
  Size: 48445
  Editor: schilly
  Comment:
-Deletions are marked like this.
+Additions are marked like this.
 Line 8:
+== Interact Examples ==
 * [:interact/graph_theory:]
-Line 41:
+Line 44:
-== 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
+=== 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
-Line 349:
+Line 373:
+=== Taylor Series ===
by Harald Schilly
{{{
var('x')
x0  = 0
f   = sin(x)*e^(-x)
p   = plot(f,-1,5, thickness=2)
dot = point((x0,f(x0)),pointsize=80,rgbcolor=(1,0,0))
@interact
def _(order=(1..12)):
    ft = f.taylor(x,x0,order)
    pt = plot(ft,-1, 5, color='green', thickness=2)
    html('$f(x)\;=\;%s$'%latex(f))
    html('$\hat{f}(x;%s)\;=\;%s+\mathcal{O}(x^{%s})$'%(x0,latex(ft),order+1))
    show(dot + p + pt, ymin = -.5, ymax = 1)
}}}
attachment:taylor_series_animated.gif
-Line 393:
+Line 435:
-by Mike Hansen
{{{
x,y = var('x,y')    
@interact
def _(f = input_box(default=x), g=input_box(default=y), xmin=input_box(default=-1), xmax=input_box(default=1), ymin=input_box(default=-1), ymax=input_box(default=1), start_x=input_box(default=0), start_y=input_box(default=0),  step_size=(0.01,(0.001, 0.2)), steps=(200,(0, 1000)) ):
+by Mike Hansen (tested and updated by William Stein)
{{{
x,y = var('x,y')
@interact
def _(f = input_box(default=y), g=input_box(default=-x*y+x^3-x), 
      xmin=input_box(default=-1), xmax=input_box(default=1), 
      ymin=input_box(default=-1), ymax=input_box(default=1), 
      start_x=input_box(default=0.5), start_y=input_box(default=0.5),  
      step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
-Line 407:
+Line 453:
-        points.append( (xx+step_size*f(xx,yy), yy+step_size*g(xx,yy)) )

    starting_point = point(points[0])
+        try:
            points.append( (xx+step_size*f(xx,yy), yy+step_size*g(xx,yy)) )
        except (ValueError, ArithmeticError, TypeError):
            break

    starting_point = point(points[0], pointsize=50)
-Line 421:
+Line 470:
+=== Vector Field with Runga-Kutta-Fehlberg ===
by Harald Schilly
{{{
# Solve ODEs using sophisticated Methods like Runga-Kutta-Fehlberg
# by Harald Schilly, April 2008
# (jacobian doesn't work, please help ...)
var('x y')
@interact
def _(fin = input_box(default=y+exp(x/10)-1/3*((x-1/2)^2+y^3)*x-x*y^3), gin=input_box(default=x^3-x+1/100*exp(y*x^2+x*y^2)-0.7*x), 
      xmin=input_box(default=-1), xmax=input_box(default=1.8), 
      ymin=input_box(default=-1.3), ymax=input_box(default=1.5), 
      x_start=(-1,(-2,2)), y_start=(0,(-2,2)), error=(0.5,(0,1)), 
      t_length=(23,(0, 100)) , num_of_points = (1500,(5,2000)),
      algorithm = selector([
         ("rkf45" , "runga-kutta-felhberg (4,5)"),
         ("rk2" , "embedded runga-kutta (2,3)"),
         ("rk4" , "4th order classical runga-kutta"),
         ("rk8pd" , 'runga-kutta prince-dormand (8,9)'),
         ("rk2imp" , "implicit 2nd order runga-kutta at gaussian points"),
         ("rk4imp" , "implicit 4th order runga-kutta at gaussian points"),
         ("bsimp" , "implicit burlisch-stoer (requires jacobian)"),
         ("gear1" , "M=1 implicit gear"),
         ("gear2" , "M=2 implicit gear")
      ])):
    f(x,y)=fin
    g(x,y)=gin    
    
    ff = f._fast_float_(*f.args())
    gg = g._fast_float_(*g.args())
    
    #solve
    path = []      
    err = error
    xerr = 0
    for yerr in [-err, 0, +err]:
      T=ode_solver()
      T.algorithm=algorithm
      T.function = lambda t, yp: [ff(yp[0],yp[1]), gg(yp[0],yp[1])]  
      T.jacobian = lambda t, yp: [[diff(fun,dval)(yp[0],yp[1]) for dval in [x,y]] for fun in [f,g]]
      T.ode_solve(y_0=[x_start + xerr, y_start + yerr],t_span=[0,t_length],num_points=num_of_points)
      path.append(line([p[1] for p in T.solution]))

    #plot
    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )
    starting_point = point([x_start, y_start], pointsize=50)
    show(vector_field + starting_point + sum(path), aspect_ratio=1, figsize=[8,9])
}}}
attachment:ode_runga_kutta.png
-Line 569:
+Line 667:
+=== Factor Trees ===
by William Stein
{{{
import random
def ftree(rows, v, i, F):
    if len(v) > 0: # add a row to g at the ith level.
        rows.append(v)
    w = []
    for i in range(len(v)):
        k, _, _ = v[i]
        if k is None or is_prime(k):
            w.append((None,None,None))
        else:
            d = random.choice(divisors(k)[1:-1])
            w.append((d,k,i))
            e = k//d
            if e == 1:
                w.append((None,None))
            else:
                w.append((e,k,i))
    if len(w) > len(v):
        ftree(rows, w, i+1, F)
def draw_ftree(rows,font):
    g = Graphics()
    for i in range(len(rows)):
        cur = rows[i]
        for j in range(len(cur)):
            e, f, k = cur[j]
            if not e is None:
                if is_prime(e):
                     c = (1,0,0)
                else:
                     c = (0,0,.4)
                g += text(str(e), (j*2-len(cur),-i), fontsize=font, rgbcolor=c)
                if not k is None and not f is None:
                    g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], 
                    alpha=0.5)
    return g

@interact
def factor_tree(n=100, font=(10, (8..20)), redraw=['Redraw']):
    n = Integer(n)
    rows = []
    v = [(n,None,0)]
    ftree(rows, v, 0, factor(n))
    show(draw_ftree(rows, font), axes=False)
}}}
attachment:factortree.png
-Line 878:
+Line 1025:
+While support for R is rapidly improving, scipy.stats has a lot of useful stuff too.  This only scratches the surface.
-Line 880:
+Line 1029:
-@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009), Update = selector(['Update'], buttons=True, label = '')):
    co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
    datalines = []
    for a_line in co2data:
        if a_line.find('Creation:') != -1:
            cdate = a_line
        if a_line[0] != '#':
            temp = a_line.replace('\n','').split(' ')
            temp = [float(q) for q in temp if q != '']
            datalines.append(temp)
    html('<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>')
    html('<h4>'+cdate+'</h4>')
+import scipy.stats as Stat
co2data = U.urlopen('ftp://ftp.cmdl.noaa.gov/ccg/co2/trends/co2_mm_mlo.txt').readlines()
datalines = []
for a_line in co2data:
    if a_line.find('Creation:') != -1:
        cdate = a_line
    if a_line[0] != '#':
        temp = a_line.replace('\n','').split(' ')
        temp = [float(q) for q in temp if q != '']
        datalines.append(temp)
trdf = RealField(16)
@interact
def mauna_loa_co2(start_date = slider(1958,2010,1,1958), end_date = slider(1958, 2010,1,2009)):
    htmls1 = '<h3>CO2 monthly averages at Mauna Loa (interpolated), from NOAA/ESRL data</h3>'
    htmls2 = '<h4>'+cdate+'</h4>'
-Line 896:
+Line 1047:
-    show(list_plot([[q[2],q[4]] for q in datalines], plotjoined=True, rgbcolor=(1,0,0)), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)
}}}
attachment:mauna_loa_co2.png
+    slope, intercept, r, ttprob, stderr = Stat.linregress(sel_data)
    html(htmls1+htmls2+'<h4>Linear regression slope: ' + str(trdf(slope)) + ' ppm/year; correlation coefficient: ' + str(trdf(r)) + '</h4>')
    var('x,y')
    show(list_plot(sel_data, plotjoined=True, rgbcolor=(1,0,0)) + plot(slope*x+intercept,start_date,end_date), xmin = start_date, ymin = c_min-2, axes = True, xmax = end_date, ymax = c_max+3, frame = False)
}}}
attachment:co2c.png

Diff for "interact"

Sage Interactions

Interact Examples

Miscellaneous

Profile a snippet of code

Evaluate a bit of code in a given system

A Random Walk

3D Random Walk

Calculus

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

A simple tangent line grapher

Function tool

Newton-Raphson Root Finding

Coordinate Transformations

Taylor Series

Differential Equations

Euler's Method in one variable

Vector Fields and Euler's Method

Vector Field with Runga-Kutta-Fehlberg

Linear Algebra

Numerical instability of the classical Gram-Schmidt algorithm

Linear transformations

Singular value decomposition

Discrete Fourier Transform

Algebra

Groebner fan of an ideal

Number Theory

Factor Trees

Continued Fraction Plotter

Illustrating the prime number thoerem

Computing Generalized Bernoulli Numbers

Fundamental Domains of SL_2(ZZ)

Computing modular forms

Computing the cuspidal subgroup

A Charpoly and Hecke Operator Graph

Demonstrating the Diffie-Hellman Key Exchange Protocol

Plotting an elliptic curve over a finite field

Web applications

Stock Market data, fetched from Yahoo and Google

CO2 data plot, fetched from NOAA

Pie Chart from the Google Chart API

Bioinformatics

Web app: protein browser

Coalescent simulator

Miscellaneous Graphics

Catalog of 3D Parametric Plots

Interactive rotatable raytracing with Tachyon3d

Interactive 3d plotting

Somewhat Silly Egg Painter