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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. | 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]. |
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== Interact Examples == * [:interact/graph_theory:] |
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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 |
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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 |
=== 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 |
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=== 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})$'%(x0,latex(ft),order+1)) show(dot + p + pt, ymin = -.5, ymax = 1) }}} attachment:taylor_series_animated.gif |
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=== Vector Fields and Euler's Method === by Mike Hansen (tested and updated by William Stein) {{{ x,y = var('x,y') @interact def _(f = input_box(default=y), g=input_box(default=-x*y+x^3-x), xmin=input_box(default=-1), xmax=input_box(default=1), ymin=input_box(default=-1), ymax=input_box(default=1), start_x=input_box(default=0.5), start_y=input_box(default=0.5), step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ): old_f = f f = f.function(x,y) old_g = g g = g.function(x,y) steps = int(steps) points = [ (start_x, start_y) ] for i in range(steps): xx, yy = points[-1] try: points.append( (xx+step_size*f(xx,yy), yy+step_size*g(xx,yy)) ) except (ValueError, ArithmeticError, TypeError): break starting_point = point(points[0], pointsize=50) solution = line(points) vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) ) result = vector_field + starting_point + solution html(r"<h2>$ \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 |
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=== 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 |
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=== 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 |
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=== 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 |
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g = k.multiplicative_generator() | if bits>100: g = k(2) else: g = k.multiplicative_generator() |
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== 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 |
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d_field = RealField(10) | |
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=== 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 |
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=== 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 |
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].
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
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