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Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions:here] or email [email protected] . | |
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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. | * [:interact/graph theory:Graph Theory] * [:interact/calculus:Calculus] * [:interact/diffeq:Differential Equations] * [:interact/linear algebra:Linear Algebra] * [:interact/algebra:Algebra] * [:interact/number theory:Number Theory] * [:interact/web:Web Applications] * [:interact/bio:Bioinformatics] * [:interact/graphics:Drawing Graphics] * [:interact/misc:Miscellaneous] |
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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. | |
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== Graphics == | == Example: Taylor Series == |
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== 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 == Linear Algebra == === Numerical instability of the classical Gram-Schmidt algorithm === {{{ 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) @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) print 'precision in bits: ' + str(precision) A = matrix([a1,a2,a3]) A = [vector(myR,x) for x in A] qn, rn = GS_classic(A) qb, rb = GS_modern(A) show(matrix(displayR,qn)), show(matrix(displayR,qb)) }}} attachment:GramSchmidt.png == Number Theory == === 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 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 |
This is the code and a mockup animation of the interact command. It defines a slider, seen on top, that can be dragged. Once dragged, it changes the value of the variable "order" and the whole block of code gets evaluated. This principle can be seen in various examples presented on the pages above! |
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# Note -- in Sage-2.10.3; multiedges are missing in plots; loops are missing in 3d plots | 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)) |
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def f(N = prime_range(11,400), p = selector(prime_range(2,12),nrows=1), three_d = ("Three Dimensional", False)): S = SupersingularModule(N) T = S.hecke_matrix(p) G = Graph(T, multiedges=True, loops=not three_d) html("<h1>Charpoly and Hecke Graph: Level %s, T_%s</h1>"%(N,p)) show(T.charpoly().factor()) if three_d: show(G.plot3d(), aspect_ratio=[1,1,1]) else: show(G.plot(),figsize=7) |
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) |
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attachment:heckegraph.png === Demonstrating the Diffie-Hellman Key Exchange Protocol === by Timothy Clemans (refereed by William Stein) {{{ @interact def diffie_hellman(button=selector(["New example"],label='',buttons=True), bits=("Number of bits of prime", (8,12,..512))): maxp = 2^bits p = random_prime(maxp) k = GF(p) g = k.multiplicative_generator() a = ZZ.random_element(10, maxp) b = ZZ.random_element(10, maxp) print """ <html> <style> .gamodp { background:yellow } .gbmodp { background:orange } .dhsame { color:green; font-weight:bold } </style> <h2>%s-Bit Diffie-Hellman Key Exchange</h2> <ol style="color:#000;font:12px Arial, Helvetica, sans-serif"> <li>Alice and Bob agree to use the prime number p=%s and base g=%s.</li> <li>Alice chooses the secret integer a=%s, then sends Bob (<span class="gamodp">g<sup>a</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gamodp">%s</span>.</li> <li>Bob chooses the secret integer b=%s, then sends Alice (<span class="gbmodp">g<sup>b</sup> mod p</span>):<br/>%s<sup>%s</sup> mod %s = <span class="gbmodp">%s</span>.</li> <li>Alice computes (<span class="gbmodp">g<sup>b</sup> mod p</span>)<sup>a</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li> <li>Bob computes (<span class="gamodp">g<sup>a</sup> mod p</span>)<sup>b</sup> mod p:<br/>%s<sup>%s</sup> mod %s = <span class="dhsame">%s</span>.</li> </ol></html> """ % (bits, p, g, a, g, a, p, (g^a), b, g, b, p, (g^b), (g^b), a, p, (g^ b)^a, g^a, b, p, (g^a)^b) }}} attachment:dh.png === Somewhat Silly Egg Painter === by Marshall Hampton (refereed by William Stein) {{{ var('s,t') g(s) = ((0.57496*sqrt(121 - 16.0*s^2))/sqrt(10.+ s)) def P(color, rng): return parametric_plot3d((cos(t)*g(s), sin(t)*g(s), s), (s,rng[0],rng[1]), (t,0,2*pi), plot_points = [150,150], rgbcolor=color, frame = False, opacity = 1) colorlist = ['red','blue','red','blue'] @interact def _(band_number = selector(range(1,5)), current_color = Color('red')): html('<h1 align=center>Egg Painter</h1>') colorlist[band_number-1] = current_color egg = sum([P(colorlist[i],[-2.75+5.5*(i/4),-2.75+5.5*(i+1)/4]) for i in range(4)]) show(egg) }}} attachment:eggpaint.png |
attachment:taylor_series_animated.gif |
Sage Interactions
Post code that demonstrates the use of the interact command in Sage here. It should be easy to just scroll through and paste examples out of here into their own sage notebooks.If you have suggestions on how to improve interact, add them [:interactSuggestions:here] or email [email protected] .
- [:interact/graph theory:Graph Theory]
- [:interact/calculus:Calculus]
- [:interact/diffeq:Differential Equations]
- [:interact/linear algebra:Linear Algebra]
- [:interact/algebra:Algebra]
- [:interact/number theory:Number Theory]
- [:interact/web:Web Applications]
- [:interact/bio:Bioinformatics]
- [:interact/graphics:Drawing Graphics]
- [:interact/misc:Miscellaneous]
Example: Taylor Series
This is the code and a mockup animation of the interact command. It defines a slider, seen on top, that can be dragged. Once dragged, it changes the value of the variable "order" and the whole block of code gets evaluated. This principle can be seen in various examples presented on the pages above!
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