.. MODULEAUTHOR:: Franco Saliola <saliola at gmail.com> and Sébastien Labbé <slabbe at gmail.com>
Integer division in Python :
sage: %python sage: 2/3 + 4/5 + 1/7 0
in Sage:
sage: 2/3 + 4/5 + 1/7 169/105
Exponent (^) in Python :
sage: %python sage: 10^14 #exclusive OR 4
in Sage :
sage: 10^14 100000000000000
The preparser
sage: preparse('2/3 + 2^3 + 3.0')
"Integer(2)/Integer(3) + Integer(2)**Integer(3) + RealNumber('3.0')"
sage: preparse('2^3')
'Integer(2)**Integer(3)'
sage: f = sin(1/x) sage: P = plot(f, -10, 10, color='red') sage: P
sage: Q = line([(3,0.9), (7,0.9), (7,1.1), (3,1.1), (3,0.9)], color='green') sage: Q
sage: R = text('$f(x) = \\sin(\\frac{1}{x})$', (5,1))
sage: R
sage: Q + R + P
L'outil interact (exemples tirés du wiki de Sage : http://wiki.sagemath.org/)
by Marshall Hampton.
sage: %hide
sage: npi = RDF(pi)
sage: from math import cos,sin
sage: def rot(t):
....: return matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
sage: def pursuit(n,x0,y0,lamb,steps = 100, threshold = .01):
....: paths = [[[x0,y0]]]
....: for i in range(1,n):
....: rx,ry = list(rot(2*npi*i/n)*vector([x0,y0]))
....: paths.append([[rx,ry]])
....: oldpath = [x[-1] for x in paths]
....: for q in range(steps):
....: diffs = [[oldpath[(j+1)%n][0]-oldpath[j][0],oldpath[(j+1)%n][1]-oldpath[j][1]] for j in range(n)]
....: npath = [[oldpath[j][0]+lamb*diffs[j][0],oldpath[j][1]+lamb*diffs[j][1]] for j in range(n)]
....: for j in range(n):
....: paths[j].append(npath[j])
....: oldpath = npath
....: return paths
sage: html('<h3>Curves of Pursuit</h3>')
sage: @interact
sage: def curves_of_pursuit(n = slider([2..20],default = 5, label="# of points"),steps = slider([floor(1.4^i) for i in range(2,18)],default = 10, label="# of steps"), stepsize = slider(srange(.01,1,.01),default = .2, label="stepsize"), colorize = selector(['BW','Line color', 'Filled'],default = 'BW')):
....: outpaths = pursuit(n,0,1,stepsize, steps = steps)
....: mcolor = (0,0,0)
....: outer = line([q[0] for q in outpaths]+[outpaths[0][0]], rgbcolor = mcolor)
....: polys = Graphics()
....: if colorize=='Line color':
....: colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))]
....: elif colorize == 'BW':
....: colors = [(0,0,0) for j in range(len(outpaths[0]))]
....: else:
....: colors = [hue(j/steps,1,1) for j in range(len(outpaths[0]))]
....: polys = sum([polygon([outpaths[(i+1)%n][j+1],outpaths[(i+1)%n][j], outpaths[i][j+1]], rgbcolor = colors[j]) for i in range(n) for j in range(len(outpaths[0])-1)])
....: #polys = polys[0]
....: colors = [(0,0,0) for j in range(len(outpaths[0]))]
....: nested = sum([line([q[j] for q in outpaths]+[outpaths[0][j]], rgbcolor = colors[j]) for j in range(len(outpaths[0]))])
....: lpaths = [line(x, rgbcolor = mcolor) for x in outpaths]
....: show(sum(lpaths)+nested+polys, axes = False, figsize = [5,5], xmin = -1, xmax = 1, ymin = -1, ymax =1)
by William Stein
sage: %hide sage: import random sage: 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) sage: 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 e is not 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 k is not None and f is not None: ....: g += line([(j*2-len(cur),-i), ((k*2)-len(rows[i-1]),-i+1)], ....: alpha=0.5) ....: return g sage: @interact sage: 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)
by William Stein
sage: @interact sage: def _(N=(100,(2..2000))): ....: html(r"<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'))
by William Stein
sage: %hide
sage: import urllib
sage: 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)
sage: 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
sage: symbols = ['bsc', 'vmw', 'sbux', 'aapl', 'amzn', 'goog', 'wfmi', 'msft', 'yhoo', 'ebay', 'java', 'rht', ]; symbols.sort()
sage: stocks = dict([(s,Stock(s)) for s in symbols])
sage: @interact
sage: 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 = sorted(Y)
....: 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>')
by Timothy Clemans and William Stein
sage: @interact
sage: def diffie_hellman(bits=slider(8, 513, 4, 8, 'Number of bits', False),
....: button=selector(["Show new example"],label='',buttons=True)):
....: 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, .gbmodp {
....: color:#000;
....: padding:5px
....: }
....: .gamodp {
....: background:#846FD8
....: }
....: .gbmodp {
....: background:#FFFC73
....: }
....: .dhsame {
....: color:#000;
....: font-weight:bold
....: }
....: </style>
....: <h2 style="color:#000;font-family:Arial, Helvetica, sans-serif">%s-Bit Diffie-Hellman Key Exchange</h2>
....: <ol style="color:#000;font-family: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)
Dessiner une fonction \mathbb{R}^2\mapsto \mathbb{R} : la commande plot3d
sage: def f(x, y): ....: return x^2 + y^2 sage: plot3d(f, (-10,10), (-10,10), viewer='tachyon')
sage: a = animate([sin(x + float(k)) for k in srange(0,2*pi,0.3)], xmin=0, xmax=2*pi, figsize=[2,1])
sage: a.show()
sage: f(x) = x^4 - 1
sage: complex_plot(f, (-2,2), (-2,2))
sage: def newton(f, z, precision=0.001) : ....: while abs(f(x=z)) >= precision: ....: z = z - f(x=z) / diff(f)(x=z) ....: return z
sage: complex_plot(lambda z : newton(f, z), (-1,1), (-1,1))
- Pour évaluer une saisie dans le Notebook de Sage, tapez la saisie dans une cellule et faites shift-entrée ou cliquer le lien evaluate
- . Essayez-le maintenant avec une expression simple (e.g., 2 + 2 ). La première évaluation d'une cellule prend plus de temps que les fois suivantes, car un processus commence.
sage: 2+3 5
sage: 4+5 9
Créez de nouvelles cellules de saisie en cliquant sur la ligne bleue qui apparaît entre les cellules lorsque vous déplacez la souris. Essayez-le maintenant.
Vous pouvez rééditer n'importe quelle cellule en cliquant dessus (ou en utilisant les flèches du clavier). Retournez plus haut et changez votre 2 + 2 en un 3 + 3 et réévaluez la cellule.
Vous pouvez aussi éditer ce texte-ci en double cliquant dessus ce qui fera apparaître un éditeur de texte TinyMCE Javascript. Vous pouvez même ajouter des expressions mathématiques telles que \sin(x) - y^3 comme avec LaTeX.
\int e^x dx = e^x + c
Vous trouvez la liste des fonctions que vous pouvez appelez sur un objet X en tappant X.<touche de tabulation> .
sage: X = 2009
- Écrivez X.
- et appuyez sur la touche de tabulation.
sage: X.factor() 7^2 * 41
Une fois que vous avez sélectionné une fonction, disons factor, tappez X.factor(<touche de tabulation> ou X.factor?<touche de tabulation> pour *obtenir de l'aide et des exemples* d'utilisation de cette fonction. Essayez-le maintenant avec X.factor .
sage: 4+5 9
- Pour obtenir l'aide complète et un tutoriel plus exhaustif, cliquez sur le lien Help
- en haut à droite de cette page, et cliquer ensuite sur Fast Static Versions of the Documentation.
sage: a,b,c,d,X = var('a,b,c,d,X')
sage: s = solve(a*X^2 + b*X + c == 0, X) sage: show(s)
\left[X = -\frac{b + \sqrt{-4 \, a c + b^{2}}}{2 \, a}, X = -\frac{b - \sqrt{-4 \, a c + b^{2}}}{2 \, a}\right]
sage: s = solve(a*X^3 + b*X^2 + c*X + d == 0, X) sage: show(s[0])
X = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3} - b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2 \, b^{3}\right)} d} \sqrt{3}}{18 \, a^{2}} - \frac{27 \, a^{2} d - 9 \, a b c + 2 \, b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} - \frac{b}{3 \, a} + \frac{{\left(-i \, \sqrt{3} + 1\right)} {\left(3 \, a c - b^{2}\right)}}{18 \, {\left(\frac{\sqrt{27 \, a^{2} d^{2} + 4 \, a c^{3} - b^{2} c^{2} - 2 \, {\left(9 \, a b c - 2 \, b^{3}\right)} d} \sqrt{3}}{18 \, a^{2}} - \frac{27 \, a^{2} d - 9 \, a b c + 2 \, b^{3}}{54 \, a^{3}}\right)}^{\left(\frac{1}{3}\right)} a^{2}}
sage: A = matrix(3, [9,4,2,4,6,1,6,4,3,2,3,4,2,7,8,6,5,3]); A [9 4 2 4 6 1] [6 4 3 2 3 4] [2 7 8 6 5 3]
sage: show(A)
\left(\begin{array}{rrrrrr}
9 & 4 & 2 & 4 & 6 & 1 \\
6 & 4 & 3 & 2 & 3 & 4 \\
2 & 7 & 8 & 6 & 5 & 3
\end{array}\right)
sage: latex(A)
\left(\begin{array}{rrrrrr}
9 & 4 & 2 & 4 & 6 & 1 \\
6 & 4 & 3 & 2 & 3 & 4 \\
2 & 7 & 8 & 6 & 5 & 3
\end{array}\right)
sage: r = random_matrix(ZZ, 200) sage: r[0] (6, 1, -4, 1, 3, 2, 0, 4, 1, 2, 1, -2, 0, 3, 1, 5, 0, 0, 3, -4, 68, 4, -1, -29, 2, 0, 1, 2, 4, -1, 1, 0, 1, 0, -22, 0, -2, 0, -1, -1, -3, -1, 0, 1, 1, 1, -32, 1, -1, -1, 0, 5, -1, -13, 0, 2, -1, -50, -1, 0, 16, 1, 1, -5, 0, -5, -3, -1, 1, 0, 1, -6, 0, -1, 1, 1, 0, 3, 0, -2, 1, 3, 0, 2, 5, -5, 3, 0, -9, 3, -1, 5, 0, -1, -1, 3, 0, 2, 0, 1, 0, 3, -1, 0, 0, 1, 0, -1, 0, 0, -7, 1, 0, 0, -3, -1, 12, 1, 0, -74, 1, 1, 0, 1, 1164, 21, -109, -5, -2, 1, 1, 3, -30, 17, -28, 1, 1, 161, -4, 1, 10, 2, -1, -1, 4, -6, 0, 17, 0, 25, -1, -1, -1, 0, -2, -1, -1, -1, 1, -6, -1, -2, 1, 2, -1, 0, -6, 1, -3, -1, 6, 0, -3, 0, -4, -1, 1, 1, 12, -7, -1, 1, -1, -1, 1, 2, 2, -25, -2, -1, 0, -1, 2, 3, 1, -3, 12, -10, 1, 0)
sage: time r.determinant() -1529834725553757938763159502025548590567911254662803196770598603331067849864395053736435397051765374245101197807489393057663130380141963203671083430967372792929619229867512126727684265591250414807452250453734959591879530432065001775694429765051483913590921267567927871370268065203061006918276079882798699436138525602103991441803398564880661084453659955387439288542429758896771118012008221672140101768416901702596791928059352838737552772934612946211933401613477671553715592 Time: CPU 0.45 s, Wall: 0.73 s
sage: r.determinant?
sage: D = graphs.DodecahedralGraph() sage: D.show()
sage: D.show3d(viewer='tachyon')
sage: D.chromatic_polynomial() x^20 - 30*x^19 + 435*x^18 - 4060*x^17 + 27393*x^16 - 142194*x^15 + 589875*x^14 - 2004600*x^13 + 5673571*x^12 - 13518806*x^11 + 27292965*x^10 - 46805540*x^9 + 68090965*x^8 - 83530946*x^7 + 85371335*x^6 - 71159652*x^5 + 46655060*x^4 - 22594964*x^3 + 7171160*x^2 - 1111968*x
sage: graph_editor(D);
sage: D.show()
sage: sloane_find([1,5,29,169],1) Searching Sloane's online database... []
sage: sloane_find([1,2,3,4,5,6],1) Searching Sloane's online database... []
The Sage notebook allows transparently editing and compiling Cython code simply by typing %cython at the top of a cell and evaluate it. Variables and functions defined in a Cython cell are imported into the running session.
Here is some simple Python code to numerically integrate the function f(x) = x^2.
sage: from math import sin sage: def f(x): ....: return sin(x**2) ....: sage: def integrate_f_py(a, b, N): ....: s = 0 ....: dx = (b-a)/N ....: for i in range(N): ....: s += f(a+i*dx) ....: return s * dx
sage: timeit('integrate_f_py(0, 1, 1000)', number=50)
50 loops, best of 3: 18.5 ms per loop
Simply compiling this in Cython gives a speedup.
sage: %cython sage: from math import sin sage: def f(x): ....: return sin(x**2) ....: sage: def integrate_f_cy0(a, b, N): ....: s = 0 ....: dx = (b-a)/N ....: for i in range(N): ....: s += f(a+i*dx) ....: return s * dx
sage: timeit('integrate_f_cy0(0, 1, 1000)', number=50)
50 loops, best of 3: 16.7 ms per loop
Adding some static type declarations makes a much greater difference.
sage: %cython sage: from math import sin sage: def f(double x): ....: return sin(x**2) ....: sage: def integrate_f_cy(double a, double b, int N): ....: cdef int i ....: cdef double s, dx ....: s = 0 ....: dx = (b-a)/N ....: for i in range(N): ....: s += f(a+i*dx) ....: return s * dx
sage: timeit('integrate_f_cy(0, 1, 1000)')
625 loops, best of 3: 489 µs per loop
sage: 18500 /489.0 37.8323108384458
Here is a Python function that computes the sum of the first n positive integers.
sage: def mysum_py(n): ....: s = 0 ....: for k in range(n): ....: s += k ....: return s
sage: time mysum_py(10^6) 499999500000 Time: CPU 2.09 s, Wall: 2.16 s
Simply compiling this function with Cython provides a speedup.
sage: %cython sage: def mysum_cy0(n): ....: s = 0 ....: for k in range(n): ....: s += k ....: return s
sage: time mysum_cy0(10^6) 499999500000L Time: CPU 0.25 s, Wall: 0.27 s
sage: 2.09 / 0.25 8.36000000000000
Adding some static type declarations makes a much greater difference.
sage: %cython sage: def mysum_cy1(n): ....: cdef int k ....: cdef long long s ....: ....: s = 0 ....: for k in range(n): ....: s += k ....: return s
sage: time mysum_cy1(10^6) 499999500000L Time: CPU 0.00 s, Wall: 0.00 s
sage: 2.09 / 0.00 +infinity
sage: timeit('mysum_cy1(10^6)')
125 loops, best of 3: 1.57 ms per loop
sage: 2.09/0.00157 1331.21019108280