# Ticket #10983: trac_10983.patch

File trac_10983.patch, 31.8 KB (added by casamayou, 7 years ago)
• ## new file sage/tests/french_book/calculus.py

```# HG changeset patch
# User Alexandre Casamayou <alexandre.casamayou@orange.fr>
# Date 1314277883 -7200
# Node ID 8f9ee6055e9eb5c109fa1d4d6327b14fdab2fef7
#10983 new doctest for french book about Sage

diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/calculus.py```
 - r""" sage: var ('a,x') ; y = cos(x+a)*(x+1) ; y (a, x) (x + 1)*cos(a + x) sage: var ('a,x') ; y = cos(x+a)*(x+1) ; y (a, x) (x + 1)*cos(a + x) sage: y.subs(a=-x) ; y.subs(x=pi/2,a=pi/3) ; y.subs(x=0.5,a=2.3) x + 1 -1/4*(pi + 2)*sqrt(3) -1.41333351100299 sage: y(a=-x) ; y(x=pi/2,a=pi/3) ; y(x=0.5,a=2.3) x + 1 -1/4*(pi + 2)*sqrt(3) -1.41333351100299 sage: var('x,y,z') ; q = x*y+y*z+z*x (x, y, z) sage: bool (q(x=y,y=z,z=x)==q), bool(q(z=y)(y=x) == 3*x^2) (True, True) sage: var('y z'); f = x^3+y^2+z; f.subs_expr(x^3==y^2, z==1) (y, z) 2*y^2 + 1 sage: f(x)=(2*x+1)^3 ; f(-3) -125 sage: f(x).expand() 8*x^3 + 12*x^2 + 6*x + 1 sage: var('y'); u = sin(x) + x*cos(y) y sage: v = u.function(x, y); v (x, y) |--> x*cos(y) + sin(x) sage: w(x, y) = u; w (x, y) |--> x*cos(y) + sin(x) sage: (x^x/x).simplify() x^(x - 1) sage: f = (e^x-1)/(1+e^(x/2)); f.simplify_exp() e^(1/2*x) - 1 sage: f = cos(x)^6 + sin(x)^6 + 3 * sin(x)^2 * cos(x)^2 sage: f.simplify_trig() 1 sage: f = cos(x)^6; f.reduce_trig() 15/32*cos(2*x) + 3/16*cos(4*x) + 1/32*cos(6*x) + 5/16 sage: f = sin(5 * x); f.expand_trig() sin(x)^5 - 10*sin(x)^3*cos(x)^2 + 5*sin(x)*cos(x)^4 sage: n = var('n'); f = factorial(n+1)/factorial(n) sage: f.simplify_factorial() n + 1 sage: f = sqrt(x^2); f.simplify_radical() abs(x) sage: f = log(x*y); f.simplify_radical() log(x) + log(y) sage: assume(x > 0); bool(sqrt(x^2) == x) True sage: forget(x > 0); bool(sqrt(x^2) == x) False sage: var('n'); assume(n, 'integer'); sin(n*pi).simplify() n 0 sage: z, phi = var('z, phi') sage: solve(z**2 -2 / cos(phi) * z + 5 / cos(phi) ** 2 - 4, z) [z == -(2*sqrt(cos(phi)^2 - 1) - 1)/cos(phi), z == (2*sqrt(cos(phi)^2 - 1) + 1)/cos(phi)] sage: var('y'); solve(y^6==y, y) y [y == e^(2/5*I*pi), y == e^(4/5*I*pi), y == e^(-4/5*I*pi), y == e^(-2/5*I*pi), y == 1, y == 0] sage: solve(x^2-1, x, solution_dict=True) [{x: -1}, {x: 1}] sage: solve([x+y == 3, 2*x+2*y == 6],x,y) [[x == -r1 + 3, y == r1]] sage: solve([cos(x)*sin(x) == 1/2, x+y == 0], x, y) [[x == 1/4*pi + pi*z30, y == -1/4*pi - pi*z30]] sage: solve(x^2+x-1>0,x) [[x < -1/2*sqrt(5) - 1/2], [x > 1/2*sqrt(5) - 1/2]] sage: x, y, z = var('x, y, z') sage: solve([x^2 * y * z == 18, x * y^3 * z == 24,\ x * y * z^4 == 6], x, y, z) [[x == 3, y == 2, z == 1], [x == (1.33721506733 - 2.68548987407*I), y == (-1.70043427146 + 1.05286432575*I), z == (0.932472229404 - 0.361241666187*I)], [x == (1.33721506733 + 2.68548987407*I), y == (-1.70043427146 - 1.05286432575*I), z == (0.932472229404 + 0.361241666187*I)], [x == (-2.55065140719 - 1.57929648863*I), y == (-0.547325980144 + 1.92365128635*I), z == (-0.982973099684 - 0.183749517817*I)], [x == (-2.55065140719 + 1.57929648863*I), y == (-0.547325980144 - 1.92365128635*I), z == (-0.982973099684 + 0.183749517817*I)], [x == (0.27680507839 - 2.98720252889*I), y == (1.47801783444 - 1.34739128729*I), z == (-0.85021713573 - 0.526432162877*I)], [x == (0.27680507839 + 2.98720252889*I), y == (1.47801783444 + 1.34739128729*I), z == (-0.85021713573 + 0.526432162877*I)], [x == (-0.820988970216 + 2.88547692952*I), y == (-1.20526927276 - 1.59603445456*I), z == (0.0922683594633 - 0.995734176295*I)], [x == (-0.820988970216 - 2.88547692952*I), y == (-1.20526927276 + 1.59603445456*I), z == (0.0922683594633 + 0.995734176295*I)], [x == (-1.80790390914 - 2.39405168184*I), y == (0.891476711553 - 1.79032658271*I), z == (0.739008917221 - 0.673695643647*I)], [x == (-1.80790390914 + 2.39405168184*I), y == (0.891476711553 + 1.79032658271*I), z == (0.739008917221 + 0.673695643647*I)], [x == (2.21702675166 + 2.02108693094*I), y == (1.86494445881 + 0.722483332374*I), z == (-0.273662990072 - 0.961825643173*I)], [x == (2.21702675166 - 2.02108693094*I), y == (1.86494445881 - 0.722483332374*I), z == (-0.273662990072 + 0.961825643173*I)], [x == (2.79741668821 - 1.08372499856*I), y == (-1.96594619937 + 0.367499035633*I), z == (-0.602634636379 - 0.79801722728*I)], [x == (2.79741668821 + 1.08372499856*I), y == (-1.96594619937 - 0.367499035633*I), z == (-0.602634636379 + 0.79801722728*I)], [x == (-2.94891929905 + 0.55124855345*I), y == (0.184536718927 + 1.99146835259*I), z == (0.445738355777 - 0.895163291355*I)], [x == (-2.94891929905 - 0.55124855345*I), y == (0.184536718927 - 1.99146835259*I), z == (0.445738355777 + 0.895163291355*I)]] sage: expr = sin(x) + sin(2 * x) + sin(3 * x) sage: solve(expr, x) [sin(3*x) == -sin(2*x) - sin(x)] sage: find_root(expr, 0.1, pi) 2.0943951023931957 sage: f = expr.simplify_trig(); f 2*(2*cos(x)^2 + cos(x))*sin(x) sage: solve(f, x) [x == 0, x == 2/3*pi, x == 1/2*pi] sage: (x^3+2*x+1).roots(x) [(-1/2*(I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3) - 1/3*(I*sqrt(3) - 1)/(1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3), 1), (-1/2*(-I*sqrt(3) + 1)*(1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3) - 1/3*(-I*sqrt(3) - 1)/(1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3), 1), ((1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3) - 2/3/(1/18*sqrt(3)*sqrt(59) - 1/2)^(1/3), 1)] sage: (x^3+2*x+1).roots(x, ring=RR) [(-0.453397651516404, 1)] sage: (x^3+2*x+1).roots(x, ring=CC) [(-0.453397651516404, 1), (0.226698825758202 - 1.46771150871022*I, 1), (0.226698825758202 + 1.46771150871022*I, 1)] sage: var('k n'); sum(k, k, 1, n).factor() (k, n) 1/2*(n + 1)*n sage: var('n y'); sum(binomial(n,k) * x^k * y^(n-k), k, 0, n) (n, y) (x + y)^n sage: var('k n'); sum(binomial(n,k), k, 0, n),\ sum(k * binomial(n, k), k, 0, n),\ sum((-1)^k*binomial(n,k), k, 0, n) (k, n) (2^n, n*2^(n - 1), 0) sage: var('a q'); sum(a*q^k, k, 0, n) (a, q) (a*q^(n + 1) - a)/(q - 1) sage: assume(abs(q) < 1) sage: sum(a*q^k, k, 0, infinity) -a/(q - 1) sage: limit((x**(1/3) -2) / ((x + 19)**(1/3) - 3), x = 8) 9/4 sage: f(x) = (cos(pi/4-x)-tan(x))/(1-sin(pi/4 + x)) sage: limit(f(x), x = pi/4, dir='minus') +Infinity sage: limit(f(x), x = pi/4, dir='plus') -Infinity sage: u(n) = n^100 / 100.^n sage: u(1);u(2);u(3);u(4);u(5);u(6);u(7);u(8);u(9);u(10) 0.0100000000000000 1.26765060022823e26 5.15377520732011e41 1.60693804425899e52 7.88860905221012e59 6.53318623500071e65 3.23447650962476e70 2.03703597633449e74 2.65613988875875e77 1.00000000000000e80 sage: plot(u(x), x, 1, 40) sage: v(x) = diff(u(x), x); sol = solve(v(x) == 0, x); sol [x == 0, x == (268850/12381)] sage: sol[1].rhs().n().floor() 21 sage: limit(u(n), n=infinity) 0 sage: n0 = find_root(u(n) - 1e-8 == 0, 22, 1000); n0 105.07496210187252 sage: taylor((1+arctan(x))**(1/x), x, 0, 3) 1/16*x^3*e + 1/8*x^2*e - 1/2*x*e + e sage: (ln(2* sin(x))).series(x==pi/6, 3) (sqrt(3))*(-1/6*pi + x) + (-2)*(-1/6*pi + x)^2 + Order(-1/216*(pi - 6*x)^3) sage: (ln(2* sin(x))).series(x==pi/6, 3).truncate() -1/18*(pi - 6*x)^2 - 1/6*(pi - 6*x)*sqrt(3) sage: taylor((x**3+x)**(1/3) - (x**3-x)**(1/3), x, infinity, 2) 2/3/x sage: tan(4*arctan(1/5)).simplify_trig() 120/119 sage: tan(pi/4+arctan(1/239)).simplify_trig() 120/119 sage: f = arctan(x).series(x, 10); f 1*x + (-1/3)*x^3 + 1/5*x^5 + (-1/7)*x^7 + 1/9*x^9 + Order(x^10) sage: (16*f.subs(x==1/5) - 4*f.subs(x==1/239)).n(); pi.n() 3.14159268240440 3.14159265358979 sage: var('k'); sum(1/k^2, k, 1, infinity),\ sum(1/k^4, k, 1, infinity),\ sum(1/k^5, k, 1, infinity) k (1/6*pi^2, 1/90*pi^4, zeta(5)) sage: s = 2*sqrt(2)/9801*(sum((factorial(4*k))*(1103+26390*k)\ /((factorial(k)) ^ 4 * 396 ^ (4 * k)) for k in (0..11))) sage: (1 / s).n(digits=100); pi.n(digits=100) 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342121432 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 sage: print "%e" % (pi - 1 / s).n(digits=100) -4.364154e-96 sage: var('n'); u = sin(pi*(sqrt(4*n^2+1)-2*n)) n sage: taylor(u, n, infinity, 3) 1/4*pi/n - 1/384*(6*pi + pi^3)/n^3 sage: diff(sin(x^2), x) 2*x*cos(x^2) sage: function('f',x); function('g',x); diff(f(g(x)), x) f(x) g(x) D[0](f)(g(x))*D[0](g)(x) sage: diff(ln(f(x)), x) D[0](f)(x)/f(x) sage: f(x,y) = x*y + sin(x^2) + e^(-x); derivative(f, x) (x, y) |--> 2*x*cos(x^2) + y - e^(-x) sage: derivative(f, y) (x, y) |--> x sage: var('x y'); f = ln(x**2+y**2) / 2 (x, y) sage: delta = diff(f,x,2)+diff(f,y,2) sage: delta.simplify_full() 0 sage: sin(x).integral(x, 0, pi/2) 1 sage: integrate(1/(1+x^2), x) arctan(x) sage: integrate(1/(1+x^2), x, -infinity, infinity) pi sage: integrate(exp(-x**2), x, 0, infinity) 1/2*sqrt(pi) sage: u = var('u'); f = x * cos(u) / (u^2 + x^2) sage: assume(x>0); f.integrate(u, 0, infinity) 1/2*pi*e^(-x) sage: forget(); assume(x<0); f.integrate(u, 0, infinity) -1/2*pi*e^x sage: integral_numerical(sin(x)/x, 0, 1) (0.946083070367182..., 1.050363207929708...e-14) sage: g = integrate(exp(-x**2), x, 0, infinity); g, g.n() (1/2*sqrt(pi), 0.886226925452...) sage: approx = integral_numerical(exp(-x**2), 0, infinity) sage: approx[0] 0.88622692545275... sage: A = matrix(QQ,[[1,2],[3,4]]); A [1 2] [3 4] sage: A = matrix(QQ, [[2, 4, 3],[-4,-6,-3],[3,3,1]]) sage: A.characteristic_polynomial() x^3 + 3*x^2 - 4 sage: A.eigenvalues() [1, -2, -2] sage: A.minimal_polynomial().factor() (x - 1) * (x + 2)^2 sage: A.eigenvectors_right() [(1, [ (1, -1, 1) ], 1), (-2, [ (1, -1, 0) ], 2)] sage: A.jordan_form(transformation=True) ( [ 1| 0  0] [--+-----]  [ 1  1  1] [ 0|-2  1]  [-1 -1  0] [ 0| 0 -2], [ 1  0 -1] ) sage: A = matrix(QQ, [[1,-1/2],[-1/2,-1]]) sage: A.minimal_polynomial() x^2 - 5/4 sage: R = QQ[sqrt(5)] sage: A = A.change_ring(R) sage: A.jordan_form(transformation=True, subdivide=False) ( [ 1/2*sqrt5          0]  [         1          1] [         0 -1/2*sqrt5], [-sqrt5 + 2  sqrt5 + 2] ) sage: K. = NumberField(x^2 - 2) sage: L. = K.extension(x^2 - 3) sage: A = matrix(L, [[2, sqrt2*sqrt3, sqrt2], [sqrt2*sqrt3, 3, sqrt3], [sqrt2, sqrt3, 1]]) sage: A.jordan_form(transformation=True) ( [6|0|0] [-+-+-] [0|0|0]  [              1               1               0] [-+-+-]  [1/2*sqrt2*sqrt3               0               1] [0|0|0], [      1/2*sqrt2          -sqrt2          -sqrt3] ) """
• ## new file sage/tests/french_book/graphique_1.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/graphique_1.py`
 - r""" sage: g = plot(x * sin(1/x), x, -2, 2, plot_points=500) sage: def p(x, n): return(taylor(sin(x), x, 0, n)) sage: xmax = 15 ; n = 15 sage: g = plot(sin(x), x, -xmax, xmax) sage: for d in range(n): ...     g += plot(p(x, 2 * d + 1), x, -xmax, xmax,\ ...       color=(1.7*d/(2*n), 1.5*d/(2*n), 1-3*d/(4*n))) sage: # g.show(ymin=-2, ymax=2) sage: a = animate([[sin(x), taylor(sin(x), x, 0, 2*k+1)]\ ...          for k in range(0, 14)], xmin=-14, xmax=14,\ ...          ymin=-3, ymax=3, figsize=[8, 4]) sage: # a.show() sage: f2(x) = 1; f1(x) = -1 sage: f = Piecewise([[(-pi,0),f1],[(0,pi),f2]]) sage: S = f.fourier_series_partial_sum(20,pi); S 4/3*sin(3*x)/pi + 4/5*sin(5*x)/pi + 4/7*sin(7*x)/pi + 4/9*sin(9*x)/pi + 4/11*sin(11*x)/pi + 4/13*sin(13*x)/pi + 4/15*sin(15*x)/pi + 4/17*sin(17*x)/pi + 4/19*sin(19*x)/pi + 4*sin(x)/pi sage: g = plot(S, x, -8, 8, color='blue') sage: scie(x) = x - 2 * pi * floor((x + pi) / (2 * pi)) sage: g += plot(scie(x) / abs(scie(x)), x, -8, 8, color='red') sage: t = var('t') sage: x = cos(t) + cos(7*t)/2 + sin(17*t)/3 sage: y = sin(t) + sin(7*t)/2 + cos(17*t)/3 sage: g = parametric_plot( (x, y), (t, 0, 2*pi)) sage: # g.show(aspect_ratio=1) sage: t = var('t'); e, n = 2, 20/19 sage: g1 = polar_plot(1+e*cos(n*t),(t,0,n*38*pi),plot_points=5000) sage: e, n = 1/3, 20/19 sage: g2 = polar_plot(1+e*cos(n*t),(t,0,n*38*pi),plot_points=5000) sage: # g1.show(aspect_ratio=1); g2.show(aspect_ratio=1) sage: z = var('z'); g1 = complex_plot(abs(cos(z^4)) - 1,\ ...             (-3, 3), (-3, 3), plot_points=400) sage: f = lambda x, y : (abs(cos((x + I * y) ** 4)) - 1) sage: g2 = implicit_plot(f, (-3, 3), (-3, 3), plot_points=400) sage: # g1.show(aspect_ratio=1); g2.show(aspect_ratio=1) sage: f(z) = z^5 + z - 1 + 1/z sage: g = complex_plot(f,\ ...      (-3, 3), (-3, 3)) """
• ## new file sage/tests/french_book/graphique_2.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/graphique_2.py`
 - r""" sage: g = bar_chart([randrange(15) for i in range(20)], color='red') sage: g = bar_chart([x^2 for x in range(1,20)], width=0.2) sage: liste = [10 + floor(10*sin(i)) for i in range(100)] sage: g = bar_chart(liste) sage: g = finance.TimeSeries(liste).plot_histogram(bins=20) sage: from random import * sage: n, l, x, y = 10000, 1, 0, 0; p = [[0, 0]] sage: for k in range(n): ...    theta = (2 * pi * random()).n(digits=5) ...    x, y = x + l * cos(theta), y + l * sin(theta) ...    p.append([x, y]) sage: g1 = line([p[n], [0, 0]], color='red', thickness=2) sage: g1 += line(p, thickness=.4); # g1.show(aspect_ratio=1) sage: length = 200; n = var('n') sage: u(n) = n * sqrt(2) sage: z(n) = exp(2 * I * pi * u(n)) sage: vertices = [CC(0, 0)] sage: for n in range(1, length): ...    vertices.append(vertices[n - 1] + CC(z(n))) sage: g = line(vertices); # g.show(aspect_ratio=1) sage: x = var('x'); y = function('y',x) sage: DE = x*diff(y, x) == 2*y + x^3 sage: desolve(DE, [y,x]) (c + x)*x^2 sage: sol = [] sage: for i in srange(-2, 2, 0.2): ...    sol.append(desolve(DE, [y, x], ics=[1, i])) ...    sol.append(desolve(DE, [y, x], ics=[-1, i])) sage: g = plot(sol, x, -2, 2) sage: y = var('y') sage: g += plot_vector_field((x, 2*y+x^3), (x, -2, 2), (y, -1, 1)) sage: # g.show(ymin=-1, ymax=1) sage: # debut variante sage: x = var('x'); y = function('y',x) sage: DE = x*diff(y, x) == 2*y + x^3 sage: g = Graphics() sage: for i in srange(-1, 1, 0.1): ...       g += line(desolve_rk4(DE, y, ics=[1, i],\ ...                         step=0.05, end_points=[0,2])) ...       g += line(desolve_rk4(DE, y, ics=[-1, i],\ ...                         step=0.05, end_points=[-2,0])) sage: y = var('y') sage: g = plot_vector_field((x, 2*y + x^3), (x,-2,2), (y,-1,1)) sage: # g.show(ymin=-1, ymax=1) sage: # fin variante """
• ## new file sage/tests/french_book/graphique_3.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/graphique_3.py`
 - r""" sage: import scipy; from scipy import integrate sage: f = lambda y, t: - cos(y * t) sage: t = srange(0, 5, 0.1); p = Graphics() sage: for k in srange(0, 10, 0.15): ...      y = integrate.odeint(f, k, t) ...      p += line(zip(t, flatten(y))) sage: t = srange(0, -5, -0.1); q = Graphics() sage: for k in srange(0, 10, 0.15): ...      y = integrate.odeint(f, k, t) ...      q += line(zip(t, flatten(y))) sage: y = var('y') sage: v = plot_vector_field((1, -cos(x * y)), (x,-5,5), (y,-2,11)) sage: g = p + q + v; # g.show() sage: import scipy; from scipy import integrate sage: a, b, c, d = 1., 0.1, 1.5, 0.75 sage: def dX_dt(X, t=0): ...    return [ a*X[0] -   b*X[0]*X[1] , ...            -c*X[1] + d*b*X[0]*X[1] ] sage: t = srange(0, 15, .01)                     # echelle de temps sage: X0 = [10, 5]  # conditions initiales : 10 lapins et 5 renards sage: X = integrate.odeint(dX_dt, X0, t)     # resolution numerique sage: lapins, renards =  X.T         # raccourcis de  X.transpose() sage: p = line(zip(t, lapins), color='red') # trace du nb de lapins sage: p += text("Lapins",(12,37), fontsize=10, color='red') sage: p += line(zip(t, renards), color='blue')# idem pr les renards sage: p += text("Renards",(12,7), fontsize=10, color='blue') sage: p.axes_labels(["temps", "population"]); # p.show(gridlines=True) sage: ### Deuxieme graphique : sage: n = 11;  L = srange(6, 18, 12 / n); R=srange(3, 9, 6 / n) sage: CI = zip(L, R)                # liste des conditions initiales sage: def g(x,y): ...       v = vector(dX_dt([x, y]))  # pour un trace plus lisible, ...       return v/v.norm()          # on norme le champ de vecteurs sage: x, y = var('x, y') sage: q = plot_vector_field(g(x, y), (x, 0, 60), (y, 0, 36)) sage: for j in range(n): ...       X = integrate.odeint(dX_dt, CI[j], t)        # resolution ...       q += line(X, color=hue(.8-float(j)/(1.8*n))) #  graphique sage: q.axes_labels(["lapins","renards"]); # q.show() sage: x, y, t = var('x, y, t') sage: alpha(t) = 1; beta(t) = t/2; gamma(t) = t + t**3/8 sage: env = solve([alpha(t)*x+beta(t)*y==gamma(t),\ ...         diff(alpha(t),t)*x+diff(beta(t),t)*y==diff(gamma(t),t)],\ ...         [x,y]) sage: f = lambda x:x^2 / 4 sage: p = plot(f, -8, 8, rgbcolor=(0.2,0.2,0.4)) # trace la parabole sage: for u in srange(0, 8, 0.1): # trace des normales a la parabole ...       p += line([[u, f(u)], [-8*u, f(u) + 18]], thickness=.3) ...       p += line([[-u, f(u)], [8*u, f(u) + 18]], thickness=.3) sage: p += parametric_plot((env[0][0].rhs(),env[0][1].rhs()),\ ...       (t, -8, 8),color='red')              # trace la developpee sage: # p.show(xmin=-8, xmax=8, ymin=-1, ymax=12, aspect_ratio=1) sage: t = var('t'); p = 2 sage: x(t) = t; y(t) = t^2 / (2 * p) sage: f(t) = [x(t), y(t)] sage: df(t) = [x(t).diff(t), y(t).diff(t)] sage: d2f(t) = [x(t).diff(t, 2), y(t).diff(t, 2)] sage: T(t) = [df(t)[0] / df(t).norm(), df[1](t) / df(t).norm()] sage: N(t) = [-df(t)[1] / df(t).norm(), df[0](t) / df(t).norm()] sage: R(t) =  (df(t).norm())^3 / \ ...        (df(t)[0] * d2f(t)[1] -df(t)[1] * d2f(t)[0]) sage: Omega(t) = [f(t)[0] + R(t)*N(t)[0], f(t)[1] + R(t)*N(t)[1]] sage: g = parametric_plot(f(t), (t, -8, 8), color='green', thickness=2) sage: for u in srange(.4, 4, .2): ...    g += line([f(t = u), Omega(t = u)], color='red', alpha = .5) ...    g += circle(Omega(t = u), R(t = u), color='blue') sage: # g.show(aspect_ratio=1, xmin=-12, xmax=7, ymin=-3, ymax=12) sage: u, v = var('u, v') sage: h = lambda u,v: u^2 + 2*v^2 sage: f = plot3d(h, (u,-1,1), (v,-1,1), aspect_ratio=[1,1,1]) sage: f(x, y) = x^2 * y / (x^4 + y^2) sage: t, theta = var('t theta') sage: limit(f(t * cos(theta), t * sin(theta)) / t, t=0) cos(theta)^2/sin(theta) sage: solve(f(x,y) == 1/2, y) [y == x^2] sage: a = var('a'); h = f(x, a*x^2).simplify_rational(); h a/(a^2 + 1) sage: g = plot(h, a, -4, 4) sage: p = plot3d(f(x, y), (x,-2,2), (y,-2,2), plot_points=[150,150]) sage: for i in range(1,4): ...    p += plot3d(-0.5 + i / 4, (x, -2, 2), (y, -2, 2),\ ...                 color=hue(i / 10), opacity=.1) sage: x, y, z = var('x, y, z'); a = 1 sage: h = lambda x, y, z:(a^2 + x^2 + y^2)^2 - 4*a^2*x^2-z^4 sage: f = implicit_plot3d(h, (x, -3, 3), (y, -3, 3), (z, -2, 2),\ ...                    plot_points=100, adaptative=True) sage: g1 = line3d([(-10*cos(t)-2*cos(5*t)+15*sin(2*t),\ ...     -15*cos(2*t)+10*sin(t)-2*sin(5*t),\ ...      10*cos(3*t)) for t in srange(0,6.4,.1)],radius=.5) """
• ## new file sage/tests/french_book/sol_calculus.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/sol_calculus.py`
 - r""" sage: var('n k'); p = 4; s = [n + 1] (n, k) sage: for k in (1..p): ...     s = s + [factor((((n + 1)^(k + 1) - sum(binomial(k + 1, j)*s[j] ...              for j in (0..k - 1))) / (k + 1)))] sage: s [n + 1, 1/2*(n + 1)*n, 1/6*(n + 1)*(2*n + 1)*n, 1/4*(n + 1)^2*n^2, 1/30*(n + 1)*(2*n + 1)*(3*n^2 + 3*n - 1)*n] sage: var('x h a'); f = function('f', x) (x, h, a) sage: g(x) = taylor(f, x, a, 3) sage: phi(h) = (g(a+3*h)-3*g(a+2*h)+3*g(a+h)-g(a))/h**3 sage: phi(h).expand() D[0, 0, 0](f)(a) sage: n = 7; var('x h a'); f = function('f', x) (x, h, a) sage: g(x) = taylor(f, x, a, n) sage: phi(h) = sum(binomial(n,k)*(-1)^(n-k)*g(a+k*h) for k in (0..n))/h**n sage: phi(h).expand() D[0, 0, 0, 0, 0, 0, 0](f)(a) sage: theta = 12 * arctan(1/38) + 20 * arctan(1/57) + 7 * arctan(1/239) + 24 * arctan(1/268) sage: x = tan(theta) sage: y = x.trig_expand() sage: y.trig_simplify() 1 sage: M = 12*(1/38)+20*(1/57)+ 7*(1/239)+24*(1/268) sage: M 37735/48039 sage: x = var('x') sage: f(x) = taylor(arctan(x), x, 0, 21) sage: approx = 4 * (12 * f(1/38) + 20 * f(1/57) + 7 * f(1/239) + 24 * f(1/268)) sage: approx.n(digits = 50); pi.n(digits = 50) 3.1415926535897932384626433832795028851616168852864 3.1415926535897932384626433832795028841971693993751 sage: approx.n(digits = 50) - pi.n(digits = 50) 9.6444748591132486785420917537404705292978817080880e-37 sage: n = var('n'); phi = lambda x: n*pi +pi/2 - arctan(1/x); x = pi * n sage: for i in range(4): x = taylor(phi(x), n, oo, 2 * i); x 1/2*pi + pi*n 1/2*pi + pi*n - 1/(pi*n) + 1/2/(pi*n^2) 1/2*pi + pi*n - 1/(pi*n) + 1/2/(pi*n^2) - 1/12*(3*pi^2 + 8)/(pi^3*n^3) + 1/8*(pi^2 + 8)/(pi^3*n^4) 1/2*pi + pi*n - 1/(pi*n) + 1/2/(pi*n^2) - 1/12*(3*pi^2 + 8)/(pi^3*n^3) + 1/8*(pi^2 + 8)/(pi^3*n^4) - 1/240*(15*pi^4 + 240*pi^2 + 208)/(pi^5*n^5) + 1/96*(3*pi^4 + 80*pi^2 + 208)/(pi^5*n^6) sage: f(x, y) = x * y * (x**2 - y**2) / (x**2 + y**2) sage: D1f(x, y) = diff(f(x,y), x) sage: limit((D1f(0,h) - 0) / h, h=0) -1 sage: D2f(x, y) = diff(f(x,y), y) sage: limit((D2f(h,0) - 0) / h, h=0) 1 sage: g = plot3d(f(x, y), (x, -3, 3), (y, -3, 3)) sage: n, t = var('n, t') sage: v(n) = (4/(8*n+1)-2/(8*n+4)-1/(8*n+5)-1/(8*n+6))*1/16^n sage: assume(8*n+1>0) sage: u(n) = integrate((4*sqrt(2)-8*t^3-4*sqrt(2)*t^4-8*t^5)\ ...                     * t^(8*n), t, 0, 1/sqrt(2)) sage: (u(n)-v(n)).simplify_full() 0 sage: J = integrate((4*sqrt(2)-8*t^3-4*sqrt(2)*t^4-8*t^5)\ ...                 / (1-t^8), t, 0, 1/sqrt(2)) sage: J.simplify_full() pi + 2*log(sqrt(2) - 1) + 2*log(sqrt(2) + 1) sage: ln(exp(J).simplify_log()) pi sage: l = sum(v(n) for n in (0..40)); l.n(digits=60); pi.n(digits=60) 3.14159265358979323846264338327950288419716939937510581474759 3.14159265358979323846264338327950288419716939937510582097494 sage: print "%e" % (l-pi).n(digits=60) -6.227358e-54 sage: var('X'); ps = lambda f,g : integral(f * g, X, -pi, pi) X sage: n = 5; Q = sin(X) sage: var('a a0 a1 a2 a3 a4 a5'); a= [a0, a1, a2, a3, a4, a5] (a, a0, a1, a2, a3, a4, a5) sage: P = sum(a[k] * X^k for k in (0..n)) sage: equ = [ps(P - Q, X^k) for k in (0..n)] sage: sol = solve(equ, a) sage: P = sum(sol[0][k].rhs() * X^k for k in (0..n)) sage: g = plot(P,X,-4,4,color='red') + plot(Q,X,-4,4,color='blue') sage: var('p e theta1 theta2 theta3') (p, e, theta1, theta2, theta3) sage: r(theta) = p / (1-e * cos(theta)) sage: r1 = r(theta1); r2 = r(theta2); r3 = r(theta3) sage: R1 = vector([r1 * cos(theta1), r1 * sin(theta1), 0]) sage: R2 = vector([r2 * cos(theta2), r2 * sin(theta2), 0]) sage: R3 = vector([r3 * cos(theta3), r3 * sin(theta3), 0]) sage: D = R1.cross_product(R2) + R2.cross_product(R3) + R3.cross_product(R1) sage: i = vector([1, 0, 0]) sage: S = (r1 - r3) * R2 + (r3 - r2) * R1 +   (r2 - r1) * R3 sage: V =  S + e * i.cross_product(D) sage: map(lambda x:x.simplify_full(), V) # rep. : [0, 0, 0] [0, 0, 0] sage: map(lambda x:x.simplify_full(), S.cross_product(D)) [(e*p^4*sin(theta1)^2*cos(theta2)^2 - 2*e*p^4*sin(theta1)*sin(theta2)*cos(theta1)*cos(theta2) + e*p^4*sin(theta2)^2*cos(theta1)^2 + (e*p^4*cos(theta1)^2 - 2*e*p^4*cos(theta1)*cos(theta2) + e*p^4*cos(theta2)^2)*sin(theta3)^2 + (e*p^4*sin(theta1)^2 - 2*e*p^4*sin(theta1)*sin(theta2) + e*p^4*sin(theta2)^2)*cos(theta3)^2 - 2*(e*p^4*sin(theta1)^2*cos(theta2) + e*p^4*sin(theta2)^2*cos(theta1) - (e*p^4*sin(theta1)*cos(theta1) + e*p^4*sin(theta1)*cos(theta2))*sin(theta2))*cos(theta3) + 2*(e*p^4*sin(theta1)*cos(theta1)*cos(theta2) - e*p^4*sin(theta1)*cos(theta2)^2 - (e*p^4*cos(theta1)^2 - e*p^4*cos(theta1)*cos(theta2))*sin(theta2) - (e*p^4*sin(theta1)*cos(theta1) - e*p^4*sin(theta1)*cos(theta2) - (e*p^4*cos(theta1) - e*p^4*cos(theta2))*sin(theta2))*cos(theta3))*sin(theta3))/(e^2*cos(theta1)^2 + (e^4*cos(theta1)^2 - 2*e^3*cos(theta1) + e^2)*cos(theta2)^2 + (e^4*cos(theta1)^2 - 2*e^3*cos(theta1) + (e^6*cos(theta1)^2 - 2*e^5*cos(theta1) + e^4)*cos(theta2)^2 - 2*(e^5*cos(theta1)^2 - 2*e^4*cos(theta1) + e^3)*cos(theta2) + e^2)*cos(theta3)^2 - 2*(e^3*cos(theta1)^2 - 2*e^2*cos(theta1) + e)*cos(theta2) - 2*(e^3*cos(theta1)^2 + (e^5*cos(theta1)^2 - 2*e^4*cos(theta1) + e^3)*cos(theta2)^2 - 2*e^2*cos(theta1) - 2*(e^4*cos(theta1)^2 - 2*e^3*cos(theta1) + e^2)*cos(theta2) + e)*cos(theta3) - 2*e*cos(theta1) + 1), 0, 0] sage: N = r3 * R1.cross_product(R2) + r1 * R2.cross_product(R3) + r2 * R3.cross_product(R1) sage: W =  p * S + e * i.cross_product(N) sage: print map(lambda x:x.simplify_full(), W)  # rep. : [0, 0, 0] [0, 0, 0] sage: R1=vector([0,1.,0]);R2=vector([2.,2.,0]);R3=vector([3.5,0,0]) sage: r1 = R1.norm(); r2 = R2.norm(); r3 = R3.norm() sage: D = R1.cross_product(R2) + R2.cross_product(R3) + R3.cross_product(R1) sage: S = (r1 - r3) * R2 + (r3 - r2) * R1 + (r2 - r1) * R3 sage: V =  S + e * i.cross_product(D) sage: N = r3 * R1.cross_product(R2) + r1 * R2.cross_product(R3) \ ...  + r2 * R3.cross_product(R1) sage: W =  p * S + e * i.cross_product(N) sage: e = S.norm() / D.norm() sage: p = N.norm() / D.norm() sage: a = p/(1-e^2) sage: c = a * e sage: b = sqrt(a^2 - c^2) sage: X = S.cross_product(D) sage: i = X / X.norm() sage: phi = atan2(i[1],i[0]) * 180 / pi.n() sage: print "%.3f %.3f %.3f %.3f %.3f %.3f" % (a, b, c, e, p, phi) 2.360 1.326 1.952 0.827 0.746 17.917 sage: A = matrix(QQ, [[2, -3, 2, -12, 33], ...                   [ 6, 1, 26, -16, 69], ...                   [10, -29, -18, -53, 32], ...                   [2, 0, 8, -18, 84]]) sage: A.right_kernel() Vector space of degree 5 and dimension 2 over Rational Field Basis matrix: [     1      0  -7/34   5/17   1/17] [     0      1  -3/34 -10/17  -2/17] sage: H = A.echelon_form() sage: A.column_space() Vector space of degree 4 and dimension 3 over Rational Field Basis matrix: [       1        0        0 1139/350] [       0        1        0    -9/50] [       0        0        1   -12/35] sage: S.=QQ[] sage: C = matrix(S, 4,1,[x,y,z,t]) sage: B = block_matrix([A,C], ncols=2) sage: C = B.echelon_form() sage: C[3,5]*350 -1139*x + 63*y + 120*z + 350*t sage: K = A.kernel(); K Vector space of degree 4 and dimension 1 over Rational Field Basis matrix: [        1  -63/1139 -120/1139 -350/1139] sage: matrix(K.0).right_kernel() Vector space of degree 4 and dimension 3 over Rational Field Basis matrix: [       1        0        0 1139/350] [       0        1        0    -9/50] [       0        0        1   -12/35] sage: A = matrix(QQ, [[-2, 1, 1], [8, 1, -5], [4, 3, -3]]) sage: C = matrix(QQ, [[1, 2, -1], [2, -1, -1], [-5, 0, 3]]) sage: B = C.solve_left(A); B [ 0 -1  0] [ 2  3  0] [ 2  1  0] sage: C.left_kernel() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 1] sage: var('x y z'); v = matrix([[1, 2, 1]]) (x, y, z) sage: B = B+(x*v).stack(y*v).stack(z*v); B [      x 2*x - 1       x] [  y + 2 2*y + 3       y] [  z + 2 2*z + 1       z] sage: A == B*C True """
• ## new file sage/tests/french_book/sol_graphiques_1.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/sol_graphiques_1.py`
 - r""" sage: t = var('t'); liste = [a + cos(t) for a in srange(0, 10, 0.1)] sage: g = polar_plot(liste, (t, 0, 2 * pi)); g.show(aspect_ratio=1) sage: f = lambda x: abs(x**2 - 1/4) sage: def liste_pts(u0, n): ...     u = u0; liste = [[u0,0]] ...     for k in range(n): ...         v, u = u, f(u) ...         liste.extend([[v,u], [u,u]]) ...     return(liste) sage: g = line(liste_pts(1.1, 8), rgbcolor = (.9,0,0)) sage: g += line(liste_pts(-.4, 8), rgbcolor = (.01,0,0)) sage: g += line(liste_pts(1.3, 3), rgbcolor = (.5,0,0)) sage: g += plot(f, -1, 3, rgbcolor = 'blue') sage: g += plot(x, -1, 3, rgbcolor = 'green') sage: # g.show(aspect_ratio=1, ymin = -.2, ymax = 3) sage: x = var('x'); y = function('y',x); DE = x^2 * diff(y, x) -  y == 0 sage: desolve(DE, [y,x]) c*e^(-1/x) sage: g = plot([c*e^(-1/x) for c in srange(-8, 8, 0.4)], (x, -3, 3)) sage: y = var('y'); g += plot_vector_field((x^2, y), (x,-3,3), (y,-5,5)) sage: # g.show(ymin = -5, ymax = 5) sage: from sage.calculus.desolvers import desolve_system_rk4 sage: f = lambda x, y:[a*x-b*x*y,-c*y+d*b*x*y] sage: x,y,t = var('x y t') sage: a, b, c, d = 1., 0.1, 1.5, 0.75 sage: P = desolve_system_rk4(f(x,y),[x,y], ...        ics=[0,10,5],ivar=t,end_points=15) sage: Ql = [ [i,j] for i,j,k in P] sage: p = line(Ql,color='red') sage: p += text("Lapins",(12,37),fontsize=10,color='red') sage: Qr = [ [i,k] for i,j,k in P] sage: p += line(Qr,color='blue') sage: p += text("Renards",(12,7),fontsize=10,color='blue') sage: p.axes_labels(["temps","population"]) sage: # p.show(gridlines=True) sage: ### Deuxieme graphique sage: n = 10;  L = srange(6, 18, 12 / n); R = srange(3, 9, 6 / n) sage: def g(x,y): ...     v = vector(f(x, y)) ...     return v/v.norm() ... sage: q = plot_vector_field(g(x, y), (x, 0, 60), (y, 0, 36)) sage: for j in range(n): ...     P = desolve_system_rk4(f(x,y),[x,y], ...            ics=[0,L[j],R[j]],ivar=t,end_points=15) ...     Q = [ [j,k] for i,j,k in P] ...     q += line(Q, color=hue(.8-j/(2*n))) ... sage: q.axes_labels(["nombre de lapins","nombre de renards"]) sage: # q.show() sage: import scipy; from scipy import integrate sage: def dX_dt(X, t=0): ...     return [X[1], 0.5*X[1] - X[0] - X[1]**3] ... sage: t = srange(0, 40, 0.01);  x0 = srange(-2, 2, 0.1) sage: y0 = 2.5 sage: CI = [[i, y0] for i in x0] + [[i, -y0] for i in x0] sage: def g(x,y): ...     v = vector(dX_dt([x, y])) ...     return v/v.norm() ... sage: x, y = var('x, y') sage: q = plot_vector_field(g(x, y), (x, -3, 3), (y, -y0, y0)) sage: for j in xrange(len(CI)): ...     X = integrate.odeint(dX_dt, CI[j], t) ...     q += line(X, color=(1.7*j/(4*n),1.5*j/(4*n),1-3*j/(8*n))) ... sage: X = integrate.odeint(dX_dt, [0.01,0], t) sage: q += line(X, color = 'red') sage: # q.show() """
• ## new file sage/tests/french_book/sol_graphiques_2.py

`diff -r 2a2abbcad325 -r 8f9ee6055e9e sage/tests/french_book/sol_graphiques_2.py`
 - r""" sage: import scipy sage: from scipy import integrate sage: # Intervalle de temps : sage: t = srange(0, 40, 0.01) sage: # Conditions initiales en cartesiennes : sage: n = 35 sage: CI_cart = [[4, .2 * i] for i in range(n)] sage: # Conditions initiales en polaires : sage: CI = map(lambda x:[sqrt(x[0]**2+x[1]**2),\ ...        pi - arctan(x[1] / x[0])], CI_cart) sage: # Choix du parametre alpha : sage: alpha = [0.1, 0.5, 1, 1.25] sage: for i in range(len(alpha)): ...     # Definition du syteme differentiel : ...     def dX_dt(X, t=0): ...         return [cos(X[1]) * (1 - 1 / X[0]^2),\ ...         -sin(X[1]) * (1 / X[0] + 1 / X[0]^3) \ ...         + 2 * alpha[i] / X[0]^2] ...     # trace du disque : ...     q = circle((0, 0), 1, fill=True, rgbcolor='purple') ...     for j in range(n): ...         # resolution du syteme autonome : ...         X = integrate.odeint(dX_dt, CI[j], t) ...         # passage en cartesiennes : ...         Y = [[u[0] * cos(u[1]), u[0] * sin(u[1])] for u in X] ...         # trace stocke dans la variable q ...         q += line(Y, xmin = -4, xmax = 4, color='blue') ...     # Trace final : ...     # q.show(aspect_ratio = 1, axes = False) """