# Ticket #10983: trac_10983_doctests_from_french_book_v3.patch

File trac_10983_doctests_from_french_book_v3.patch, 23.9 KB (added by zimmerma, 8 years ago)
• ## new file sage/tests/french_book/calculus_doctest.py

```# HG changeset patch
# User Paul Zimmermann <zimmerma@loria.fr>
# Date 1383756682 -3600
# Node ID 4743a01e63430fa0eb6bc4a07eb009da8cc8440b
# Parent  f0ee3538887fe739601babb54e177ec5e1133b7a
#10983: new doctest for french book about Sage (chapters calculus and graphics)

diff --git a/sage/tests/french_book/calculus_doctest.py b/sage/tests/french_book/calculus_doctest.py
new file mode 100644```
 - ## -*- encoding: utf-8 -*- """ Doctests from French Sage book Test file for chapter "Analyse et algèbre avec Sage" ("Calculus and algebra with Sage") Tests extracted from ./calculus.tex. Sage example in ./calculus.tex, line 37:: sage: bool(x^2 + 3*x + 1 == (x+1)*(x+2)) False Sage example in ./calculus.tex, line 74:: sage: a, x = var('a, x'); y = cos(x+a) * (x+1); y (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*sqrt(3)*(pi + 2) -1.41333351100299 sage: y(a=-x); y(x=pi/2, a=pi/3); y(x=0.5, a=2.3) x + 1 -1/4*sqrt(3)*(pi + 2) -1.41333351100299 Sage example in ./calculus.tex, line 91:: sage: x, y, z = var('x, y, z') ; q = x*y + y*z + z*x sage: bool(q(x=y, y=z, z=x) == q), bool(q(z=y)(y=x) == 3*x^2) (True, True) Sage example in ./calculus.tex, line 99:: sage: y, z = var('y, z'); f = x^3 + y^2 + z sage: f.subs_expr(x^3 == y^2, z==1) 2*y^2 + 1 Sage example in ./calculus.tex, line 110:: sage: f(x)=(2*x+1)^3 ; f(-3) -125 sage: f.expand() x |--> 8*x^3 + 12*x^2 + 6*x + 1 Sage example in ./calculus.tex, line 122:: sage: y = var('y'); u = sin(x) + x*cos(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 example in ./calculus.tex, line 153:: sage: x, y = SR.var('x,y') sage: p = (x+y)*(x+1)^2 sage: p2 = p.expand(); p2 x^3 + x^2*y + 2*x^2 + 2*x*y + x + y Sage example in ./calculus.tex, line 160:: sage: p2.collect(x) x^3 + x^2*(y + 2) + x*(2*y + 1) + y Sage example in ./calculus.tex, line 165:: sage: ((x+y+sin(x))^2).expand().collect(sin(x)) x^2 + 2*x*y + y^2 + 2*(x + y)*sin(x) + sin(x)^2 Sage example in ./calculus.tex, line 254:: sage: (x^x/x).simplify() x^(x - 1) Sage example in ./calculus.tex, line 260:: sage: f = (e^x-1) / (1+e^(x/2)); f.simplify_exp() e^(1/2*x) - 1 Sage example in ./calculus.tex, line 266:: sage: f = cos(x)^6 + sin(x)^6 + 3 * sin(x)^2 * cos(x)^2 sage: f.simplify_trig() 1 Sage example in ./calculus.tex, line 273:: sage: f = cos(x)^6; f.reduce_trig() 1/32*cos(6*x) + 3/16*cos(4*x) + 15/32*cos(2*x) + 5/16 sage: f = sin(5 * x); f.expand_trig() 5*cos(x)^4*sin(x) - 10*cos(x)^2*sin(x)^3 + sin(x)^5 Sage example in ./calculus.tex, line 306:: sage: n = var('n'); f = factorial(n+1)/factorial(n) sage: f.simplify_factorial() n + 1 Sage example in ./calculus.tex, line 318:: sage: f = sqrt(abs(x)^2); f.simplify_radical() abs(x) sage: f = log(x*y); f.simplify_radical() log(x) + log(y) Sage example in ./calculus.tex, line 371:: sage: assume(x > 0); bool(sqrt(x^2) == x) True sage: forget(x > 0); bool(sqrt(x^2) == x) False sage: n = var('n'); assume(n, 'integer'); sin(n*pi).simplify() 0 Sage example in ./calculus.tex, line 420:: sage: a = var('a') sage: c = (a+1)^2 - (a^2+2*a+1) Sage example in ./calculus.tex, line 425:: sage: eq =  c * x == 0 Sage example in ./calculus.tex, line 430:: sage: eq2 = eq / c; eq2 x == 0 sage: solve(eq2, x) [x == 0] Sage example in ./calculus.tex, line 437:: sage: solve(eq, x) [x == x] Sage example in ./calculus.tex, line 444:: sage: expand(c) 0 Sage example in ./calculus.tex, line 452:: sage: c = cos(a)^2 + sin(a)^2 - 1 sage: eq = c*x == 0 sage: solve(eq, x) [x == 0] Sage example in ./calculus.tex, line 460:: sage: c.simplify_trig() 0 sage: c.is_zero() True Sage example in ./calculus.tex, line 516:: sage: z, phi = var('z, phi') sage: eq =  z**2 - 2/cos(phi)*z + 5/cos(phi)**2 - 4 == 0; eq z^2 - 2*z/cos(phi) + 5/cos(phi)^2 - 4 == 0 Sage example in ./calculus.tex, line 523:: sage: eq.lhs() z^2 - 2*z/cos(phi) + 5/cos(phi)^2 - 4 sage: eq.rhs() 0 Sage example in ./calculus.tex, line 531:: sage: solve(eq, z) [z == -(2*sqrt(cos(phi)^2 - 1) - 1)/cos(phi), z ==  (2*sqrt(cos(phi)^2 - 1) + 1)/cos(phi)] Sage example in ./calculus.tex, line 537:: sage: y = var('y'); solve(y^6==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 example in ./calculus.tex, line 544:: sage: solve(x^2-1, x, solution_dict=True) [{x: -1}, {x: 1}] Sage example in ./calculus.tex, line 550:: sage: solve([x+y == 3, 2*x+2*y == 6], x, y) [[x == -r1 + 3, y == r1]] Sage example in ./calculus.tex, line 560:: sage: solve([cos(x)*sin(x) == 1/2, x+y == 0], x, y) [[x == 1/4*pi + pi*z..., y == -1/4*pi - pi*z...]] Sage example in ./calculus.tex, line 565:: sage: solve(x^2+x-1 > 0, x) [[x < -1/2*sqrt(5) - 1/2], [x > 1/2*sqrt(5) - 1/2]] Sage example in ./calculus.tex, line 583:: sage: x, y, z = var('x, y, z') sage: solve([x^2 * y * z == 18, x * y^3 * z == 24,\ ....:        x * y * z^4 == 3], x, y, z) [[x == (-2.76736473308 - 1.71347969911*I), y == (-0.570103503963 + 2.00370597877*I), z == (-0.801684337646 - 0.14986077496*I)], ...] Sage example in ./calculus.tex, line 597:: sage: expr = sin(x) + sin(2 * x) + sin(3 * x) sage: solve(expr, x) [sin(3*x) == -sin(2*x) - sin(x)] Sage example in ./calculus.tex, line 605:: sage: find_root(expr, 0.1, pi) 2.0943951023931957 Sage example in ./calculus.tex, line 610:: 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 example in ./calculus.tex, line 629:: sage: (x^3+2*x+1).roots(x) [(-1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(I*sqrt(3) + 1) - 1/3*(I*sqrt(3) - 1)/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3), 1), (-1/2*(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3)*(-I*sqrt(3) + 1) - 1/3*(-I*sqrt(3) - 1)/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3), 1), ((1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3) - 2/3/(1/18*sqrt(59)*sqrt(3) - 1/2)^(1/3), 1)] Sage example in ./calculus.tex, line 658:: sage: (x^3+2*x+1).roots(x, ring=RR) [(-0.453397651516404, 1)] Sage example in ./calculus.tex, line 662:: 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 example in ./calculus.tex, line 680:: sage: solve(x^(1/x)==(1/x)^x, x) [(1/x)^x == x^(1/x)] Sage example in ./calculus.tex, line 706:: sage: y = function('y', x) sage: desolve(diff(y,x,x) + x*diff(y,x) + y == 0, y, [0,0,1]) -1/2*I*sqrt(2)*sqrt(pi)*erf(1/2*I*sqrt(2)*x)*e^(-1/2*x^2) Sage example in ./calculus.tex, line 733:: sage: k, n = var('k, n') sage: sum(k, k, 1, n).factor() 1/2*(n + 1)*n Sage example in ./calculus.tex, line 739:: sage: n, k, y = var('n, k, y') sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n) (x + y)^n Sage example in ./calculus.tex, line 745:: sage: k, n = var('k, n') sage: sum(binomial(n,k), k, 0, n),\ ....: sum(k * binomial(n, k), k, 0, n),\ ....: sum((-1)^k*binomial(n,k), k, 0, n) (2^n, 2^(n - 1)*n, 0) Sage example in ./calculus.tex, line 753:: sage: a, q, k, n = var('a, q, k, n') sage: sum(a*q^k, k, 0, n) (a*q^(n + 1) - a)/(q - 1) Sage example in ./calculus.tex, line 760:: sage: assume(abs(q) < 1) sage: sum(a*q^k, k, 0, infinity) -a/(q - 1) Sage example in ./calculus.tex, line 766:: sage: forget(); assume(q > 1); sum(a*q^k, k, 0, infinity) Traceback (most recent call last): ... ValueError: Sum is divergent. Sage example in ./calculus.tex, line 842:: 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) Infinity Sage example in ./calculus.tex, line 855:: sage: limit(f(x), x = pi/4, dir='minus') +Infinity sage: limit(f(x), x = pi/4, dir='plus') -Infinity Sage example in ./calculus.tex, line 898:: sage: u(n) = n^100 / 100^n sage: u(2.);u(3.);u(4.);u(5.);u(6.);u(7.);u(8.);u(9.);u(10.) 1.26765060022823e26 5.15377520732011e41 1.60693804425899e52 7.88860905221012e59 6.53318623500071e65 3.23447650962476e70 2.03703597633449e74 2.65613988875875e77 1.00000000000000e80 Sage example in ./calculus.tex, line 914:: sage: plot(u(x), x, 1, 40) Sage example in ./calculus.tex, line 929:: sage: v(x) = diff(u(x), x); sol = solve(v(x) == 0, x); sol [x == 100/log(100), x == 0] sage: floor(sol[0].rhs()) 21 Sage example in ./calculus.tex, line 938:: sage: limit(u(n), n=infinity) 0 sage: n0 = find_root(u(n) - 1e-8 == 0, 22, 1000); n0 105.07496210187252 Sage example in ./calculus.tex, line 988:: 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 example in ./calculus.tex, line 993:: 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 example in ./calculus.tex, line 1002:: sage: (ln(2*sin(x))).series(x==pi/6, 3).truncate() -1/18*(pi - 6*x)^2 - 1/6*sqrt(3)*(pi - 6*x) Sage example in ./calculus.tex, line 1017:: sage: taylor((x**3+x)**(1/3) - (x**3-x)**(1/3), x, infinity, 2) 2/3/x Sage example in ./calculus.tex, line 1041:: sage: tan(4*arctan(1/5)).simplify_trig() 120/119 sage: tan(pi/4+arctan(1/239)).simplify_trig() 120/119 Sage example in ./calculus.tex, line 1052:: 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 example in ./calculus.tex, line 1093:: sage: k = var('k') sage: sum(1/k^2, k, 1, infinity),\ ....: sum(1/k^4, k, 1, infinity),\ ....: sum(1/k^5, k, 1, infinity) (1/6*pi^2, 1/90*pi^4, zeta(5)) Sage example in ./calculus.tex, line 1111:: 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) 3.141592653589793238462643383279502884197169399375105820974... sage: (pi-1/s).n(digits=100).n() -4.36415445739398e-96 Sage example in ./calculus.tex, line 1139:: sage: n = var('n'); u = sin(pi*(sqrt(4*n^2+1)-2*n)) sage: taylor(u, n, infinity, 3) 1/4*pi/n - 1/384*(6*pi + pi^3)/n^3 Sage example in ./calculus.tex, line 1163:: 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 example in ./calculus.tex, line 1180:: 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 example in ./calculus.tex, line 1195:: sage: x, y = var('x, y'); f = ln(x**2+y**2) / 2 sage: delta = diff(f,x,2) + diff(f,y,2) sage: delta.simplify_full() 0 Sage example in ./calculus.tex, line 1231:: 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 example in ./calculus.tex, line 1241:: sage: integrate(exp(-x), x, -infinity, infinity) Traceback (most recent call last): ... ValueError: Integral is divergent. Sage example in ./calculus.tex, line 1254:: 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 example in ./calculus.tex, line 1270:: sage: integral_numerical(sin(x)/x, 0, 1)   # abs tol 1e-12 (0.94608307036718287, 1.0503632079297086e-14) sage: g = integrate(exp(-x**2), x, 0, infinity) sage: g, g.n()                             # abs tol 1e-12 (1/2*sqrt(pi), 0.886226925452758) sage: approx = integral_numerical(exp(-x**2), 0, infinity) sage: approx                               # abs tol 1e-12 (0.88622692545275705, 1.7147744320162414e-08) sage: approx[0]-g.n()                      # abs tol 1e-12 -8.88178419700125e-16 Sage example in ./calculus.tex, line 1482:: sage: A = matrix(QQ, [[1,2],[3,4]]); A [1 2] [3 4] Sage example in ./calculus.tex, line 1629:: 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 example in ./calculus.tex, line 1641:: sage: A.eigenvectors_right() [(1, [ (1, -1, 1) ], 1), (-2, [ (1, -1, 0) ], 2)] Sage example in ./calculus.tex, line 1652:: 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 example in ./calculus.tex, line 1686:: sage: A = matrix(QQ, [[1,-1/2],[-1/2,-1]]) sage: A.jordan_form() Traceback (most recent call last): ... RuntimeError: Some eigenvalue does not exist in Rational Field. Sage example in ./calculus.tex, line 1695:: sage: A = matrix(QQ, [[1,-1/2],[-1/2,-1]]) sage: A.minimal_polynomial() x^2 - 5/4 Sage example in ./calculus.tex, line 1701:: 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 example in ./calculus.tex, line 1734:: 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] ) """ """ Tests extracted from sol/calculus.tex. Sage example in ./sol/calculus.tex, line 3:: sage: reset() Sage example in ./sol/calculus.tex, line 9:: sage: n, k = var('n, k'); p = 4; s = [n + 1] sage: for k in (1..p): ....:  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*(2*n + 1)*(n + 1)*n, 1/4*(n + 1)^2*n^2, 1/30*(3*n^2 + 3*n - 1)*(2*n + 1)*(n + 1)*n] Sage example in ./sol/calculus.tex, line 34:: sage: x, h, a = var('x, h, a'); f = function('f') sage: g(x) = taylor(f(x), 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 example in ./sol/calculus.tex, line 57:: sage: n = 7; x, h, a = var('x h a') sage: f = function('f') sage: g(x) = taylor(f(x), 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 example in ./sol/calculus.tex, line 82:: 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 example in ./sol/calculus.tex, line 94:: sage: M = 12*(1/38)+20*(1/57)+ 7*(1/239)+24*(1/268) sage: M 37735/48039 Sage example in ./sol/calculus.tex, line 113:: 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 example in ./sol/calculus.tex, line 143:: sage: n = var('n') sage: phi = lambda x: n*pi+pi/2-arctan(1/x) sage: 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 example in ./sol/calculus.tex, line 192:: sage: h = var('h') 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 example in ./sol/calculus.tex, line 230:: 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_radical() 0 Sage example in ./sol/calculus.tex, line 258:: sage: t = var('t') 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_radical() pi + 2*log(sqrt(2) + 1) + 2*log(sqrt(2) - 1) Sage example in ./sol/calculus.tex, line 272:: sage: ln(exp(J).simplify_log()) pi Sage example in ./sol/calculus.tex, line 281:: sage: l = sum(v(n) for n in (0..40)); l.n(digits=60) 3.14159265358979323846264338327950288419716939937510581474759 sage: pi.n(digits=60) 3.14159265358979323846264338327950288419716939937510582097494 sage: print("%e" % (l-pi).n(digits=60)) -6.227358e-54 Sage example in ./sol/calculus.tex, line 302:: sage: X = var('X') sage: ps = lambda f,g : integral(f * g, X, -pi, pi) sage: n = 5; Q = sin(X) sage: a, a0, a1, a2, a3, a4, a5 = var('a a0 a1 a2 a3 a4 a5') sage: 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,-6,6,color='red') + plot(Q,X,-6,6,color='blue') Sage example in ./sol/calculus.tex, line 353:: sage: p, e = var('p e') sage: theta1, theta2, theta3 = var('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 example in ./sol/calculus.tex, line 365:: 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) [0, 0, 0] Sage example in ./sol/calculus.tex, line 390:: 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)) [0, 0, 0] Sage example in ./sol/calculus.tex, line 409:: 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: i = vector([1, 0, 0]); W =  p * S + e * i.cross_product(N) sage: e = S.norm() / D.norm(); p = N.norm() / D.norm() sage: a = p/(1-e^2); c = a * e; b = sqrt(a^2 - c^2) sage: X = S.cross_product(D); 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 example in ./sol/calculus.tex, line 445:: 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 example in ./sol/calculus.tex, line 463:: sage: H = A.echelon_form() Sage example in ./sol/calculus.tex, line 484:: 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 example in ./sol/calculus.tex, line 496:: 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 example in ./sol/calculus.tex, line 511:: 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 example in ./sol/calculus.tex, line 519:: 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 example in ./sol/calculus.tex, line 533:: 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 example in ./sol/calculus.tex, line 540:: sage: B = C.solve_left(A); B [ 0 -1  0] [ 2  3  0] [ 2  1  0] Sage example in ./sol/calculus.tex, line 548:: sage: C.left_kernel() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [1 2 1] Sage example in ./sol/calculus.tex, line 560:: sage: x, y, z = var('x, y, z'); v = matrix([[1, 2, 1]]) 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 example in ./sol/calculus.tex, line 568:: sage: A == B*C True """ # This file was *autogenerated* from the file calculus_doctest.sage.