#14318: More doctests from the book "Calcul mathématique avec Sage" c…

…hapter "integration"

src/sage/tests/french_book/integration_doctest.py

@@ -0,0 +1,218 @@
+## -*- encoding: utf-8 -*-
+"""
+This file (./integration_doctest.sage) was *autogenerated* from ./integration.tex,
+with sagetex.sty version 2011/05/27 v2.3.1.
+It contains the contents of all the sageexample environments from this file.
+You should be able to doctest this file with:
+sage -t ./integration_doctest.sage
+It is always safe to delete this file; it is not used in typesetting your
+document.
+
+Sage example in ./integration.tex, line 44::
+
+    sage: x = var('x'); f(x) = exp(-x^2) * log(x)
+    sage: N(integrate(f, x, 1, 3))
+    0.035860294991267694
+    sage: plot(f, 1, 3, fill='axis')
+
+Sage example in ./integration.tex, line 103::
+
+    sage: fp = plot(f, 1, 3, color='red')
+    sage: n = 4
+    sage: interp_points = [(1+2*u/(n-1), N(f(1+2*u/(n-1))))
+    ....:                  for u in xrange(n)]
+    sage: A = PolynomialRing(RR, 'x')
+    sage: pp = plot(A.lagrange_polynomial(interp_points), 1, 3, fill='axis')
+    sage: show(fp+pp)
+
+Sage example in ./integration.tex, line 346::
+
+    sage: N(integrate(exp(-x^2)*log(x), x, 17, 42)) # rel tol 7e-15
+    2.5657285006962035e-127
+
+Sage example in ./integration.tex, line 355::
+
+    sage: integrate(log(1+x)*x, x, 0, 1)
+    1/4
+    sage: N(integrate(log(1+x)*x, x, 0, 1))
+    0.250000000000000
+
+Sage example in ./integration.tex, line 372::
+
+    sage: numerical_integral(exp(-x^2)*log(x), 17, 42) # rel tol 7e-12
+    (2.5657285006962035e-127, 3.3540254049238093e-128)
+
+Sage example in ./integration.tex, line 394::
+
+    sage: numerical_integral(exp(-x^100), 0, 1.1)
+    (0.99432585119150..., 4.0775730...e-09)
+    sage: numerical_integral(exp(-x^100), 0, 1.1, algorithm='qng')
+    (0.994327538576531..., 0.016840666914...)
+
+Sage example in ./integration.tex, line 404::
+
+    sage: integrate(exp(-x^2)*log(x), x, 17, 42)
+    integrate(e^(-x^2)*log(x), x, 17, 42)
+
+Sage example in ./integration.tex, line 412::
+
+    sage: N(integrate(exp(-x^2)*log(x), x, 17, 42), 200) # rel tol 7e-15
+    2.5657285006962035e-127
+
+Sage example in ./integration.tex, line 417::
+
+    sage: N(integrate(sin(x)*exp(cos(x)), x, 0, pi), 200)
+    2.3504023872876029137647637011912016303114359626681917404591
+
+Sage example in ./integration.tex, line 430::
+
+    sage: sage.calculus.calculus.nintegral(sin(sin(x)), x, 0, 1)
+    (0.430606103120690..., 4.780688102287053...e-15, 21, 0)
+
+Sage example in ./integration.tex, line 436::
+
+    sage: g(x) = sin(sin(x))
+    sage: g.nintegral(x, 0, 1)
+    (0.430606103120690..., 4.780688102287053...e-15, 21, 0)
+
+Sage example in ./integration.tex, line 465::
+
+    sage: gp('intnum(x=17, 42, exp(-x^2)*log(x))') # rel tol 8e-29
+    2.5657285005610514829173563961304785900 E-127
+
+Sage example in ./integration.tex, line 474::
+
+    sage: gp('intnum(x=0, 1, sin(sin(x)))')
+    0.430606103120690604912377355...
+    sage: old_prec = gp.set_precision(50)
+    sage: gp('intnum(x=0, 1, sin(sin(x)))')
+    0.43060610312069060491237735524846578643360804182200
+
+Sage example in ./integration.tex, line 490::
+
+    sage: p = gp.set_precision(old_prec) # on remet la précision par défaut
+    sage: gp('intnum(x=0, 1, x^(-1/2))')
+    1.999999999999999999999...
+
+Sage example in ./integration.tex, line 496::
+
+    sage: gp('intnum(x=[0, -1/2], 1, x^(-1/2))')
+    2.000000000000000000000000000...
+
+Sage example in ./integration.tex, line 504::
+
+    sage: gp('intnum(x=[0, -1/42], 1, x^(-1/2))')
+    1.999999999999999999999...
+
+Sage example in ./integration.tex, line 518::
+
+    sage: import mpmath
+    sage: mpmath.mp.prec = 53
+    sage: mpmath.quad(lambda x: mpmath.sin(mpmath.sin(x)), [0, 1])
+    mpf('0.43060610312069059')
+
+Sage example in ./integration.tex, line 526::
+
+    sage: mpmath.mp.prec = 113
+    sage: mpmath.quad(lambda x: mpmath.sin(mpmath.sin(x)), [0, 1])
+    mpf('0.430606103120690604912377355248465809')
+    sage: mpmath.mp.prec = 114
+    sage: mpmath.quad(lambda x: mpmath.sin(mpmath.sin(x)), [0, 1])
+    mpf('0.430606103120690604912377355248465785')
+
+Sage example in ./integration.tex, line 550::
+
+    sage: mpmath.quad(sin(sin(x)), [0, 1])
+    Traceback (most recent call last):
+    ...
+    TypeError: no canonical coercion from
+    <type 'sage.libs.mpmath.ext_main.mpf'> to Symbolic Ring
+
+Sage example in ./integration.tex, line 565::
+
+    sage: g(x) = max_symbolic(sin(x), cos(x))
+    sage: mpmath.mp.prec = 100
+    sage: mpmath.quadts(lambda x: g(N(x, 100)), [0, 1])
+    mpf('0.873912416263035435957979086252')
+
+Sage example in ./integration.tex, line 574::
+
+    sage: mpmath.mp.prec = 170
+    sage: mpmath.quadts(lambda x: g(N(x, 190)), [0, 1])
+    mpf('0.87391090757400975205393005981962476344054148354188794')
+    sage: N(sqrt(2) - cos(1), 100)
+    0.87391125650495533140075211677
+
+Sage example in ./integration.tex, line 585::
+
+    sage: mpmath.quadts(lambda x: g(N(x, 170)), [0, mpmath.pi / 4, 1])
+    mpf('0.87391125650495533140075211676672147483736145475902551')
+
+Sage example in ./integration.tex, line 750::
+
+    sage: T = ode_solver()
+
+Sage example in ./integration.tex, line 761::
+
+    sage: def f_1(t,y,params): return [y[1],params[0]*(1-y[0]^2)*y[1]-y[0]]
+    sage: T.function = f_1
+
+Sage example in ./integration.tex, line 776::
+
+    sage: def j_1(t,y,params):
+    ....:     return [[0, 1],
+    ....:             [-2*params[0]*y[0]*y[1]-1, params[0]*(1-y[0]^2)],
+    ....:             [0,0]]
+    sage: T.jacobian = j_1
+
+Sage example in ./integration.tex, line 786::
+
+    sage: T.algorithm = "rk8pd"
+    sage: T.ode_solve(y_0=[1,0], t_span=[0,100], params=[10],
+    ....:             num_points=1000)
+    sage: f = T.interpolate_solution()
+
+Sage example in ./integration.tex, line 801::
+
+    sage: plot(f, 0, 100)
+
+Sage example in ./integration.tex, line 838::
+
+    sage: t, y = var('t, y')
+    sage: desolve_rk4(t*y*(2-y), y, ics=[0,1], end_points=[0, 1], step=0.5)
+    [[0, 1], [0.5, 1.12419127425], [1.0, 1.46159016229]]
+
+Sage example in ./integration.tex, line 861::
+
+    sage: import mpmath
+    sage: mpmath.mp.prec = 53
+    sage: sol = mpmath.odefun(lambda t, y: y, 0, 1)
+    sage: sol(1)
+    mpf('2.7182818284590451')
+    sage: mpmath.mp.prec = 100
+    sage: sol(1)
+    mpf('2.7182818284590452353602874802307')
+    sage: N(exp(1), 100)
+    2.7182818284590452353602874714
+
+Sage example in ./integration.tex, line 889::
+
+    sage: mpmath.mp.prec = 53
+    sage: f = mpmath.odefun(lambda t, y: [-y[1], y[0]], 0, [1, 0])
+    sage: f(3)
+    [mpf('-0.98999249660044542'), mpf('0.14112000805986721')]
+    sage: (cos(3.), sin(3.))
+    (-0.989992496600445, 0.141120008059867)
+
+Sage example in ./integration.tex, line 939::
+
+    sage: mpmath.mp.prec = 10
+    sage: sol = mpmath.odefun(lambda t, y: y, 0, 1)
+    sage: sol(1)
+    mpf('2.7148')
+    sage: mpmath.mp.prec = 100
+    sage: sol(1)
+    mpf('2.7135204235459511323824699502438')
+
+"""
+# This file was *autogenerated* from the file integration_doctest.sage.