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## -*- 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') Graphics object consisting of 2 graphics primitives
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 range(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..., 4.78068810228705...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..., 4.78068810228705...e-15, 21, 0)
Sage example in ./integration.tex, line 465::
sage: gp('intnum(x=17, 42, exp(-x^2)*log(x))') # rel tol 1e-17 2.5657285005610514829176211363206621657 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))') 1.99999999999999999999...
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))') 1.99999999999999999999...
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) Graphics object consisting of 1 graphics primitive
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, 1.12419127424558], [1.0, 1.461590162288825]]
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. |