ore_algebra.analytic.function¶
D-Finite analytic functions
Classes
DFiniteFunction(dop, ini[, name, max_prec, …]) |
At the moment, this class just provides a simple caching mechanism for repeated evaluations of a D-Finite function on the real line. |
RealPolApprox(pol, prec) |
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class
ore_algebra.analytic.function.DFiniteFunction(dop, ini, name='dfinitefun', max_prec=256, max_rad=[+/- inf])¶ At the moment, this class just provides a simple caching mechanism for repeated evaluations of a D-Finite function on the real line. It may evolve to support evaluations on the complex plane, branch cuts, ring operations on D-Finite functions, and more. Do not expect any API stability.
TESTS:
sage: from ore_algebra import * sage: from ore_algebra.analytic.function import DFiniteFunction sage: from ore_algebra.analytic.path import Point sage: from sage.rings.infinity import AnInfinity sage: Dops, x, Dx = DifferentialOperators() sage: f = DFiniteFunction(Dx - 1, [1], max_rad=7/8) sage: f._disk(Point(pi, dop=Dx-1)) (7/2, 0.5000000000000000) sage: f = DFiniteFunction(Dx^2 - x, ....: [1/(gamma(2/3)*3^(2/3)), -1/(gamma(1/3)*3^(1/3))], ....: name='my_Ai') sage: f.plot((-5,5)) Graphics object consisting of 1 graphics primitive sage: f._known_bound(RBF(RIF(1/3, 2/3)), post_transform=Dops.one()) [0.2...] sage: f._known_bound(RBF(RIF(-1, 5.99)), post_transform=Dops.one()) [+/- ...] sage: f._known_bound(RBF(RIF(-1, 6.01)), Dops.one()) [+/- inf] sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1]) sage: f.plot((0, 4)) Graphics object consisting of 1 graphics primitive sage: f._known_bound(RBF(RIF(1,4)), Dops.one()) [1e+0 +/- 0.3...] sage: f.approx(1, post_transform=Dx), f.approx(2, post_transform=Dx) ([0.5000000000000...], [0.2000000000000...]) sage: f._known_bound(RBF(RIF(1.1, 2.9)), post_transform=Dx) [+/- 0.4...] sage: f._known_bound(RBF(AnInfinity()), Dops.one()) [+/- inf] sage: f = DFiniteFunction((x + 1)*Dx^2 + 2*x*Dx, [0, 1]) sage: f._known_bound(RBF(RIF(1,10)),Dops.one()) nan sage: f._known_bound(RBF(AnInfinity()), Dops.one()) nan sage: f = DFiniteFunction(Dx^2 + 2*x*Dx, [1, -2/sqrt(pi)], name='my_erfc') sage: f._known_bound(RBF(RIF(-1/2,1/2)), post_transform=Dx^2) [+/- inf] sage: _ = f.approx(1/2, post_transform=Dx^2) sage: _ = f.approx(-1/2, post_transform=Dx^2) sage: f._known_bound(RBF(RIF(-1/2,1/2)), post_transform=Dx^2) [+/- 1.5...]
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approx(pt, prec=None, post_transform=None)¶ TESTS:
sage: from ore_algebra import * sage: from ore_algebra.analytic.function import DFiniteFunction sage: DiffOps, x, Dx = DifferentialOperators() sage: h = DFiniteFunction(Dx^3-1, [0, 0, 1]) sage: h.approx(0, post_transform=Dx^2) [2.0000000000000...] sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1], max_prec=20) sage: f.approx(1/3, prec=10) [0.32...] sage: f.approx(1/3, prec=40) [0.321750554396...] sage: f.approx(1/3, prec=10, post_transform=Dx) [0.9...] sage: f.approx(1/3, prec=40, post_transform=Dx) [0.900000000000...] sage: f.approx(1/3, prec=10, post_transform=Dx^2) [-0.54...] sage: f.approx(1/3, prec=40, post_transform=Dx^2) [-0.540000000000...]
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plot(x_range, **options)¶ Plot this function.
The plot is intended to give an idea of the accuracy of the evaluations that led to it. However, it may not be a rigorous enclosure of the graph of the function.
EXAMPLES:
sage: from ore_algebra import * sage: from ore_algebra.analytic.function import DFiniteFunction sage: DiffOps, x, Dx = DifferentialOperators() sage: f = DFiniteFunction(Dx^2 - x, ....: [1/(gamma(2/3)*3^(2/3)), -1/(gamma(1/3)*3^(1/3))]) sage: plot(f, (-10, 5), color='black') Graphics object consisting of 1 graphics primitive
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plot_known()¶ TESTS:
sage: from ore_algebra import * sage: from ore_algebra.analytic.function import DFiniteFunction sage: DiffOps, x, Dx = DifferentialOperators() sage: f = DFiniteFunction((x^2 + 1)*Dx^2 + 2*x*Dx, [0, 1]) sage: f(-10, 100) # long time [-1.4711276743037345918528755717...] sage: f.approx(5, post_transform=Dx) # long time [0.038461538461538...] sage: f.plot_known() # long time Graphics object consisting of ... graphics primitives
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