Operations on holonomic functions
Addition and Multiplication
Two holonomic functions can be added or multiplied with the result also a holonomic functions.
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy.polys.domains import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx')
pandqhere are holonomic representation of \(e^x\) and \(\sin(x)\) respectively.>>> p = HolonomicFunction(Dx - 1, x, 0, [1]) >>> q = HolonomicFunction(Dx**2 + 1, x, 0, [0, 1])Holonomic representation of \(e^x+\sin(x)\)
>>> p + q HolonomicFunction((-1) + (1)*Dx + (-1)*Dx**2 + (1)*Dx**3, x, 0, [1, 2, 1])Holonomic representation of \(e^x \cdot \sin(x)\)
>>> p * q HolonomicFunction((2) + (-2)*Dx + (1)*Dx**2, x, 0, [0, 1])
Integration and Differentiation
- HolonomicFunction.integrate(limits, initcond=False)[source]
Integrates the given holonomic function.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).integrate((x, 0, x)) # e^x - 1 HolonomicFunction((-1)*Dx + (1)*Dx**2, x, 0, [0, 1]) >>> HolonomicFunction(Dx**2 + 1, x, 0, [1, 0]).integrate((x, 0, x)) HolonomicFunction((1)*Dx + (1)*Dx**3, x, 0, [0, 1, 0])
- HolonomicFunction.diff(*args, **kwargs)[source]
Differentiation of the given Holonomic function.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).diff().to_expr() cos(x) >>> HolonomicFunction(Dx - 2, x, 0, [1]).diff().to_expr() 2*exp(2*x)
See also
Composition with polynomials
- HolonomicFunction.composition(expr, *args, **kwargs)[source]
Returns function after composition of a holonomic function with an algebraic function. The method cannot compute initial conditions for the result by itself, so they can be also be provided.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x).composition(x**2, 0, [1]) # e^(x**2) HolonomicFunction((-2*x) + (1)*Dx, x, 0, [1]) >>> HolonomicFunction(Dx**2 + 1, x).composition(x**2 - 1, 1, [1, 0]) HolonomicFunction((4*x**3) + (-1)*Dx + (x)*Dx**2, x, 1, [1, 0])
See also
Convert to holonomic sequence
- HolonomicFunction.to_sequence(lb=True)[source]
Finds recurrence relation for the coefficients in the series expansion of the function about \(x_0\), where \(x_0\) is the point at which the initial condition is stored.
Explanation
If the point \(x_0\) is ordinary, solution of the form \([(R, n_0)]\) is returned. Where \(R\) is the recurrence relation and \(n_0\) is the smallest
nfor which the recurrence holds true.If the point \(x_0\) is regular singular, a list of solutions in the format \((R, p, n_0)\) is returned, i.e. \([(R, p, n_0), ... ]\). Each tuple in this vector represents a recurrence relation \(R\) associated with a root of the indicial equation
p. Conditions of a different format can also be provided in this case, see the docstring of HolonomicFunction class.If it’s not possible to numerically compute a initial condition, it is returned as a symbol \(C_j\), denoting the coefficient of \((x - x_0)^j\) in the power series about \(x_0\).
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols, S >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).to_sequence() [(HolonomicSequence((-1) + (n + 1)Sn, n), u(0) = 1, 0)] >>> HolonomicFunction((1 + x)*Dx**2 + Dx, x, 0, [0, 1]).to_sequence() [(HolonomicSequence((n**2) + (n**2 + n)Sn, n), u(0) = 0, u(1) = 1, u(2) = -1/2, 2)] >>> HolonomicFunction(-S(1)/2 + x*Dx, x, 0, {S(1)/2: [1]}).to_sequence() [(HolonomicSequence((n), n), u(0) = 1, 1/2, 1)]
See also
References
Series expansion
- HolonomicFunction.series(n=6, coefficient=False, order=True, _recur=None)[source]
Finds the power series expansion of given holonomic function about \(x_0\).
Explanation
A list of series might be returned if \(x_0\) is a regular point with multiple roots of the indicial equation.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx') >>> HolonomicFunction(Dx - 1, x, 0, [1]).series() # e^x 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6) >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).series(n=8) # sin(x) x - x**3/6 + x**5/120 - x**7/5040 + O(x**8)
See also
Numerical evaluation
- HolonomicFunction.evalf(points, method='RK4', h=0.05, derivatives=False)[source]
Finds numerical value of a holonomic function using numerical methods. (RK4 by default). A set of points (real or complex) must be provided which will be the path for the numerical integration.
Explanation
The path should be given as a list \([x_1, x_2, \dots x_n]\). The numerical values will be computed at each point in this order \(x_1 \rightarrow x_2 \rightarrow x_3 \dots \rightarrow x_n\).
Returns values of the function at \(x_1, x_2, \dots x_n\) in a list.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import QQ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(QQ.old_poly_ring(x),'Dx')
A straight line on the real axis from (0 to 1)
>>> r = [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1]
Runge-Kutta 4th order on e^x from 0.1 to 1. Exact solution at 1 is 2.71828182845905
>>> HolonomicFunction(Dx - 1, x, 0, [1]).evalf(r) [1.10517083333333, 1.22140257085069, 1.34985849706254, 1.49182424008069, 1.64872063859684, 1.82211796209193, 2.01375162659678, 2.22553956329232, 2.45960141378007, 2.71827974413517]
Euler’s method for the same
>>> HolonomicFunction(Dx - 1, x, 0, [1]).evalf(r, method='Euler') [1.1, 1.21, 1.331, 1.4641, 1.61051, 1.771561, 1.9487171, 2.14358881, 2.357947691, 2.5937424601]
One can also observe that the value obtained using Runge-Kutta 4th order is much more accurate than Euler’s method.
Convert to a linear combination of hypergeometric functions
- HolonomicFunction.to_hyper(as_list=False, _recur=None)[source]
Returns a hypergeometric function (or linear combination of them) representing the given holonomic function.
Explanation
Returns an answer of the form: \(a_1 \cdot x^{b_1} \cdot{hyper()} + a_2 \cdot x^{b_2} \cdot{hyper()} \dots\)
This is very useful as one can now use
hyperexpandto find the symbolic expressions/functions.Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> # sin(x) >>> HolonomicFunction(Dx**2 + 1, x, 0, [0, 1]).to_hyper() x*hyper((), (3/2,), -x**2/4) >>> # exp(x) >>> HolonomicFunction(Dx - 1, x, 0, [1]).to_hyper() hyper((), (), x)
See also
Convert to a linear combination of Meijer G-functions
- HolonomicFunction.to_meijerg()[source]
Returns a linear combination of Meijer G-functions.
Examples
>>> from sympy.holonomic import expr_to_holonomic >>> from sympy import sin, cos, hyperexpand, log, symbols >>> x = symbols('x') >>> hyperexpand(expr_to_holonomic(cos(x) + sin(x)).to_meijerg()) sin(x) + cos(x) >>> hyperexpand(expr_to_holonomic(log(x)).to_meijerg()).simplify() log(x)
See also
Convert to expressions
- HolonomicFunction.to_expr()[source]
Converts a Holonomic Function back to elementary functions.
Examples
>>> from sympy.holonomic.holonomic import HolonomicFunction, DifferentialOperators >>> from sympy import ZZ >>> from sympy import symbols, S >>> x = symbols('x') >>> R, Dx = DifferentialOperators(ZZ.old_poly_ring(x),'Dx') >>> HolonomicFunction(x**2*Dx**2 + x*Dx + (x**2 - 1), x, 0, [0, S(1)/2]).to_expr() besselj(1, x) >>> HolonomicFunction((1 + x)*Dx**3 + Dx**2, x, 0, [1, 1, 1]).to_expr() x*log(x + 1) + log(x + 1) + 1