477 lines
17 KiB
Python
477 lines
17 KiB
Python
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"""
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Finite difference weights
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=========================
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This module implements an algorithm for efficient generation of finite
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difference weights for ordinary differentials of functions for
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derivatives from 0 (interpolation) up to arbitrary order.
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The core algorithm is provided in the finite difference weight generating
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function (``finite_diff_weights``), and two convenience functions are provided
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for:
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- estimating a derivative (or interpolate) directly from a series of points
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is also provided (``apply_finite_diff``).
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- differentiating by using finite difference approximations
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(``differentiate_finite``).
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"""
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from sympy.core.function import Derivative
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from sympy.core.singleton import S
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from sympy.core.function import Subs
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from sympy.core.traversal import preorder_traversal
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from sympy.utilities.exceptions import sympy_deprecation_warning
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from sympy.utilities.iterables import iterable
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def finite_diff_weights(order, x_list, x0=S.One):
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"""
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Calculates the finite difference weights for an arbitrarily spaced
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one-dimensional grid (``x_list``) for derivatives at ``x0`` of order
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0, 1, ..., up to ``order`` using a recursive formula. Order of accuracy
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is at least ``len(x_list) - order``, if ``x_list`` is defined correctly.
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Parameters
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==========
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order: int
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Up to what derivative order weights should be calculated.
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0 corresponds to interpolation.
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x_list: sequence
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Sequence of (unique) values for the independent variable.
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It is useful (but not necessary) to order ``x_list`` from
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nearest to furthest from ``x0``; see examples below.
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x0: Number or Symbol
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Root or value of the independent variable for which the finite
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difference weights should be generated. Default is ``S.One``.
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Returns
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=======
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list
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A list of sublists, each corresponding to coefficients for
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increasing derivative order, and each containing lists of
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coefficients for increasing subsets of x_list.
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Examples
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========
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>>> from sympy import finite_diff_weights, S
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>>> res = finite_diff_weights(1, [-S(1)/2, S(1)/2, S(3)/2, S(5)/2], 0)
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>>> res
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[[[1, 0, 0, 0],
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[1/2, 1/2, 0, 0],
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[3/8, 3/4, -1/8, 0],
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[5/16, 15/16, -5/16, 1/16]],
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[[0, 0, 0, 0],
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[-1, 1, 0, 0],
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[-1, 1, 0, 0],
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[-23/24, 7/8, 1/8, -1/24]]]
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>>> res[0][-1] # FD weights for 0th derivative, using full x_list
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[5/16, 15/16, -5/16, 1/16]
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>>> res[1][-1] # FD weights for 1st derivative
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[-23/24, 7/8, 1/8, -1/24]
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>>> res[1][-2] # FD weights for 1st derivative, using x_list[:-1]
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[-1, 1, 0, 0]
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>>> res[1][-1][0] # FD weight for 1st deriv. for x_list[0]
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-23/24
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>>> res[1][-1][1] # FD weight for 1st deriv. for x_list[1], etc.
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7/8
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Each sublist contains the most accurate formula at the end.
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Note, that in the above example ``res[1][1]`` is the same as ``res[1][2]``.
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Since res[1][2] has an order of accuracy of
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``len(x_list[:3]) - order = 3 - 1 = 2``, the same is true for ``res[1][1]``!
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>>> res = finite_diff_weights(1, [S(0), S(1), -S(1), S(2), -S(2)], 0)[1]
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>>> res
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[[0, 0, 0, 0, 0],
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[-1, 1, 0, 0, 0],
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[0, 1/2, -1/2, 0, 0],
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[-1/2, 1, -1/3, -1/6, 0],
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[0, 2/3, -2/3, -1/12, 1/12]]
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>>> res[0] # no approximation possible, using x_list[0] only
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[0, 0, 0, 0, 0]
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>>> res[1] # classic forward step approximation
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[-1, 1, 0, 0, 0]
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>>> res[2] # classic centered approximation
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[0, 1/2, -1/2, 0, 0]
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>>> res[3:] # higher order approximations
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[[-1/2, 1, -1/3, -1/6, 0], [0, 2/3, -2/3, -1/12, 1/12]]
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Let us compare this to a differently defined ``x_list``. Pay attention to
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``foo[i][k]`` corresponding to the gridpoint defined by ``x_list[k]``.
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>>> foo = finite_diff_weights(1, [-S(2), -S(1), S(0), S(1), S(2)], 0)[1]
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>>> foo
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[[0, 0, 0, 0, 0],
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[-1, 1, 0, 0, 0],
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[1/2, -2, 3/2, 0, 0],
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[1/6, -1, 1/2, 1/3, 0],
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[1/12, -2/3, 0, 2/3, -1/12]]
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>>> foo[1] # not the same and of lower accuracy as res[1]!
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[-1, 1, 0, 0, 0]
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>>> foo[2] # classic double backward step approximation
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[1/2, -2, 3/2, 0, 0]
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>>> foo[4] # the same as res[4]
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[1/12, -2/3, 0, 2/3, -1/12]
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Note that, unless you plan on using approximations based on subsets of
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``x_list``, the order of gridpoints does not matter.
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The capability to generate weights at arbitrary points can be
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used e.g. to minimize Runge's phenomenon by using Chebyshev nodes:
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>>> from sympy import cos, symbols, pi, simplify
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>>> N, (h, x) = 4, symbols('h x')
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>>> x_list = [x+h*cos(i*pi/(N)) for i in range(N,-1,-1)] # chebyshev nodes
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>>> print(x_list)
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[-h + x, -sqrt(2)*h/2 + x, x, sqrt(2)*h/2 + x, h + x]
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>>> mycoeffs = finite_diff_weights(1, x_list, 0)[1][4]
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>>> [simplify(c) for c in mycoeffs] #doctest: +NORMALIZE_WHITESPACE
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[(h**3/2 + h**2*x - 3*h*x**2 - 4*x**3)/h**4,
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(-sqrt(2)*h**3 - 4*h**2*x + 3*sqrt(2)*h*x**2 + 8*x**3)/h**4,
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(6*h**2*x - 8*x**3)/h**4,
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(sqrt(2)*h**3 - 4*h**2*x - 3*sqrt(2)*h*x**2 + 8*x**3)/h**4,
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(-h**3/2 + h**2*x + 3*h*x**2 - 4*x**3)/h**4]
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Notes
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=====
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If weights for a finite difference approximation of 3rd order
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derivative is wanted, weights for 0th, 1st and 2nd order are
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calculated "for free", so are formulae using subsets of ``x_list``.
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This is something one can take advantage of to save computational cost.
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Be aware that one should define ``x_list`` from nearest to furthest from
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``x0``. If not, subsets of ``x_list`` will yield poorer approximations,
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which might not grand an order of accuracy of ``len(x_list) - order``.
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See also
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========
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sympy.calculus.finite_diff.apply_finite_diff
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References
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==========
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.. [1] Generation of Finite Difference Formulas on Arbitrarily Spaced
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Grids, Bengt Fornberg; Mathematics of computation; 51; 184;
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(1988); 699-706; doi:10.1090/S0025-5718-1988-0935077-0
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"""
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# The notation below closely corresponds to the one used in the paper.
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order = S(order)
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if not order.is_number:
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raise ValueError("Cannot handle symbolic order.")
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if order < 0:
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raise ValueError("Negative derivative order illegal.")
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if int(order) != order:
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raise ValueError("Non-integer order illegal")
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M = order
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N = len(x_list) - 1
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delta = [[[0 for nu in range(N+1)] for n in range(N+1)] for
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m in range(M+1)]
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delta[0][0][0] = S.One
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c1 = S.One
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for n in range(1, N+1):
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c2 = S.One
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for nu in range(n):
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c3 = x_list[n] - x_list[nu]
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c2 = c2 * c3
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if n <= M:
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delta[n][n-1][nu] = 0
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for m in range(min(n, M)+1):
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delta[m][n][nu] = (x_list[n]-x0)*delta[m][n-1][nu] -\
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m*delta[m-1][n-1][nu]
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delta[m][n][nu] /= c3
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for m in range(min(n, M)+1):
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delta[m][n][n] = c1/c2*(m*delta[m-1][n-1][n-1] -
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(x_list[n-1]-x0)*delta[m][n-1][n-1])
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c1 = c2
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return delta
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def apply_finite_diff(order, x_list, y_list, x0=S.Zero):
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"""
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Calculates the finite difference approximation of
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the derivative of requested order at ``x0`` from points
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provided in ``x_list`` and ``y_list``.
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Parameters
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==========
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order: int
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order of derivative to approximate. 0 corresponds to interpolation.
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x_list: sequence
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Sequence of (unique) values for the independent variable.
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y_list: sequence
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The function value at corresponding values for the independent
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variable in x_list.
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x0: Number or Symbol
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At what value of the independent variable the derivative should be
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evaluated. Defaults to 0.
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Returns
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=======
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sympy.core.add.Add or sympy.core.numbers.Number
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The finite difference expression approximating the requested
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derivative order at ``x0``.
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Examples
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========
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>>> from sympy import apply_finite_diff
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>>> cube = lambda arg: (1.0*arg)**3
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>>> xlist = range(-3,3+1)
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>>> apply_finite_diff(2, xlist, map(cube, xlist), 2) - 12 # doctest: +SKIP
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-3.55271367880050e-15
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we see that the example above only contain rounding errors.
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apply_finite_diff can also be used on more abstract objects:
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>>> from sympy import IndexedBase, Idx
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>>> x, y = map(IndexedBase, 'xy')
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>>> i = Idx('i')
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>>> x_list, y_list = zip(*[(x[i+j], y[i+j]) for j in range(-1,2)])
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>>> apply_finite_diff(1, x_list, y_list, x[i])
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((x[i + 1] - x[i])/(-x[i - 1] + x[i]) - 1)*y[i]/(x[i + 1] - x[i]) -
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(x[i + 1] - x[i])*y[i - 1]/((x[i + 1] - x[i - 1])*(-x[i - 1] + x[i])) +
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(-x[i - 1] + x[i])*y[i + 1]/((x[i + 1] - x[i - 1])*(x[i + 1] - x[i]))
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Notes
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=====
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Order = 0 corresponds to interpolation.
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Only supply so many points you think makes sense
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to around x0 when extracting the derivative (the function
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need to be well behaved within that region). Also beware
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of Runge's phenomenon.
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See also
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========
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sympy.calculus.finite_diff.finite_diff_weights
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References
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==========
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Fortran 90 implementation with Python interface for numerics: finitediff_
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.. _finitediff: https://github.com/bjodah/finitediff
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"""
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# In the original paper the following holds for the notation:
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# M = order
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# N = len(x_list) - 1
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N = len(x_list) - 1
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if len(x_list) != len(y_list):
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raise ValueError("x_list and y_list not equal in length.")
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delta = finite_diff_weights(order, x_list, x0)
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derivative = 0
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for nu in range(len(x_list)):
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derivative += delta[order][N][nu]*y_list[nu]
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return derivative
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def _as_finite_diff(derivative, points=1, x0=None, wrt=None):
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"""
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Returns an approximation of a derivative of a function in
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the form of a finite difference formula. The expression is a
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weighted sum of the function at a number of discrete values of
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(one of) the independent variable(s).
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Parameters
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==========
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derivative: a Derivative instance
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points: sequence or coefficient, optional
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If sequence: discrete values (length >= order+1) of the
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independent variable used for generating the finite
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difference weights.
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If it is a coefficient, it will be used as the step-size
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for generating an equidistant sequence of length order+1
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centered around ``x0``. default: 1 (step-size 1)
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x0: number or Symbol, optional
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the value of the independent variable (``wrt``) at which the
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derivative is to be approximated. Default: same as ``wrt``.
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wrt: Symbol, optional
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"with respect to" the variable for which the (partial)
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derivative is to be approximated for. If not provided it
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is required that the Derivative is ordinary. Default: ``None``.
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Examples
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========
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>>> from sympy import symbols, Function, exp, sqrt, Symbol
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>>> from sympy.calculus.finite_diff import _as_finite_diff
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>>> x, h = symbols('x h')
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>>> f = Function('f')
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>>> _as_finite_diff(f(x).diff(x))
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-f(x - 1/2) + f(x + 1/2)
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The default step size and number of points are 1 and ``order + 1``
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respectively. We can change the step size by passing a symbol
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as a parameter:
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>>> _as_finite_diff(f(x).diff(x), h)
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-f(-h/2 + x)/h + f(h/2 + x)/h
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We can also specify the discretized values to be used in a sequence:
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>>> _as_finite_diff(f(x).diff(x), [x, x+h, x+2*h])
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-3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h)
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The algorithm is not restricted to use equidistant spacing, nor
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do we need to make the approximation around ``x0``, but we can get
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an expression estimating the derivative at an offset:
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>>> e, sq2 = exp(1), sqrt(2)
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>>> xl = [x-h, x+h, x+e*h]
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>>> _as_finite_diff(f(x).diff(x, 1), xl, x+h*sq2)
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2*h*((h + sqrt(2)*h)/(2*h) - (-sqrt(2)*h + h)/(2*h))*f(E*h + x)/((-h + E*h)*(h + E*h)) +
|
||
|
|
(-(-sqrt(2)*h + h)/(2*h) - (-sqrt(2)*h + E*h)/(2*h))*f(-h + x)/(h + E*h) +
|
||
|
|
(-(h + sqrt(2)*h)/(2*h) + (-sqrt(2)*h + E*h)/(2*h))*f(h + x)/(-h + E*h)
|
||
|
|
|
||
|
|
Partial derivatives are also supported:
|
||
|
|
|
||
|
|
>>> y = Symbol('y')
|
||
|
|
>>> d2fdxdy=f(x,y).diff(x,y)
|
||
|
|
>>> _as_finite_diff(d2fdxdy, wrt=x)
|
||
|
|
-Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y)
|
||
|
|
|
||
|
|
See also
|
||
|
|
========
|
||
|
|
|
||
|
|
sympy.calculus.finite_diff.apply_finite_diff
|
||
|
|
sympy.calculus.finite_diff.finite_diff_weights
|
||
|
|
|
||
|
|
"""
|
||
|
|
if derivative.is_Derivative:
|
||
|
|
pass
|
||
|
|
elif derivative.is_Atom:
|
||
|
|
return derivative
|
||
|
|
else:
|
||
|
|
return derivative.fromiter(
|
||
|
|
[_as_finite_diff(ar, points, x0, wrt) for ar
|
||
|
|
in derivative.args], **derivative.assumptions0)
|
||
|
|
|
||
|
|
if wrt is None:
|
||
|
|
old = None
|
||
|
|
for v in derivative.variables:
|
||
|
|
if old is v:
|
||
|
|
continue
|
||
|
|
derivative = _as_finite_diff(derivative, points, x0, v)
|
||
|
|
old = v
|
||
|
|
return derivative
|
||
|
|
|
||
|
|
order = derivative.variables.count(wrt)
|
||
|
|
|
||
|
|
if x0 is None:
|
||
|
|
x0 = wrt
|
||
|
|
|
||
|
|
if not iterable(points):
|
||
|
|
if getattr(points, 'is_Function', False) and wrt in points.args:
|
||
|
|
points = points.subs(wrt, x0)
|
||
|
|
# points is simply the step-size, let's make it a
|
||
|
|
# equidistant sequence centered around x0
|
||
|
|
if order % 2 == 0:
|
||
|
|
# even order => odd number of points, grid point included
|
||
|
|
points = [x0 + points*i for i
|
||
|
|
in range(-order//2, order//2 + 1)]
|
||
|
|
else:
|
||
|
|
# odd order => even number of points, half-way wrt grid point
|
||
|
|
points = [x0 + points*S(i)/2 for i
|
||
|
|
in range(-order, order + 1, 2)]
|
||
|
|
others = [wrt, 0]
|
||
|
|
for v in set(derivative.variables):
|
||
|
|
if v == wrt:
|
||
|
|
continue
|
||
|
|
others += [v, derivative.variables.count(v)]
|
||
|
|
if len(points) < order+1:
|
||
|
|
raise ValueError("Too few points for order %d" % order)
|
||
|
|
return apply_finite_diff(order, points, [
|
||
|
|
Derivative(derivative.expr.subs({wrt: x}), *others) for
|
||
|
|
x in points], x0)
|
||
|
|
|
||
|
|
|
||
|
|
def differentiate_finite(expr, *symbols,
|
||
|
|
points=1, x0=None, wrt=None, evaluate=False):
|
||
|
|
r""" Differentiate expr and replace Derivatives with finite differences.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
==========
|
||
|
|
|
||
|
|
expr : expression
|
||
|
|
\*symbols : differentiate with respect to symbols
|
||
|
|
points: sequence, coefficient or undefined function, optional
|
||
|
|
see ``Derivative.as_finite_difference``
|
||
|
|
x0: number or Symbol, optional
|
||
|
|
see ``Derivative.as_finite_difference``
|
||
|
|
wrt: Symbol, optional
|
||
|
|
see ``Derivative.as_finite_difference``
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy import sin, Function, differentiate_finite
|
||
|
|
>>> from sympy.abc import x, y, h
|
||
|
|
>>> f, g = Function('f'), Function('g')
|
||
|
|
>>> differentiate_finite(f(x)*g(x), x, points=[x-h, x+h])
|
||
|
|
-f(-h + x)*g(-h + x)/(2*h) + f(h + x)*g(h + x)/(2*h)
|
||
|
|
|
||
|
|
``differentiate_finite`` works on any expression, including the expressions
|
||
|
|
with embedded derivatives:
|
||
|
|
|
||
|
|
>>> differentiate_finite(f(x) + sin(x), x, 2)
|
||
|
|
-2*f(x) + f(x - 1) + f(x + 1) - 2*sin(x) + sin(x - 1) + sin(x + 1)
|
||
|
|
>>> differentiate_finite(f(x, y), x, y)
|
||
|
|
f(x - 1/2, y - 1/2) - f(x - 1/2, y + 1/2) - f(x + 1/2, y - 1/2) + f(x + 1/2, y + 1/2)
|
||
|
|
>>> differentiate_finite(f(x)*g(x).diff(x), x)
|
||
|
|
(-g(x) + g(x + 1))*f(x + 1/2) - (g(x) - g(x - 1))*f(x - 1/2)
|
||
|
|
|
||
|
|
To make finite difference with non-constant discretization step use
|
||
|
|
undefined functions:
|
||
|
|
|
||
|
|
>>> dx = Function('dx')
|
||
|
|
>>> differentiate_finite(f(x)*g(x).diff(x), points=dx(x))
|
||
|
|
-(-g(x - dx(x)/2 - dx(x - dx(x)/2)/2)/dx(x - dx(x)/2) +
|
||
|
|
g(x - dx(x)/2 + dx(x - dx(x)/2)/2)/dx(x - dx(x)/2))*f(x - dx(x)/2)/dx(x) +
|
||
|
|
(-g(x + dx(x)/2 - dx(x + dx(x)/2)/2)/dx(x + dx(x)/2) +
|
||
|
|
g(x + dx(x)/2 + dx(x + dx(x)/2)/2)/dx(x + dx(x)/2))*f(x + dx(x)/2)/dx(x)
|
||
|
|
|
||
|
|
"""
|
||
|
|
if any(term.is_Derivative for term in list(preorder_traversal(expr))):
|
||
|
|
evaluate = False
|
||
|
|
|
||
|
|
Dexpr = expr.diff(*symbols, evaluate=evaluate)
|
||
|
|
if evaluate:
|
||
|
|
sympy_deprecation_warning("""
|
||
|
|
The evaluate flag to differentiate_finite() is deprecated.
|
||
|
|
|
||
|
|
evaluate=True expands the intermediate derivatives before computing
|
||
|
|
differences, but this usually not what you want, as it does not
|
||
|
|
satisfy the product rule.
|
||
|
|
""",
|
||
|
|
deprecated_since_version="1.5",
|
||
|
|
active_deprecations_target="deprecated-differentiate_finite-evaluate",
|
||
|
|
)
|
||
|
|
return Dexpr.replace(
|
||
|
|
lambda arg: arg.is_Derivative,
|
||
|
|
lambda arg: arg.as_finite_difference(points=points, x0=x0, wrt=wrt))
|
||
|
|
else:
|
||
|
|
DFexpr = Dexpr.as_finite_difference(points=points, x0=x0, wrt=wrt)
|
||
|
|
return DFexpr.replace(
|
||
|
|
lambda arg: isinstance(arg, Subs),
|
||
|
|
lambda arg: arg.expr.as_finite_difference(
|
||
|
|
points=points, x0=arg.point[0], wrt=arg.variables[0]))
|