"""
Finite difference weights
=========================
This module implements an algorithm for efficient generation of finite
difference weights for ordinary differentials of functions for
derivatives from 0 (interpolation) up to arbitrary order.
The core algorithm is provided in the finite difference weight generating
function (finite_diff_weights), and two convenience functions are provided
for:
- estimating a derivative (or interpolate) directly from a series of points
is also provided (``apply_finite_diff``).
- making a finite difference approximation of a Derivative instance
(``as_finite_diff``).
"""
from sympy import S
from sympy.core.compatibility import iterable, range
[docs]def finite_diff_weights(order, x_list, x0=S(0)):
"""
Calculates the finite difference weights for an arbitrarily
spaced one-dimensional grid (x_list) for derivatives at 'x0'
of order 0, 1, ..., up to 'order' using a recursive formula.
Order of accuracy is at least len(x_list) - order, if x_list
is defined accurately.
Parameters
==========
order: int
Up to what derivative order weights should be calculated.
0 corresponds to interpolation.
x_list: sequence
Sequence of (unique) values for the independent variable.
It is usefull (but not necessary) to order x_list from
nearest to farest from x0; see examples below.
x0: Number or Symbol
Root or value of the independent variable for which the finite
difference weights should be generated. Defaults to S(0).
Returns
=======
list
A list of sublists, each corresponding to coefficients for
increasing derivative order, and each containing lists of
coefficients for increasing subsets of x_list.
Examples
========
>>> from sympy import S
>>> from sympy.calculus import finite_diff_weights
>>> res = finite_diff_weights(1, [-S(1)/2, S(1)/2, S(3)/2, S(5)/2], 0)
>>> res
[[[1, 0, 0, 0],
[1/2, 1/2, 0, 0],
[3/8, 3/4, -1/8, 0],
[5/16, 15/16, -5/16, 1/16]],
[[0, 0, 0, 0],
[-1, 1, 0, 0],
[-1, 1, 0, 0],
[-23/24, 7/8, 1/8, -1/24]]]
>>> res[0][-1] # FD weights for 0th derivative, using full x_list
[5/16, 15/16, -5/16, 1/16]
>>> res[1][-1] # FD weights for 1st derivative
[-23/24, 7/8, 1/8, -1/24]
>>> res[1][-2] # FD weights for 1st derivative, using x_list[:-1]
[-1, 1, 0, 0]
>>> res[1][-1][0] # FD weight for 1st deriv. for x_list[0]
-23/24
>>> res[1][-1][1] # FD weight for 1st deriv. for x_list[1], etc.
7/8
Each sublist contains the most accurate formula at the end.
Note, that in the above example res[1][1] is the same as res[1][2].
Since res[1][2] has an order of accuracy of
len(x_list[:3]) - order = 3 - 1 = 2, the same is true for res[1][1]!
>>> from sympy import S
>>> from sympy.calculus import finite_diff_weights
>>> res = finite_diff_weights(1, [S(0), S(1), -S(1), S(2), -S(2)], 0)[1]
>>> res
[[0, 0, 0, 0, 0],
[-1, 1, 0, 0, 0],
[0, 1/2, -1/2, 0, 0],
[-1/2, 1, -1/3, -1/6, 0],
[0, 2/3, -2/3, -1/12, 1/12]]
>>> res[0] # no approximation possible, using x_list[0] only
[0, 0, 0, 0, 0]
>>> res[1] # classic forward step approximation
[-1, 1, 0, 0, 0]
>>> res[2] # classic centered approximation
[0, 1/2, -1/2, 0, 0]
>>> res[3:] # higher order approximations
[[-1/2, 1, -1/3, -1/6, 0], [0, 2/3, -2/3, -1/12, 1/12]]
Let us compare this to a differently defined x_list. Pay attention to
foo[i][k] corresponding to the gridpoint defined by x_list[k].
>>> from sympy import S
>>> from sympy.calculus import finite_diff_weights
>>> foo = finite_diff_weights(1, [-S(2), -S(1), S(0), S(1), S(2)], 0)[1]
>>> foo
[[0, 0, 0, 0, 0],
[-1, 1, 0, 0, 0],
[1/2, -2, 3/2, 0, 0],
[1/6, -1, 1/2, 1/3, 0],
[1/12, -2/3, 0, 2/3, -1/12]]
>>> foo[1] # not the same and of lower accuracy as res[1]!
[-1, 1, 0, 0, 0]
>>> foo[2] # classic double backward step approximation
[1/2, -2, 3/2, 0, 0]
>>> foo[4] # the same as res[4]
[1/12, -2/3, 0, 2/3, -1/12]
Note that, unless you plan on using approximations based on subsets of
x_list, the order of gridpoints does not matter.
The capability to generate weights at arbitrary points can be
used e.g. to minimize Runge's phenomenon by using Chebyshev nodes:
>>> from sympy import cos, symbols, pi, simplify
>>> from sympy.calculus import finite_diff_weights
>>> N, (h, x) = 4, symbols('h x')
>>> x_list = [x+h*cos(i*pi/(N)) for i in range(N,-1,-1)] # chebyshev nodes
>>> print(x_list)
[-h + x, -sqrt(2)*h/2 + x, x, sqrt(2)*h/2 + x, h + x]
>>> mycoeffs = finite_diff_weights(1, x_list, 0)[1][4]
>>> [simplify(c) for c in mycoeffs] #doctest: +NORMALIZE_WHITESPACE
[(h**3/2 + h**2*x - 3*h*x**2 - 4*x**3)/h**4,
(-sqrt(2)*h**3 - 4*h**2*x + 3*sqrt(2)*h*x**2 + 8*x**3)/h**4,
6*x/h**2 - 8*x**3/h**4,
(sqrt(2)*h**3 - 4*h**2*x - 3*sqrt(2)*h*x**2 + 8*x**3)/h**4,
(-h**3/2 + h**2*x + 3*h*x**2 - 4*x**3)/h**4]
Notes
=====
If weights for a finite difference approximation of 3rd order
derivative is wanted, weights for 0th, 1st and 2nd order are
calculated "for free", so are formulae using subsets of x_list.
This is something one can take advantage of to save computational cost.
Be aware that one should define x_list from nearest to farest from
x_list. If not, subsets of x_list will yield poorer approximations,
which might not grand an order of accuracy of len(x_list) - order.
See also
========
sympy.calculus.finite_diff.apply_finite_diff
References
==========
.. [1] Generation of Finite Difference Formulas on Arbitrarily Spaced
Grids, Bengt Fornberg; Mathematics of computation; 51; 184;
(1988); 699-706; doi:10.1090/S0025-5718-1988-0935077-0
"""
# The notation below closely corresponds to the one used in the paper.
if order < 0:
raise ValueError("Negative derivative order illegal.")
if int(order) != order:
raise ValueError("Non-integer order illegal")
M = order
N = len(x_list) - 1
delta = [[[0 for nu in range(N+1)] for n in range(N+1)] for
m in range(M+1)]
delta[0][0][0] = S(1)
c1 = S(1)
for n in range(1, N+1):
c2 = S(1)
for nu in range(0, n):
c3 = x_list[n]-x_list[nu]
c2 = c2 * c3
if n <= M:
delta[n][n-1][nu] = 0
for m in range(0, min(n, M)+1):
delta[m][n][nu] = (x_list[n]-x0)*delta[m][n-1][nu] -\
m*delta[m-1][n-1][nu]
delta[m][n][nu] /= c3
for m in range(0, min(n, M)+1):
delta[m][n][n] = c1/c2*(m*delta[m-1][n-1][n-1] -
(x_list[n-1]-x0)*delta[m][n-1][n-1])
c1 = c2
return delta
[docs]def apply_finite_diff(order, x_list, y_list, x0=S(0)):
"""
Calculates the finite difference approximation of
the derivative of requested order at x0 from points
provided in x_list and y_list.
Parameters
==========
order: int
order of derivative to approximate. 0 corresponds to interpolation.
x_list: sequence
Sequence of (unique) values for the independent variable.
y_list: sequence
The function value at corresponding values for the independent
variable in x_list.
x0: Number or Symbol
At what value of the independent variable the derivative should be
evaluated. Defaults to S(0).
Returns
=======
sympy.core.add.Add or sympy.core.numbers.Number
The finite difference expression approximating the requested
derivative order at x0.
Examples
========
>>> from sympy.calculus import apply_finite_diff
>>> cube = lambda arg: (1.0*arg)**3
>>> xlist = range(-3,3+1)
>>> apply_finite_diff(2, xlist, map(cube, xlist), 2) - 12 # doctest: +SKIP
-3.55271367880050e-15
we see that the example above only contain rounding errors.
apply_finite_diff can also be used on more abstract objects:
>>> from sympy import IndexedBase, Idx
>>> from sympy.calculus import apply_finite_diff
>>> x, y = map(IndexedBase, 'xy')
>>> i = Idx('i')
>>> x_list, y_list = zip(*[(x[i+j], y[i+j]) for j in range(-1,2)])
>>> apply_finite_diff(1, x_list, y_list, x[i])
(-1 + (x[i + 1] - x[i])/(-x[i - 1] + x[i]))*y[i]/(x[i + 1] - x[i]) + \
(-x[i - 1] + x[i])*y[i + 1]/((-x[i - 1] + x[i + 1])*(x[i + 1] - x[i])) - \
(x[i + 1] - x[i])*y[i - 1]/((-x[i - 1] + x[i + 1])*(-x[i - 1] + x[i]))
Notes
=====
Order = 0 corresponds to interpolation.
Only supply so many points you think makes sense
to around x0 when extracting the derivative (the function
need to be well behaved within that region). Also beware
of Runge's phenomenon.
See also
========
sympy.calculus.finite_diff.finite_diff_weights
References
==========
Fortran 90 implementation with Python interface for numerics: finitediff_
.. _finitediff: https://github.com/bjodah/finitediff
"""
# In the original paper the following holds for the notation:
# M = order
# N = len(x_list) - 1
N = len(x_list) - 1
if len(x_list) != len(y_list):
raise ValueError("x_list and y_list not equal in length.")
delta = finite_diff_weights(order, x_list, x0)
derivative = 0
for nu in range(0, len(x_list)):
derivative += delta[order][N][nu]*y_list[nu]
return derivative
[docs]def as_finite_diff(derivative, points=1, x0=None, wrt=None):
"""
Returns an approximation of a derivative of a function in
the form of a finite difference formula. The expression is a
weighted sum of the function at a number of discrete values of
(one of) the independent variable(s).
Parameters
==========
derivative: a Derivative instance (needs to have an variables
and expr attribute).
points: sequence or coefficient, optional
If sequence: discrete values (length >= order+1) of the
independent variable used for generating the finite
difference weights.
If it is a coefficient, it will be used as the step-size
for generating an equidistant sequence of length order+1
centered around x0. default: 1 (step-size 1)
x0: number or Symbol, optional
the value of the independent variable (wrt) at which the
derivative is to be approximated. default: same as wrt
wrt: Symbol, optional
"with respect to" the variable for which the (partial)
derivative is to be approximated for. If not provided it
is required that the Derivative is ordinary. default: None
Examples
========
>>> from sympy import symbols, Function, exp, sqrt, Symbol, as_finite_diff
>>> x, h = symbols('x h')
>>> f = Function('f')
>>> as_finite_diff(f(x).diff(x))
-f(x - 1/2) + f(x + 1/2)
The default step size and number of points are 1 and ``order + 1``
respectively. We can change the step size by passing a symbol
as a parameter:
>>> as_finite_diff(f(x).diff(x), h)
-f(-h/2 + x)/h + f(h/2 + x)/h
We can also specify the discretized values to be used in a sequence:
>>> as_finite_diff(f(x).diff(x), [x, x+h, x+2*h])
-3*f(x)/(2*h) + 2*f(h + x)/h - f(2*h + x)/(2*h)
The algorithm is not restricted to use equidistant spacing, nor
do we need to make the approximation around x0, but we can get
an expression estimating the derivative at an offset:
>>> e, sq2 = exp(1), sqrt(2)
>>> xl = [x-h, x+h, x+e*h]
>>> as_finite_diff(f(x).diff(x, 1), xl, x+h*sq2)
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)
-f(x - 1/2, y) + f(x + 1/2, y)
See also
========
sympy.calculus.finite_diff.apply_finite_diff
sympy.calculus.finite_diff.finite_diff_weights
"""
if wrt is None:
wrt = derivative.variables[0]
# we need Derivative to be univariate to guess wrt
if any(v != wrt for v in derivative.variables):
raise ValueError('if the function is not univariate' +
' then `wrt` must be given')
order = derivative.variables.count(wrt)
if x0 is None:
x0 = wrt
if not iterable(points):
# 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*i/S(2) for i
in range(-order, order + 1, 2)]
if len(points) < order+1:
raise ValueError("Too few points for order %d" % order)
return apply_finite_diff(order, points, [
derivative.expr.subs({wrt: x}) for x in points], x0)
