# Calculus¶

Calculus-related methods.

This module implements a method to find Euler-Lagrange Equations for given Lagrangian.

sympy.calculus.euler.euler_equations(L, funcs=(), vars=())[source]

Find the Euler-Lagrange equations [R22] for a given Lagrangian.

Parameters: L : Expr The Lagrangian that should be a function of the functions listed in the second argument and their derivatives. For example, in the case of two functions $$f(x,y)$$, $$g(x,y)$$ and two independent variables $$x$$, $$y$$ the Lagrangian would have the form: $L\left(f(x,y),g(x,y),\frac{\partial f(x,y)}{\partial x}, \frac{\partial f(x,y)}{\partial y}, \frac{\partial g(x,y)}{\partial x}, \frac{\partial g(x,y)}{\partial y},x,y\right)$ In many cases it is not necessary to provide anything, except the Lagrangian, it will be auto-detected (and an error raised if this couldn’t be done). funcs : Function or an iterable of Functions The functions that the Lagrangian depends on. The Euler equations are differential equations for each of these functions. vars : Symbol or an iterable of Symbols The Symbols that are the independent variables of the functions. eqns : list of Eq The list of differential equations, one for each function.

References

Examples

>>> from sympy import Symbol, Function
>>> from sympy.calculus.euler import euler_equations
>>> x = Function('x')
>>> t = Symbol('t')
>>> L = (x(t).diff(t))**2/2 - x(t)**2/2
>>> euler_equations(L, x(t), t)
[Eq(-x(t) - Derivative(x(t), t, t), 0)]
>>> u = Function('u')
>>> x = Symbol('x')
>>> L = (u(t, x).diff(t))**2/2 - (u(t, x).diff(x))**2/2
>>> euler_equations(L, u(t, x), [t, x])
[Eq(-Derivative(u(t, x), t, t) + Derivative(u(t, x), x, x), 0)]


## Singularities¶

This module implements algorithms for finding singularities for a function and identifying types of functions.

The differential calculus methods in this module include methods to identify the following function types in the given Interval: - Increasing - Strictly Increasing - Decreasing - Strictly Decreasing - Monotonic

sympy.calculus.singularities.is_decreasing(expression, interval=S.Reals, symbol=None)[source]

Return whether the function is decreasing in the given interval.

Examples

>>> from sympy import is_decreasing
>>> from sympy.abc import x, y
>>> from sympy import S, Interval, oo
>>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3))
True
>>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo))
True
>>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2))
False
>>> is_decreasing(-x**2, Interval(-oo, 0))
False
>>> is_decreasing(-x**2 + y, Interval(-oo, 0), x)
False

sympy.calculus.singularities.is_increasing(expression, interval=S.Reals, symbol=None)[source]

Return whether the function is increasing in the given interval.

Examples

>>> from sympy import is_increasing
>>> from sympy.abc import x, y
>>> from sympy import S, Interval, oo
>>> is_increasing(x**3 - 3*x**2 + 4*x, S.Reals)
True
>>> is_increasing(-x**2, Interval(-oo, 0))
True
>>> is_increasing(-x**2, Interval(0, oo))
False
>>> is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3))
False
>>> is_increasing(x**2 + y, Interval(1, 2), x)
True

sympy.calculus.singularities.is_monotonic(expression, interval=S.Reals, symbol=None)[source]

Return whether the function is monotonic in the given interval.

Examples

>>> from sympy import is_monotonic
>>> from sympy.abc import x, y
>>> from sympy import S, Interval, oo
>>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3))
True
>>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo))
True
>>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals)
True
>>> is_monotonic(-x**2, S.Reals)
False
>>> is_monotonic(x**2 + y + 1, Interval(1, 2), x)
True

sympy.calculus.singularities.is_strictly_decreasing(expression, interval=S.Reals, symbol=None)[source]

Return whether the function is strictly decreasing in the given interval.

Examples

>>> from sympy import is_strictly_decreasing
>>> from sympy.abc import x, y
>>> from sympy import S, Interval, oo
>>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo))
True
>>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2))
False
>>> is_strictly_decreasing(-x**2, Interval(-oo, 0))
False
>>> is_strictly_decreasing(-x**2 + y, Interval(-oo, 0), x)
False

sympy.calculus.singularities.is_strictly_increasing(expression, interval=S.Reals, symbol=None)[source]

Return whether the function is strictly increasing in the given interval.

Examples

>>> from sympy import is_strictly_increasing
>>> from sympy.abc import x, y
>>> from sympy import Interval, oo
>>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Ropen(-oo, -2))
True
>>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Lopen(3, oo))
True
>>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3))
False
>>> is_strictly_increasing(-x**2, Interval(0, oo))
False
>>> is_strictly_increasing(-x**2 + y, Interval(-oo, 0), x)
False

sympy.calculus.singularities.monotonicity_helper(expression, predicate, interval=S.Reals, symbol=None)[source]

Helper function for functions checking function monotonicity.

It returns a boolean indicating whether the interval in which the function’s derivative satisfies given predicate is a superset of the given interval.

sympy.calculus.singularities.singularities(expression, symbol)[source]

Find singularities of a given function.

Currently supported functions are: - univariate rational (real or complex) functions

Notes

This function does not find nonisolated singularities nor does it find branch points of the expression.

References

Examples

>>> from sympy.calculus.singularities import singularities
>>> from sympy import Symbol
>>> x = Symbol('x', real=True)
>>> y = Symbol('y', real=False)
>>> singularities(x**2 + x + 1, x)
EmptySet()
>>> singularities(1/(x + 1), x)
{-1}
>>> singularities(1/(y**2 + 1), y)
{-I, I}
>>> singularities(1/(y**3 + 1), y)
{-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2}


## 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).
• differentiating by using finite difference approximations
(differentiate_finite).
sympy.calculus.finite_diff.apply_finite_diff(order, x_list, y_list, x0=0)[source]

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). sympy.core.add.Add or sympy.core.numbers.Number The finite difference expression approximating the requested derivative order at x0.

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.

References

Fortran 90 implementation with Python interface for numerics: finitediff

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
-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])
((x[i + 1] - x[i])/(-x[i - 1] + x[i]) - 1)*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]))

sympy.calculus.finite_diff.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 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
>>> from sympy.utilities.exceptions import SymPyDeprecationWarning
>>> import warnings
>>> warnings.simplefilter("ignore", SymPyDeprecationWarning)
>>> 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)
-Derivative(f(x - 1/2, y), y) + Derivative(f(x + 1/2, y), y)

sympy.calculus.finite_diff.differentiate_finite(expr, *symbols, **kwargs)[source]

Differentiate expr and replace Derivatives with finite differences.

Parameters: expr : expression *symbols : differentiate with respect to symbols points: sequence or coefficient, optional see Derivative.as_finite_difference x0: number or Symbol, optional see Derivative.as_finite_difference wrt: Symbol, optional see Derivative.as_finite_difference evaluate : bool kwarg passed on to diff, whether or not to evaluate the Derivative intermediately (default: False).

Examples

>>> from sympy import cos, 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)


Note that the above form preserves the product rule in discrete form. If we want we can pass evaluate=True to get another form (which is usually not what we want):

>>> differentiate_finite(f(x)*g(x), x, points=[x-h, x+h], evaluate=True).simplify()
-((f(-h + x) - f(h + x))*g(x) + (g(-h + x) - g(h + x))*f(x))/(2*h)


differentiate_finite works on any expression:

>>> 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) + sin(x), x, 2, evaluate=True)
-2*f(x) + f(x - 1) + f(x + 1) - sin(x)
>>> 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)

sympy.calculus.finite_diff.finite_diff_weights(order, x_list, x0=1)[source]

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 correctly.

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 useful (but not necessary) to order x_list from nearest to furthest 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. Default is S.One. 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.

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 x0. If not, subsets of x_list will yield poorer approximations, which might not grand an order of accuracy of len(x_list) - order.

References

 [R24] 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

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]
[(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]