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 [R26] 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.

Returns

eqns : list of Eq

The list of differential equations, one for each function.

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, 2)), 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, 2)) + Derivative(u(t, x), (x, 2)), 0)]

References

R26(1,2)

https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation

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=Reals, symbol=None)[source]

Return whether the function is decreasing in the given interval.

Parameters

expression : Expr

The target function which is being checked.

interval : Set, optional

The range of values in which we are testing (defaults to set of all real numbers).

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

Returns

Boolean

True if expression is decreasing (either strictly decreasing or constant) in the given interval, False otherwise.

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=Reals, symbol=None)[source]

Return whether the function is increasing in the given interval.

Parameters

expression : Expr

The target function which is being checked.

interval : Set, optional

The range of values in which we are testing (defaults to set of all real numbers).

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

Returns

Boolean

True if expression is increasing (either strictly increasing or constant) in the given interval, False otherwise.

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=Reals, symbol=None)[source]

Return whether the function is monotonic in the given interval.

Parameters

expression : Expr

The target function which is being checked.

interval : Set, optional

The range of values in which we are testing (defaults to set of all real numbers).

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

Returns

Boolean

True if expression is monotonic in the given interval, False otherwise.

Raises

NotImplementedError

Monotonicity check has not been implemented for the queried function.

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=Reals, symbol=None)[source]

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

Parameters

expression : Expr

The target function which is being checked.

interval : Set, optional

The range of values in which we are testing (defaults to set of all real numbers).

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

Returns

Boolean

True if expression is strictly decreasing in the given interval, False otherwise.

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=Reals, symbol=None)[source]

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

Parameters

expression : Expr

The target function which is being checked.

interval : Set, optional

The range of values in which we are testing (defaults to set of all real numbers).

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

Returns

Boolean

True if expression is strictly increasing in the given interval, False otherwise.

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=Reals, symbol=None)[source]

Helper function for functions checking function monotonicity.

Parameters

expression : Expr

The target function which is being checked

predicate : function

The property being tested for. The function takes in an integer and returns a boolean. The integer input is the derivative and the boolean result should be true if the property is being held, and false otherwise.

interval : Set, optional

The range of values in which we are testing, defaults to all reals.

symbol : Symbol, optional

The symbol present in expression which gets varied over the given range.

It returns a boolean indicating whether the interval in which

the function’s derivative satisfies given predicate is a superset

of the given interval.

Returns

Boolean

True if predicate is true for all the derivatives when symbol is varied in range, False otherwise.

sympy.calculus.singularities.singularities(expression, symbol, domain=None)[source]

Find singularities of a given function.

Parameters

expression : Expr

The target function in which singularities need to be found.

symbol : Symbol

The symbol over the values of which the singularity in expression in being searched for.

Returns

Set

A set of values for symbol for which expression has a singularity. An EmptySet is returned if expression has no singularities for any given value of Symbol.

Raises

NotImplementedError

Methods for determining the singularities of this function have not been developed.

Notes

This function does not find non-isolated singularities nor does it find branch points of the expression.

Currently supported functions are:

  • univariate continuous (real or complex) functions

Examples

>>> from sympy.calculus.singularities import singularities
>>> from sympy import Symbol, log
>>> 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}
>>> singularities(log(x), x)
{0}

References

R27

https://en.wikipedia.org/wiki/Mathematical_singularity

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 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 
-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]))

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

sympy.calculus.finite_diff.as_finite_diff(derivative, points=1, x0=None, wrt=None)[source]

Deprecated function. Use Diff.as_finite_difference instead.

sympy.calculus.finite_diff.differentiate_finite(expr, *symbols, points=1, x0=None, wrt=None, evaluate=False)[source]

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)
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.

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] 
[(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*h**2*x - 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 furthest 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.

See also

sympy.calculus.finite_diff.apply_finite_diff

References

R28

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

sympy.calculus.util.AccumBounds[source]

alias of sympy.calculus.util.AccumulationBounds

class sympy.calculus.util.AccumulationBounds(min, max)[source]

# Note AccumulationBounds has an alias: AccumBounds

AccumulationBounds represent an interval \([a, b]\), which is always closed at the ends. Here \(a\) and \(b\) can be any value from extended real numbers.

The intended meaning of AccummulationBounds is to give an approximate location of the accumulation points of a real function at a limit point.

Let \(a\) and \(b\) be reals such that a <= b.

\(\left\langle a, b\right\rangle = \{x \in \mathbb{R} \mid a \le x \le b\}\)

\(\left\langle -\infty, b\right\rangle = \{x \in \mathbb{R} \mid x \le b\} \cup \{-\infty, \infty\}\)

\(\left\langle a, \infty \right\rangle = \{x \in \mathbb{R} \mid a \le x\} \cup \{-\infty, \infty\}\)

\(\left\langle -\infty, \infty \right\rangle = \mathbb{R} \cup \{-\infty, \infty\}\)

\(oo\) and \(-oo\) are added to the second and third definition respectively, since if either \(-oo\) or \(oo\) is an argument, then the other one should be included (though not as an end point). This is forced, since we have, for example, \(1/AccumBounds(0, 1) = AccumBounds(1, oo)\), and the limit at \(0\) is not one-sided. As x tends to \(0-\), then \(1/x -> -oo\), so \(-oo\) should be interpreted as belonging to \(AccumBounds(1, oo)\) though it need not appear explicitly.

In many cases it suffices to know that the limit set is bounded. However, in some other cases more exact information could be useful. For example, all accumulation values of cos(x) + 1 are non-negative. (AccumBounds(-1, 1) + 1 = AccumBounds(0, 2))

A AccumulationBounds object is defined to be real AccumulationBounds, if its end points are finite reals.

Let \(X\), \(Y\) be real AccumulationBounds, then their sum, difference, product are defined to be the following sets:

\(X + Y = \{ x+y \mid x \in X \cap y \in Y\}\)

\(X - Y = \{ x-y \mid x \in X \cap y \in Y\}\)

\(X * Y = \{ x*y \mid x \in X \cap y \in Y\}\)

When an AccumBounds is raised to a negative power, if 0 is contained between the bounds then an infinite range is returned, otherwise if an endpoint is 0 then a semi-infinite range with consistent sign will be returned.

AccumBounds in expressions behave a lot like Intervals but the semantics are not necessarily the same. Division (or exponentiation to a negative integer power) could be handled with intervals by returning a union of the results obtained after splitting the bounds between negatives and positives, but that is not done with AccumBounds. In addition, bounds are assumed to be independent of each other; if the same bound is used in more than one place in an expression, the result may not be the supremum or infimum of the expression (see below). Finally, when a boundary is 1, exponentiation to the power of oo yields oo, neither 1 nor nan.

Examples

>>> from sympy import AccumBounds, sin, exp, log, pi, E, S, oo
>>> from sympy.abc import x

>>> AccumBounds(0, 1) + AccumBounds(1, 2)
AccumBounds(1, 3)

>>> AccumBounds(0, 1) - AccumBounds(0, 2)
AccumBounds(-2, 1)

>>> AccumBounds(-2, 3)*AccumBounds(-1, 1)
AccumBounds(-3, 3)

>>> AccumBounds(1, 2)*AccumBounds(3, 5)
AccumBounds(3, 10)

The exponentiation of AccumulationBounds is defined as follows:

If 0 does not belong to \(X\) or \(n > 0\) then

\(X^n = \{ x^n \mid x \in X\}\)

>>> AccumBounds(1, 4)**(S(1)/2)
AccumBounds(1, 2)

otherwise, an infinite or semi-infinite result is obtained:

>>> 1/AccumBounds(-1, 1)
AccumBounds(-oo, oo)
>>> 1/AccumBounds(0, 2)
AccumBounds(1/2, oo)
>>> 1/AccumBounds(-oo, 0)
AccumBounds(-oo, 0)

A boundary of 1 will always generate all nonnegatives:

>>> AccumBounds(1, 2)**oo
AccumBounds(0, oo)
>>> AccumBounds(0, 1)**oo
AccumBounds(0, oo)

If the exponent is itself an AccumulationBounds or is not an integer then unevaluated results will be returned unless the base values are positive:

>>> AccumBounds(2, 3)**AccumBounds(-1, 2)
AccumBounds(1/3, 9)
>>> AccumBounds(-2, 3)**AccumBounds(-1, 2)
AccumBounds(-2, 3)**AccumBounds(-1, 2)

>>> AccumBounds(-2, -1)**(S(1)/2)
sqrt(AccumBounds(-2, -1))

Note: \(<a, b>^2\) is not same as \(<a, b>*<a, b>\)

>>> AccumBounds(-1, 1)**2
AccumBounds(0, 1)

>>> AccumBounds(1, 3) < 4
True

>>> AccumBounds(1, 3) < -1
False

Some elementary functions can also take AccumulationBounds as input. A function \(f\) evaluated for some real AccumulationBounds \(<a, b>\) is defined as \(f(\left\langle a, b\right\rangle) = \{ f(x) \mid a \le x \le b \}\)

>>> sin(AccumBounds(pi/6, pi/3))
AccumBounds(1/2, sqrt(3)/2)

>>> exp(AccumBounds(0, 1))
AccumBounds(1, E)

>>> log(AccumBounds(1, E))
AccumBounds(0, 1)

Some symbol in an expression can be substituted for a AccumulationBounds object. But it doesn’t necessarily evaluate the AccumulationBounds for that expression.

The same expression can be evaluated to different values depending upon the form it is used for substitution since each instance of an AccumulationBounds is considered independent. For example:

>>> (x**2 + 2*x + 1).subs(x, AccumBounds(-1, 1))
AccumBounds(-1, 4)

>>> ((x + 1)**2).subs(x, AccumBounds(-1, 1))
AccumBounds(0, 4)

Notes

Do not use AccumulationBounds for floating point interval arithmetic calculations, use mpmath.iv instead.

References

R29

https://en.wikipedia.org/wiki/Interval_arithmetic

R30

http://fab.cba.mit.edu/classes/S62.12/docs/Hickey_interval.pdf

property delta

Returns the difference of maximum possible value attained by AccumulationBounds object and minimum possible value attained by AccumulationBounds object.

Examples

>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).delta
2
intersection(other)[source]

Returns the intersection of ‘self’ and ‘other’. Here other can be an instance of FiniteSet or AccumulationBounds.

Parameters

other: AccumulationBounds

Another AccumulationBounds object with which the intersection has to be computed.

Returns

AccumulationBounds

Intersection of ‘self’ and ‘other’.

Examples

>>> from sympy import AccumBounds, FiniteSet
>>> AccumBounds(1, 3).intersection(AccumBounds(2, 4))
AccumBounds(2, 3)

>>> AccumBounds(1, 3).intersection(AccumBounds(4, 6))
EmptySet

>>> AccumBounds(1, 4).intersection(FiniteSet(1, 2, 5))
{1, 2}
property max

Returns the maximum possible value attained by AccumulationBounds object.

Examples

>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).max
3
property mid

Returns the mean of maximum possible value attained by AccumulationBounds object and minimum possible value attained by AccumulationBounds object.

Examples

>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).mid
2
property min

Returns the minimum possible value attained by AccumulationBounds object.

Examples

>>> from sympy import AccumBounds
>>> AccumBounds(1, 3).min
1
sympy.calculus.util.continuous_domain(f, symbol, domain)[source]

Returns the intervals in the given domain for which the function is continuous. This method is limited by the ability to determine the various singularities and discontinuities of the given function.

Parameters

f : Expr

The concerned function.

symbol : Symbol

The variable for which the intervals are to be determined.

domain : Interval

The domain over which the continuity of the symbol has to be checked.

Returns

Interval

Union of all intervals where the function is continuous.

Raises

NotImplementedError

If the method to determine continuity of such a function has not yet been developed.

Examples

>>> from sympy import Symbol, S, tan, log, pi, sqrt
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import continuous_domain
>>> x = Symbol('x')
>>> continuous_domain(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> continuous_domain(tan(x), x, Interval(0, pi))
Union(Interval.Ropen(0, pi/2), Interval.Lopen(pi/2, pi))
>>> continuous_domain(sqrt(x - 2), x, Interval(-5, 5))
Interval(2, 5)
>>> continuous_domain(log(2*x - 1), x, S.Reals)
Interval.open(1/2, oo)
sympy.calculus.util.function_range(f, symbol, domain)[source]

Finds the range of a function in a given domain. This method is limited by the ability to determine the singularities and determine limits.

Parameters

f : Expr

The concerned function.

symbol : Symbol

The variable for which the range of function is to be determined.

domain : Interval

The domain under which the range of the function has to be found.

Returns

Interval

Union of all ranges for all intervals under domain where function is continuous.

Raises

NotImplementedError

If any of the intervals, in the given domain, for which function is continuous are not finite or real, OR if the critical points of the function on the domain can’t be found.

Examples

>>> from sympy import Symbol, S, exp, log, pi, sqrt, sin, tan
>>> from sympy.sets import Interval
>>> from sympy.calculus.util import function_range
>>> x = Symbol('x')
>>> function_range(sin(x), x, Interval(0, 2*pi))
Interval(-1, 1)
>>> function_range(tan(x), x, Interval(-pi/2, pi/2))
Interval(-oo, oo)
>>> function_range(1/x, x, S.Reals)
Union(Interval.open(-oo, 0), Interval.open(0, oo))
>>> function_range(exp(x), x, S.Reals)
Interval.open(0, oo)
>>> function_range(log(x), x, S.Reals)
Interval(-oo, oo)
>>> function_range(sqrt(x), x , Interval(-5, 9))
Interval(0, 3)
sympy.calculus.util.is_convex(f, *syms, domain=Reals)[source]

Determines the convexity of the function passed in the argument.

Parameters

f : Expr

The concerned function.

syms : Tuple of symbols

The variables with respect to which the convexity is to be determined.

domain : Interval, optional

The domain over which the convexity of the function has to be checked. If unspecified, S.Reals will be the default domain.

Returns

Boolean

The method returns \(True\) if the function is convex otherwise it returns \(False\).

Raises

NotImplementedError

The check for the convexity of multivariate functions is not implemented yet.

Notes

To determine concavity of a function pass \(-f\) as the concerned function. To determine logarithmic convexity of a function pass log(f) as concerned function. To determine logartihmic concavity of a function pass -log(f) as concerned function.

Currently, convexity check of multivariate functions is not handled.

Examples

>>> from sympy import symbols, exp, oo, Interval
>>> from sympy.calculus.util import is_convex
>>> x = symbols('x')
>>> is_convex(exp(x), x)
True
>>> is_convex(x**3, x, domain = Interval(-1, oo))
False

References

R31

https://en.wikipedia.org/wiki/Convex_function

R32

http://www.ifp.illinois.edu/~angelia/L3_convfunc.pdf

R33

https://en.wikipedia.org/wiki/Logarithmically_convex_function

R34

https://en.wikipedia.org/wiki/Logarithmically_concave_function

R35

https://en.wikipedia.org/wiki/Concave_function

sympy.calculus.util.lcim(numbers)[source]

Returns the least common integral multiple of a list of numbers.

The numbers can be rational or irrational or a mixture of both. \(None\) is returned for incommensurable numbers.

Parameters

numbers : list

Numbers (rational and/or irrational) for which lcim is to be found.

Returns

number

lcim if it exists, otherwise \(None\) for incommensurable numbers.

Examples

>>> from sympy import S, pi
>>> from sympy.calculus.util import lcim
>>> lcim([S(1)/2, S(3)/4, S(5)/6])
15/2
>>> lcim([2*pi, 3*pi, pi, pi/2])
6*pi
>>> lcim([S(1), 2*pi])
sympy.calculus.util.maximum(f, symbol, domain=Reals)[source]

Returns the maximum value of a function in the given domain.

Parameters

f : Expr

The concerned function.

symbol : Symbol

The variable for maximum value needs to be determined.

domain : Interval

The domain over which the maximum have to be checked. If unspecified, then Global maximum is returned.

Returns

number

Maximum value of the function in given domain.

Examples

>>> from sympy import Symbol, S, sin, cos, pi, maximum
>>> from sympy.sets import Interval
>>> x = Symbol('x')

>>> f = -x**2 + 2*x + 5
>>> maximum(f, x, S.Reals)
6

>>> maximum(sin(x), x, Interval(-pi, pi/4))
sqrt(2)/2

>>> maximum(sin(x)*cos(x), x)
1/2
sympy.calculus.util.minimum(f, symbol, domain=Reals)[source]

Returns the minimum value of a function in the given domain.

Parameters

f : Expr

The concerned function.

symbol : Symbol

The variable for minimum value needs to be determined.

domain : Interval

The domain over which the minimum have to be checked. If unspecified, then Global minimum is returned.

Returns

number

Minimum value of the function in the given domain.

Examples

>>> from sympy import Symbol, S, sin, cos, minimum
>>> from sympy.sets import Interval
>>> x = Symbol('x')

>>> f = x**2 + 2*x + 5
>>> minimum(f, x, S.Reals)
4

>>> minimum(sin(x), x, Interval(2, 3))
sin(3)

>>> minimum(sin(x)*cos(x), x)
-1/2
sympy.calculus.util.not_empty_in(finset_intersection, *syms)[source]

Finds the domain of the functions in \(finite_set\) in which the \(finite_set\) is not-empty

Parameters

finset_intersection : The unevaluated intersection of FiniteSet containing

real-valued functions with Union of Sets

syms : Tuple of symbols

Symbol for which domain is to be found

Raises

NotImplementedError

The algorithms to find the non-emptiness of the given FiniteSet are not yet implemented.

ValueError

The input is not valid.

RuntimeError

It is a bug, please report it to the github issue tracker (https://github.com/sympy/sympy/issues).

Examples

>>> from sympy import FiniteSet, Interval, not_empty_in, oo
>>> from sympy.abc import x
>>> not_empty_in(FiniteSet(x/2).intersect(Interval(0, 1)), x)
Interval(0, 2)
>>> not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x)
Union(Interval(1, 2), Interval(-sqrt(2), -1))
>>> not_empty_in(FiniteSet(x**2/(x + 2)).intersect(Interval(1, oo)), x)
Union(Interval.Lopen(-2, -1), Interval(2, oo))
sympy.calculus.util.periodicity(f, symbol, check=False)[source]

Tests the given function for periodicity in the given symbol.

Parameters

f : Expr.

The concerned function.

symbol : Symbol

The variable for which the period is to be determined.

check : Boolean, optional

The flag to verify whether the value being returned is a period or not.

Returns

period

The period of the function is returned. \(None\) is returned when the function is aperiodic or has a complex period. The value of \(0\) is returned as the period of a constant function.

Raises

NotImplementedError

The value of the period computed cannot be verified.

Notes

Currently, we do not support functions with a complex period. The period of functions having complex periodic values such as \(exp\), \(sinh\) is evaluated to \(None\).

The value returned might not be the “fundamental” period of the given function i.e. it may not be the smallest periodic value of the function.

The verification of the period through the \(check\) flag is not reliable due to internal simplification of the given expression. Hence, it is set to \(False\) by default.

Examples

>>> from sympy import Symbol, sin, cos, tan, exp
>>> from sympy.calculus.util import periodicity
>>> x = Symbol('x')
>>> f = sin(x) + sin(2*x) + sin(3*x)
>>> periodicity(f, x)
2*pi
>>> periodicity(sin(x)*cos(x), x)
pi
>>> periodicity(exp(tan(2*x) - 1), x)
pi/2
>>> periodicity(sin(4*x)**cos(2*x), x)
pi
>>> periodicity(exp(x), x)
sympy.calculus.util.stationary_points(f, symbol, domain=Reals)[source]

Returns the stationary points of a function (where derivative of the function is 0) in the given domain.

Parameters

f : Expr

The concerned function.

symbol : Symbol

The variable for which the stationary points are to be determined.

domain : Interval

The domain over which the stationary points have to be checked. If unspecified, S.Reals will be the default domain.

Returns

Set

A set of stationary points for the function. If there are no stationary point, an EmptySet is returned.

Examples

>>> from sympy import Symbol, S, sin, pi, pprint, stationary_points
>>> from sympy.sets import Interval
>>> x = Symbol('x')

>>> stationary_points(1/x, x, S.Reals)
EmptySet

>>> pprint(stationary_points(sin(x), x), use_unicode=False)
          pi                              3*pi
{2*n*pi + -- | n in Integers} U {2*n*pi + ---- | n in Integers}
          2                                2

>>> stationary_points(sin(x),x, Interval(0, 4*pi))
{pi/2, 3*pi/2, 5*pi/2, 7*pi/2}