API: smoothmath

exception smoothmath.DomainError: Raised when evaluating or differentiating at a point where an Expression is undefined.

exception smoothmath.CoordinateMissing: Raised when a Point does not have a required coordinate.

class smoothmath.Point(**kwargs)

A point. Can have any number of coordinates.

>>> from smoothmath import Point
>>> Point(x=3, y=4.5)
Point(x=3, y=4.5)

Parameters:: **kwargs (float) – a real number for each coordinate name

coordinate(variable)

Retrieves a coordinate.

Raises CoordinateMissing if the point has no entry for the coordinate name.

>>> from smoothmath import Point
>>> point = Point(x=3, y=4.5)
>>> point.coordinate("x")
3
>>> point.coordinate("y")
4.5

Parameters:: variable (Variable | str) – selects which coordinate
Return type:: float

class smoothmath.Expression

An abstract base class for all expresssions. See the smoothmath.expression module for concrete expression classes.

at(point)

Evaluates the expression at a point.

Will raise an exception if the point parameter is a float and the expression has more than one variable.

Parameters:: point (Point | float) – where to evaluate
Return type:: float

class smoothmath.Derivative(expression, compute_early=False)

The derivative of an expression.

>>> from smoothmath import Derivative
>>> from smoothmath.expression import Variable, NthPower
>>> Derivative(NthPower(Variable("x"), n=2))
Derivative(NthPower(Variable("x"), n=2))

NOTE: The expression must have only one variable. For alternatives that support expressions with multiple variables, see the Differential, Partial, and LocatedDifferential classes.

Parameters:

expression (Expression) – an expression with one variable
compute_early (bool) – whether to do extra work on initialization to have faster evaluation afterwards

at(point)

Evaluates the derivative.

Parameters:: point (Point | float) – where to evaluate the derivative
Return type:: float

as_expression()

Writes the derivative as an expression.

NOTE: Writing the derivative as an expression may enlargen the domain. For example, Derivative(Logarithm(Variable("x"))) is not defined at negative numbers, but as an expression, Reciprocal(Variable("x")) is defined at negative numbers.

Return type:: Expression

class smoothmath.Differential(expression, compute_early=False)

The differential of an expression.

>>> from smoothmath import Differential
>>> from smoothmath.expression import Variable, Multiply
>>> Differential(Multiply(Variable("x"), Variable("y")))
Differential(Multiply(Variable("x"), Variable("y")))

Parameters:

expression (Expression) – an expression
compute_early (bool) – whether to do extra work on initialization to have faster evaluation afterwards

component(variable)

Retrieves a component of the differential.

NOTE: The components of the differential are the partials of the original expression.

Parameters:: variable (Variable | str) – selects which component
Return type:: Partial

at(point)

Evaluates the differential at a point.

Parameters:: point (Point) – where to evaluate
Return type:: LocatedDifferential

component_at(variable, point)

Retrieves a component of the differential and evaluates it at a point.

Parameters:

variable (Variable | str) – selects which component
point (Point) – where to evaluate

Return type:

float

class smoothmath.Partial(expression, variable, compute_early=False)

The partial derivative of an expression.

>>> from smoothmath import Partial
>>> from smoothmath.expression import Variable, Multiply
>>> Partial(Multiply(Variable("x"), Variable("y")), Variable("x"))
Partial(Multiply(Variable("x"), Variable("y")), Variable("x"))

Parameters:

expression (Expression) – an expression
variable (Variable | str) – the partial is taken with respect to this variable
compute_early (bool) – whether to do extra work on initialization to have faster evaluation afterwards

at(point)

Evaluates the partial at a point.

Parameters:: point (Point) – where to evaluate the partial
Return type:: float

as_expression()

Writes the partial as an expression.

NOTE: Writing the partial as an expression may enlargen the domain. For example, Partial(Logarithm(Variable("x")), Variable("x")) is not defined at negative numbers, but as an expression, Reciprocal(Variable("x")) is defined at negative numbers.

Return type:: Expression

class smoothmath.LocatedDifferential(expression, point)

The differential of an expression located at a point.

>>> from smoothmath import Point, LocatedDifferential
>>> from smoothmath.expression import Variable, Multiply
>>> LocatedDifferential(Multiply(Variable("x"), Variable("y")), Point(x=2, y=4))
LocatedDifferential(Multiply(Variable("x"), Variable("y")), Point(x=2, y=4))

Parameters:

expression (Expression) – an expression
point (Point) – where to locate the differential

component(variable)

Retrieves a component of the differential.

NOTE: The components of the differential are the partials of the original expression.

Parameters:: variable (Variable | str) – selects which component
Return type:: float