4.6.5. pde.tools.expressions module

Handling mathematical expressions with sympy

This module provides classes representing expressions that can be provided as human-readable strings and are converted to numpy and numba representations using sympy.

parse_number

return a number compiled from an expression

ScalarExpression

describes a mathematical expression of a scalar quantity

TensorExpression

describes a mathematical expression of a tensorial quantity

evaluate

evaluate an expression involving fields

class ExpressionBase(expression: sympy.core.basic.Basic, signature: Optional[Sequence[Union[str, List[str]]]] = None, *, user_funcs: Optional[Dict[str, Callable]] = None, consts: Optional[Dict[str, NumberOrArray]] = None)[source]

Bases: object

abstract base class for handling expressions

Warning

This implementation uses exec() and should therefore not be used in a context where malicious input could occur.

Parameters
  • expression (sympy.core.basic.Basic) – A sympy expression or array. This could for instance be an instance of Expr or NDimArray.

  • signature (list of str, optional) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as list, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.

  • user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expression.

  • consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. The values of these constants should either be numbers or ndarray.

classmethod check_reserved_symbols(symbols: Iterable[str], strict: bool = True) None[source]

throws an error if reserved symbols are found

Parameters
  • symbols (iterable) – A sequence or set of strings with symbols to check.

  • strict (bool) – Flag determining whether an exception is raised

property complex: bool

whether the expression contains the imaginary unit I

Type

bool

property constant: bool

whether the expression is a constant

Type

bool

depends_on(variable: str) bool[source]

determine whether the expression depends on variable

Parameters

variable (str) – the name of the variable to check for

Returns

whether the variable appears in the expression

Return type

bool

property expression: str

the expression in string form

Type

str

get_compiled(single_arg: bool = False) Callable[[...], NumberOrArray][source]

return numba function evaluating expression

Parameters

single_arg (bool) – Determines whether the returned function accepts all variables in a single argument as an array or whether all variables need to be supplied separately

Returns

the compiled function

Return type

function

property rank: int

the rank of the expression

Type

int

abstract property shape: Tuple[int, ...]
class ScalarExpression(expression: Union[float, str, numpy.ndarray, sympy.core.basic.Basic, ExpressionBase] = 0, signature: Optional[Sequence[Union[str, List[str]]]] = None, *, user_funcs: Optional[Dict[str, Callable]] = None, consts: Optional[Dict[str, NumberOrArray]] = None, allow_indexed: bool = False)[source]

Bases: pde.tools.expressions.ExpressionBase

describes a mathematical expression of a scalar quantity

Warning

This implementation uses exec() and should therefore not be used in a context where malicious input could occur.

Parameters
  • expression (str or float) – The expression, which is either a number or a string that sympy can parse

  • signature (list of str) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as lists, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.

  • user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expression

  • consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. The values of these constants should either be numbers or ndarray.

  • allow_indexed (bool) – Whether to allow indexing of variables. If enabled, array variables are allowed to be indexed using square bracket notation.

copy() ScalarExpression[source]

return a copy of the current expression

derivatives

differentiate the expression with respect to all variables

differentiate(var: str) ScalarExpression[source]

return the expression differentiated with respect to var

property is_zero: bool

returns whether the expression is zero

Type

bool

shape: Tuple[int, ...] = ()
property value: Union[int, float, complex]

the value for a constant expression

Type

float

class TensorExpression(expression: Union[float, str, numpy.ndarray, sympy.core.basic.Basic, ExpressionBase], signature: Optional[Sequence[Union[str, List[str]]]] = None, *, user_funcs: Optional[Dict[str, Callable]] = None, consts: Optional[Dict[str, NumberOrArray]] = None)[source]

Bases: pde.tools.expressions.ExpressionBase

describes a mathematical expression of a tensorial quantity

Warning

This implementation uses exec() and should therefore not be used in a context where malicious input could occur.

Parameters
  • expression (str or float) – The expression, which is either a number or a string that sympy can parse

  • signature (list of str) – The signature defines which variables are expected in the expression. This is typically a list of strings identifying the variable names. Individual names can be specified as list, in which case any of these names can be used. The first item in such a list is the definite name and if another name of the list is used, the associated variable is renamed to the definite name. If signature is None, all variables in expressions are allowed.

  • user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expression.

  • consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. The values of these constants should either be numbers or ndarray.

derivatives

differentiate the expression with respect to all variables

differentiate(var: str) TensorExpression[source]

return the expression differentiated with respect to var

get_compiled_array(single_arg: bool = True) Callable[[numpy.ndarray, numpy.ndarray], numpy.ndarray][source]

compile the tensor expression such that a numpy array is returned

Parameters

single_arg (bool) – Whether the compiled function expects all arguments as a single array or whether they are supplied individually.

property shape: Tuple[int, ...]

the shape of the tensor

Type

tuple

property value

the value for a constant expression

evaluate(expression: str, fields: Dict[str, DataFieldBase], *, bc: BoundaryConditionData = 'auto_periodic_neumann', bc_ops: Optional[Dict[str, BoundaryConditionData]] = None, user_funcs: Optional[Dict[str, Callable]] = None, consts: Optional[Dict[str, NumberOrArray]] = None, label: str = None) DataFieldBase[source]

evaluate an expression involving fields

Warning

This implementation uses exec() and should therefore not be used in a context where malicious input could occur.

Parameters
  • expression (str) – The expression, which is parsed by sympy. The expression may contain variables (i.e., fields and spatial coordinates of the grid), standard local mathematical operators defined by sympy, and the operators defined in the pde package. Note that operators need to be specified with their full name, i.e., laplace for a scalar Laplacian and vector_laplace for a Laplacian operating on a vector field. Moreover, the dot product between two vector fields can be denoted by using dot(field1, field2) in the expression, and outer(field1, field2) calculates an outer product. More information can be found in the expression documentation.

  • fields (dict) – Dictionary of the fields involved in the expression.

  • bc – Boundary conditions for the operators used in the expression. The conditions here are applied to all operators that do not have a specialized condition given in bc_ops. Boundary conditions are generally given as a list with one condition for each axis. For periodic axis, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non- periodic axes, different boundary conditions can be specified for the lower and upper end (using a tuple of two conditions). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

  • bc_ops (dict) – Special boundary conditions for some operators. The keys in this dictionary specify the operator to which the boundary condition will be applied.

  • user_funcs (dict, optional) – A dictionary with user defined functions that can be used in the expressions in rhs.

  • consts (dict, optional) – A dictionary with user defined constants that can be used in the expression. These can be either scalar numbers or fields defined on the same grid as the actual simulation.

  • label (str) – Name of the field that is returned.

Returns

The resulting field. The rank of the returned field (and thus the precise class) is determined automatically.

Return type

pde.fields.base.DataFieldBase

parse_number(expression: Union[str, int, float, complex], variables: Optional[Mapping[str, Number]] = None) Number[source]

return a number compiled from an expression

Warning

This implementation uses exec() and should therefore not be used in a context where malicious input could occur.

Parameters
  • expression (str or Number) – An expression that can be interpreted as a number

  • variables (dict) – A dictionary of values that replace variables in the expression

Returns

the calculated value

Return type

Number