Poly

class Poly

The polynomial class used for representing objective functions and constraints.

Methods

__init__	__init__
as_dict	Return the dictionary representation of the polynomial.
as_variable	Return the variable representation of the polynomial.
asdict	alias of as_dict()
decode	Equivalent to evaluate().
degree	Return the degree of the polynomial.
evaluate	Evaluate the polynomial by substituting variables using the given values.
is_linear	Return True if the polynomial is linear.
is_number	Return True if the polynomial is a constant polynomial.
is_quadratic	Return True if the polynomial is quadratic.
is_variable	Return True if the polynomial represents a single variable.
substitute	Assign numbers or other polynomials to variables in the polynomial.

Attributes

id	Variable ID.
lower_bound	Lower bound of the variable.
name	Variable name.
type	Variable type.
upper_bound	Upper bound of the variable.
variables	List of all variables in the polynomial.

class Iterator

__init__(*args, **kwargs)

__iter__(self) → Iterator

Return type:

Iterator

__next__(self) → tuple[tuple[Poly, ...], float]

Return type:

tuple[tuple[Poly, …], float]

__add__(self, arg: Poly | float, /) → Poly

__add__(self, arg: ndarray, /) → PolyArray[Dim]

__add__(self, arg: Constraint | ConstraintList, /) → Model

Add a polynomial, a number, an array, or a constraint to the polynomial.

The result is a Poly for a polynomial or a number, a PolyArray for an array, and a Model for a Constraint or a ConstraintList.

Overload 1.

Args:

• arg (amplify.Poly | float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

Overload 3.

Args:

• arg (amplify.Constraint | amplify.ConstraintList)

Returns:

amplify.Model:

__and__(self, arg: Poly | bool, /) → Poly

__and__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical AND of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 & q_1 is equivalent to q_0 * q_1.

Overload 1.

Args:

• arg (amplify.Poly | bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__bool__(self) → bool

Cast a polynomial to a boolean value. Only valid for constant polynomials that take values 0 or 1.

Return type:

bool

__eq__(self, arg: object, /) → bool

Tell whether the two sides are the same polynomial.

This compares the terms, so it gives a bool and not a constraint. Use equal_to() to make a constraint that asks for equality.

Parameters:

arg (object)

Return type:

bool

__float__(self) → float

Cast a polynomial to a float value. Only valid for constant polynomials.

Return type:

float

Raises:

ValueError – The polynomial is not a constant.

__iadd__(self, arg: Poly | float, /) → Poly

Parameters:

arg (Poly | float)

Return type:

Poly

__iand__(self, arg: Poly | bool, /) → Poly

Parameters:

arg (Poly | bool)

Return type:

Poly

__imul__(self, arg: Poly | float, /) → Poly

Parameters:

arg (Poly | float)

Return type:

Poly

__index__(self) → int

Cast a polynomial to an integer. Only valid for constant polynomials.

Return type:

int

Raises:

ValueError – The polynomial is not a constant.

__init__(self) → None

__init__(self, arg: float, /) → None

__init__(self, arg: Variable, /) → None

__init__

Overload 2.

Args:

• arg (float)

Overload 3.

Args:

• arg (amplify.Variable)

__int__(self) → int

Cast a polynomial to an integer. Only valid for constant polynomials.

Return type:

int

Raises:

ValueError – The polynomial is not a constant.

__invert__(self) → Poly

Logical NOT of a polynomial. This is meaningful only for polynomials that take values 0 or 1.

~q_0 is equivalent to 1 - q_0.

Return type:

Poly

__ior__(self, arg: Poly | bool, /) → Poly

Parameters:

arg (Poly | bool)

Return type:

Poly

__ipow__(self, arg: int, /) → Poly

Raise the polynomial to a power in place. The exponent must be an integer that is 0 or larger.

Parameters:

arg (int)

Return type:

Poly

Raises:

ValueError – The exponent is negative.

__isub__(self, arg: Poly | float, /) → Poly

Parameters:

arg (Poly | float)

Return type:

Poly

__iter__(self) → Iterator

Iterate over the terms in the polynomial.

Each term is represented as a tuple of a tuple of variables and a coefficient. For example, the polynomial 3 * x_0 * x_1 - 2 * x_0 + 5 is iterated as ((Poly(x_0), Poly(x_1)), 3.0), ((Poly(x_0),), -2.0), and ((), 5.0).

Return type:

Iterator

__itruediv__(self, arg: float, /) → Poly

Parameters:

arg (float)

Return type:

Poly

__ixor__(self, arg: Poly | bool, /) → Poly

Parameters:

arg (Poly | bool)

Return type:

Poly

__len__(self) → int

Return the number of terms in the polynomial.

Return type:

int

__mul__(self, arg: Poly | float, /) → Poly

__mul__(self, arg: ndarray, /) → PolyArray[Dim]

Multiply the polynomial by a polynomial or by a number.

Overload 1.

Args:

• arg (amplify.Poly | float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__ne__(self, arg: object, /) → bool

Tell whether the two sides are different polynomials.

This is the negation of ==. It gives a bool and not a constraint.

Parameters:

arg (object)

Return type:

bool

__neg__(self) → Poly

Give the polynomial in which every coefficient has the opposite sign.

Return type:

Poly

__or__(self, arg: Poly | bool, /) → Poly

__or__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical OR of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 | q_1 is equivalent to -q_0 * q_1 + q_0 + q_1.

Overload 1.

Args:

• arg (amplify.Poly | bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__pos__(self) → Poly

Give a copy of the polynomial.

Return type:

Poly

__pow__(self, arg: int, /) → Poly

Raise the polynomial to a power. The exponent must be an integer that is 0 or larger.

Parameters:

arg (int)

Return type:

Poly

Raises:

ValueError – The exponent is negative.

__radd__(self, arg: float, /) → Poly

__radd__(self, arg: ndarray, /) → PolyArray[Dim]

__radd__(self, arg: Constraint | ConstraintList, /) → Model

Add a polynomial, a number, an array, or a constraint to the polynomial.

The result is a Poly for a polynomial or a number, a PolyArray for an array, and a Model for a Constraint or a ConstraintList.

Overload 1.

Args:

• arg (float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

Overload 3.

Args:

• arg (amplify.Constraint | amplify.ConstraintList)

Returns:

amplify.Model:

__rand__(self, arg: bool, /) → Poly

__rand__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical AND of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 & q_1 is equivalent to q_0 * q_1.

Overload 1.

Args:

• arg (bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__repr__(self) → str

Return type:

str

__rmul__(self, arg: float, /) → Poly

__rmul__(self, arg: ndarray, /) → PolyArray[Dim]

Multiply the polynomial by a polynomial or by a number.

Overload 1.

Args:

• arg (float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__ror__(self, arg: bool, /) → Poly

__ror__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical OR of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 | q_1 is equivalent to -q_0 * q_1 + q_0 + q_1.

Overload 1.

Args:

• arg (bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__rsub__(self, arg: float, /) → Poly

__rsub__(self, arg: ndarray, /) → PolyArray[Dim]

Subtract the polynomial from a number.

Overload 1.

Args:

• arg (float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__rxor__(self, arg: bool, /) → Poly

__rxor__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical XOR of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 ^ q_1 is equivalent to -2 * q_0 * q_1 + q_0 + q_1.

Overload 1.

Args:

• arg (bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__str__(self) → str

Return type:

str

__sub__(self, arg: Poly | float, /) → Poly

__sub__(self, arg: ndarray, /) → PolyArray[Dim]

Subtract a polynomial or a number from the polynomial.

Overload 1.

Args:

• arg (amplify.Poly | float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__truediv__(self, arg: float, /) → Poly

__truediv__(self, arg: ndarray, /) → PolyArray[Dim]

Divide the polynomial by a number.

Overload 1.

Args:

• arg (float)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

__xor__(self, arg: Poly | bool, /) → Poly

__xor__(self, arg: ndarray[dtype=bool], /) → PolyArray[Dim]

Logical XOR of two polynomials. This is meaningful only for polynomials that take values 0 or 1.

q_0 ^ q_1 is equivalent to -2 * q_0 * q_1 + q_0 + q_1.

Overload 1.

Args:

• arg (amplify.Poly | bool)

Returns:

amplify.Poly:

Overload 2.

Args:

• arg (ndarray)

Returns:

amplify.PolyArray[Dim]:

as_dict(self) → dict[tuple[int, ...], float]

Return the dictionary representation of the polynomial.

The polynomial is represented as a dictionary where keys are tuples of variable indices and values are coefficients. For example, the polynomial 3 * x_0 - 2 * x_0 * x_1 + 5 will be represented as {(): 5, (0,): 3, (0, 1): -2}.

Return type:

dict[tuple[int, …], float]

as_variable(self) → Variable

Return the variable representation of the polynomial. Only valid for polynomials that represent a single variable.

Return type:

Variable

Raises:

RuntimeError – The polynomial is not a single variable.

asdict(self) → dict[tuple[int, ...], float]

alias of as_dict()

Return type:

dict[tuple[int, …], float]

decode(self, values: Values) → float

decode(self, values: Values, default: float) → float

decode(self, values: Values, default: None) → Poly

Equivalent to evaluate().

This method is deprecated since Amplify v1.0.0 and will be removed in a future version. Please see the migration guide for details: https://amplify.fixstars.com/docs/amplify/v1/migration.html

Overload 1.

Args:

• values (amplify.Values)

Returns:

float:

Overload 2.

Args:

• values (amplify.Values)

• default (float)

Returns:

float:

Overload 3.

Args:

• values (amplify.Values)

• default (None)

Returns:

amplify.Poly:

degree(self) → int

Return the degree of the polynomial.

Return type:

int

evaluate(self, values: Values) → float

evaluate(self, values: Values, default: float) → float

evaluate(self, values: Values, default: None) → Poly

Evaluate the polynomial by substituting variables using the given values.

This method is usually used to obtain optimization results in array form.

When the keyword argument default is absent, a variable that values does not contain takes the default value of its type:

• a binary variable takes 0;

• an Ising variable takes -1;

• an integer or a real variable takes 0 when its bounds contain 0, the upper bound when they do not, and the lower bound when it has no upper bound.

If the keyword argument default is specified with a float value, variables that are not specified in values will be substituted by that value.

If the keyword argument default is specified with None, variables that are not specified in values will remain as symbolic variables without substitution.

Overload 1.

Args:

• values (amplify.Values): Values obtained from optimization results, e.g. result.best.values.

Returns:

float:

Raises:

ValueError: values comes from another VariableGenerator than the one that made the variables of this polynomial.

Overload 2.

Args:

• values (amplify.Values): Values that can be obtained from optimization results, e.g. result.best.values.

• default (float): The default value for variables not specified in values.

Returns:

float:

Overload 3.

Args:

• values (amplify.Values): Values obtained from optimization results, e.g. result.best.values.

• default (None): Set to None to keep variables not specified in values symbolic.

Returns:

amplify.Poly:

is_linear(self) → bool

Return True if the polynomial is linear.

Return type:

bool

is_number(self) → bool

Return True if the polynomial is a constant polynomial.

Return type:

bool

is_quadratic(self) → bool

Return True if the polynomial is quadratic.

Return type:

bool

is_variable(self) → bool

Return True if the polynomial represents a single variable.

Return type:

bool

substitute(

self,

mapping: collections.abc.Mapping[Poly, Poly | float | int] | collections.abc.Mapping[Variable, Poly | float | int],

) → Poly

Assign numbers or other polynomials to variables in the polynomial.

Parameters:

mapping (collections.abc.Mapping[Poly, Poly | float | int] | collections.abc.Mapping[Variable, Poly | float | int]) – Key-value pairs of variables and replacement values. Each value can be a number or another polynomial.

Return type:

Poly

property id

Variable ID. Only valid for polynomials that represent a single variable.

Return type:

int

Raises:

RuntimeError – The polynomial is not a single variable.

property lower_bound

Lower bound of the variable. Only valid for polynomials that represent a single variable.

Return type:

float | None

Raises:

RuntimeError – The polynomial is not a single variable.

property name

Variable name. Only valid for polynomials that represent a single variable.

Return type:

str

Raises:

RuntimeError – The polynomial is not a single variable.

property type

Variable type. Only valid for polynomials that represent a single variable.

Return type:

VariableType

Raises:

RuntimeError – The polynomial is not a single variable.

property upper_bound

Upper bound of the variable. Only valid for polynomials that represent a single variable.

Return type:

float | None

Raises:

RuntimeError – The polynomial is not a single variable.

property variables

List of all variables in the polynomial.

Return type:

list[Variable]