Struct Monomial

Represents a single term of a Polynomial, of the form c * x^n, where c is the Coefficient and n is the Degree. Supports evaluation, differentiation, integration, and scalar/monomial arithmetic.

Implements

IEquatable<Monomial>

Inherited Members

object.Equals(object, object)

object.GetType()

object.ReferenceEquals(object, object)

Namespace: Scylla.Core.Util.Math

Assembly: ScyllaCore.dll

Syntax

[Serializable]
public struct Monomial : IEquatable<Monomial>

Remarks

A monomial is the building block of a Polynomial. Adding or subtracting two monomials via the + and - operators produces a Polynomial because the result may contain two distinct terms. Multiplying or dividing monomials always produces another Monomial because the degrees add or subtract.

This struct is serializable and can be exposed in the Unity Inspector. Equality comparisons use Approximately(float, float, float) to tolerate floating-point rounding errors in the coefficient while requiring exact degree equality.

Constructors

Monomial(float, int)

Initializes a new Monomial representing the term coefficient * x^degree.

Declaration

public Monomial(float coefficient, int degree)

Parameters

Type	Name	Description
float	coefficient	The scalar multiplier of the term. A value of zero creates a trivial monomial that always evaluates to zero.
int	degree	The exponent applied to the variable. Zero produces a constant term, negative values produce inverse-power terms.

Fields

Coefficient

The coefficient c of the term c * x^n. A coefficient of zero makes the monomial trivial (evaluates to zero for all x).

Declaration

[SerializeField]
public float Coefficient

Field Value

Type	Description
float

Degree

The degree (exponent) n of the term c * x^n. A degree of zero makes the monomial a constant equal to Coefficient. Negative degrees are permitted and represent negative powers of x.

Declaration

[SerializeField]
public int Degree

Field Value

Type	Description
int

Methods

Equals(Monomial)

Determines whether this Monomial is equal to another Monomial. The coefficients are compared using Approximately(float, float, float) to tolerate floating-point rounding errors, while the degrees are compared exactly.

Declaration

public bool Equals(Monomial other)

Parameters

Type	Name	Description
Monomial	other	The Monomial to compare with this instance.

Returns

Type	Description
bool	true if both the coefficient and the degree of other match those of this instance; otherwise false.

Equals(object)

Determines whether this Monomial is equal to another object.

Declaration

public override bool Equals(object obj)

Parameters

Type	Name	Description
object	obj	The object to compare with this instance.

Returns

Type	Description
bool	true if obj is a Monomial with approximately equal coefficient and exactly equal degree; otherwise false.

Overrides

ValueType.Equals(object)

Evaluate(float)

Evaluates this monomial at the given value of the variable, returning Coefficient * x^Degree.

Declaration

public float Evaluate(float x)

Parameters

Type	Name	Description
float	x	The value at which to evaluate the monomial.

Returns

Type	Description
float	The result of Coefficient * pow(x, Degree). Returns 0 when Coefficient is zero, regardless of x.

GetAntiderivative()

Computes the symbolic antiderivative (indefinite integral) of this monomial, excluding the constant of integration. The antiderivative of c * x^n is (c / (n+1)) * x^(n+1).

Declaration

public Monomial GetAntiderivative()

Returns

Type	Description
Monomial	A new Monomial with coefficient Coefficient / (Degree + 1) and degree Degree + 1.

Remarks

This method does not guard against Degree equal to -1 (the logarithmic case). Calling it on a monomial with degree -1 will produce a division by zero in the coefficient, resulting in Infinity or NaN.

See Also

GetDerivative()

Integrate(float, float)

GetDerivative()

Computes the symbolic derivative of this monomial using the power rule. The derivative of c * x^n is (n*c) * x^(n-1).

Declaration

public Monomial GetDerivative()

Returns

Type	Description
Monomial	A new Monomial with coefficient Degree * Coefficient and degree Degree - 1. For a constant term (Degree = 0) the result has coefficient 0 and degree -1.

See Also

GetAntiderivative()

GetHashCode()

Returns a hash code for this Monomial based on both the Coefficient and the Degree.

Declaration

public override int GetHashCode()

Returns

Type	Description
int	An int hash code suitable for use in hash-based collections.

Overrides

ValueType.GetHashCode()

Integrate(float, float)

Computes the definite integral of this monomial over the interval [a, b] using the fundamental theorem of calculus: F(b) - F(a), where F is the antiderivative.

Declaration

public float Integrate(float a, float b)

Parameters

Type	Name	Description
float	a	The lower limit of integration.
float	b	The upper limit of integration.

Returns

Type	Description
float	The signed area under the monomial curve between a and b. A negative result indicates the curve lies below the x-axis over the integration interval, or that b is less than a.

Remarks

Delegates to GetAntiderivative() and therefore inherits its restriction: do not use on monomials with Degree equal to -1.

See Also

GetAntiderivative()

ToString()

Returns a string representation of this monomial using "x" as the variable name.

Declaration

public override string ToString()

Returns

Type	Description
string	A string such as "3x^2", "5x", or "7" depending on the degree of this monomial.

Overrides

ValueType.ToString()

See Also

ToString(string)

ToString(string)

Returns a string representation of this monomial using the specified variable name.

Declaration

public string ToString(string variableName)

Parameters

Type	Name	Description
string	variableName	The name of the variable to use in the output (e.g., "x", "t", "z").

Returns

Type	Description
string	If Degree is 0: returns the coefficient as a string (e.g., "7"). If Degree is 1: returns the coefficient concatenated with the variable name (e.g., "3x"). Otherwise: returns "coefficient variable^degree" (e.g., "3x^2").

Operators

operator +(Monomial, Monomial)

Adds two monomials together, returning a Polynomial containing both terms. If both monomials share the same Degree, the Polynomial will combine them into a single term with a summed coefficient.

Declaration

public static Polynomial operator +(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The left-hand addend.
Monomial	b	The right-hand addend.

Returns

Type	Description
Polynomial	A Polynomial representing the sum a + b.

operator /(Monomial, Monomial)

Divides one monomial by another. The resulting monomial has the quotient of their coefficients and the difference of their degrees: (a * x^n) / (b * x^m) = (a/b) * x^(n-m).

Declaration

public static Monomial operator /(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The dividend monomial.
Monomial	b	The divisor monomial. Its coefficient must not be zero.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to a.Coefficient / b.Coefficient and Degree equal to a.Degree - b.Degree.

operator /(Monomial, float)

Divides a monomial by a scalar, reducing its Coefficient while leaving the Degree unchanged.

Declaration

public static Monomial operator /(Monomial m, float k)

Parameters

Type	Name	Description
Monomial	m	The monomial dividend.
float	k	The scalar divisor. Must not be zero.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to m.Coefficient / k and the same Degree.

operator ==(Monomial, Monomial)

Returns true if two Monomial values are equal, comparing coefficients approximately and degrees exactly.

Declaration

public static bool operator ==(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The left-hand operand.
Monomial	b	The right-hand operand.

Returns

Type	Description
bool	true if a and b are equal; otherwise false.

implicit operator Monomial(float)

Implicitly converts a float constant to a Monomial of degree 0 with a coefficient equal to the constant. This allows scalar values to be used wherever a constant term is expected.

Declaration

public static implicit operator Monomial(float constant)

Parameters

Type	Name	Description
float	constant	The scalar constant to convert.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to constant and Degree equal to 0.

operator !=(Monomial, Monomial)

Returns true if two Monomial values are not equal.

Declaration

public static bool operator !=(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The left-hand operand.
Monomial	b	The right-hand operand.

Returns

Type	Description
bool	true if a and b are not equal; otherwise false.

operator *(Monomial, Monomial)

Multiplies two monomials together. The resulting monomial has the product of their coefficients and the sum of their degrees: (a * x^n) * (b * x^m) = (a*b) * x^(n+m).

Declaration

public static Monomial operator *(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The left-hand factor.
Monomial	b	The right-hand factor.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to a.Coefficient * b.Coefficient and Degree equal to a.Degree + b.Degree.

operator *(Monomial, float)

Multiplies a monomial by a scalar, scaling its Coefficient while leaving the Degree unchanged.

Declaration

public static Monomial operator *(Monomial m, float k)

Parameters

Type	Name	Description
Monomial	m	The monomial to scale.
float	k	The scalar multiplier.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to m.Coefficient * k and the same Degree.

operator *(float, Monomial)

Multiplies a scalar by a monomial, scaling its Coefficient while leaving the Degree unchanged.

Declaration

public static Monomial operator *(float k, Monomial m)

Parameters

Type	Name	Description
float	k	The scalar multiplier.
Monomial	m	The monomial to scale.

Returns

Type	Description
Monomial	A Monomial with Coefficient equal to k * m.Coefficient and the same Degree.

operator -(Monomial, Monomial)

Subtracts one monomial from another, returning a Polynomial containing both terms. If both monomials share the same Degree, the Polynomial will combine them into a single term with the difference as its coefficient.

Declaration

public static Polynomial operator -(Monomial a, Monomial b)

Parameters

Type	Name	Description
Monomial	a	The minuend monomial.
Monomial	b	The subtrahend monomial.

Returns

Type	Description
Polynomial	A Polynomial representing the difference a - b.

operator -(Monomial)

Unary negation operator. Returns the additive inverse of the monomial by negating its Coefficient while preserving the Degree.

Declaration

public static Monomial operator -(Monomial m)

Parameters

Type	Name	Description
Monomial	m	The monomial to negate.

Returns

Type	Description
Monomial	A Monomial with the same Degree and negated Coefficient.

operator +(Monomial)

Unary plus operator. Returns the monomial unchanged.

Declaration

public static Monomial operator +(Monomial m)

Parameters

Type	Name	Description
Monomial	m	The operand.

Returns

Type	Description
Monomial	The same Monomial value as m.

Implements

IEquatable<T>

See Also

Polynomial