TedToolkit.Mathematics.Asymptotic 2026.6.23.1

TedToolkit.Mathematics.Asymptotic

TedToolkit.Mathematics.Asymptotic models polynomial-like asymptotic expressions as x -> +infinity.

Use it when your code needs explicit reasoning about:

• normalized terms
• leading-order growth
• limit classification
• arithmetic over asymptotic expressions

Installation

dotnet add package TedToolkit.Mathematics.Asymptotic

Core Types

PolynomialTerm<TCoefficient, TExponent>

Represents a single term such as 3ω², -5ω, 7, or 2ω⁻¹.

• TCoefficient must satisfy INumber<TCoefficient>
• TExponent must satisfy ISignedNumber<TExponent> and INumber<TExponent>
• a zero coefficient is normalized to exponent 0

LimitPolynomial<TCoefficient, TExponent>

Represents a normalized collection of terms ordered by descending exponent.

During construction:

• terms with the same exponent are merged
• zero-coefficient terms are removed
• terms are stored in descending exponent order
• an all-zero input becomes a single zero term

LimitResult<TValue> and LimitState

Represents the limit classification returned by LimitValue.

Possible states are:

• FINITE
• POSITIVE_INFINITY
• NEGATIVE_INFINITY
• POSITIVE_INFINITESIMAL
• NEGATIVE_INFINITESIMAL

Basic Usage

using TedToolkit.Mathematics.Asymptotic;

var expression = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(3, 2),
    new PolynomialTerm<double, int>(-4, 1),
    new PolynomialTerm<double, int>(10, 0),
]);

Console.WriteLine(expression);

If multiple terms use the same exponent, they are merged automatically:

var simplified = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(3, 2),
    new PolynomialTerm<double, int>(5, 2),
    new PolynomialTerm<double, int>(-8, 2),
    new PolynomialTerm<double, int>(1, 0),
]);

Console.WriteLine(simplified); // 1

Limit Evaluation

LimitValue is determined by the leading term.

var polynomial = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(-2, 3),
    new PolynomialTerm<double, int>(5, 0),
]);

Console.WriteLine(polynomial.LimitValue);

Examples of the returned classification:

• positive coefficient with positive exponent → positive infinity
• negative coefficient with positive exponent → negative infinity
• exponent 0 → finite constant
• positive coefficient with negative exponent → positive infinitesimal
• negative coefficient with negative exponent → negative infinitesimal

Growth Comparison

Use CompareGrowthRateTo when only the leading term matters:

var left = new LimitPolynomial<double, int>(new PolynomialTerm<double, int>(2, 3));
var right = new LimitPolynomial<double, int>(new PolynomialTerm<double, int>(1, 2));

Console.WriteLine(left.CompareGrowthRateTo(right) > 0); // True

Use EqualsGrowthRate when two expressions should be treated as having the same leading-order behavior:

var a = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(2, 3),
    new PolynomialTerm<double, int>(1, 0),
]);

var b = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(2, 3),
    new PolynomialTerm<double, int>(-100, 1),
]);

Console.WriteLine(a.EqualsGrowthRate(b)); // True

Arithmetic and Comparison

LimitPolynomial<TCoefficient, TExponent> supports:

• comparison operators: ==, !=, <, <=, >, >=
• arithmetic operators: +, -, unary +, unary -, *, /, %

var x2 = new LimitPolynomial<double, int>([new PolynomialTerm<double, int>(1, 2)]);
var x = new LimitPolynomial<double, int>([new PolynomialTerm<double, int>(1, 1)]);

var sum = x2 + x;
var product = x2 * x;
var quotient = product / x2;

Console.WriteLine(sum);
Console.WriteLine(product);
Console.WriteLine(quotient);

Formatting

PolynomialTerm<TCoefficient, TExponent> and LimitPolynomial<TCoefficient, TExponent> support formatted string output.

var polynomial = new LimitPolynomial<double, int>(
[
    new PolynomialTerm<double, int>(12.5, 12),
    new PolynomialTerm<double, int>(-3.25, -2),
]);

Console.WriteLine(polynomial.ToString("0.00", "00", null));

The default string form uses ω notation, for example (2ω³ - 4ω + 6).

Notes

• This package models behavior only for x -> +infinity.
• LimitPolynomial<TCoefficient, TExponent> implements numeric-style interfaces and exposes parse members, but parsing is not currently supported.
• Generic parameters must satisfy the .NET numeric interface constraints used by the public types.

Target Frameworks

• net7.0
• net8.0
• net9.0
• net10.0