Engineering

Algebraic Numbers and Minimal Polynomials

Algebraic numbers, minimal polynomials, algebraic integers, and the computational tests that distinguish them.

Algebras over a Ring

Algebras as rings carrying a compatible module structure, and the examples that matter computationally.

Algorithm Notation and Complexity Conventions

The notation, cost model and complexity conventions used throughout this collection, including the sub-exponential L-function.

Analysis of Berlekamp's Algorithm

Cost analysis of Berlekamp's algorithm and the variants that improve its dependence on field size.

Analysis of the Cantor-Zassenhaus Algorithm

The complete Cantor-Zassenhaus factorisation algorithm, its three stages, and its overall cost.

Applications of the Hermite Normal Form

Solving integer linear systems, computing indices and intersections, and testing module membership using the Hermite normal form.

Approximating Functions by Random Sampling

Estimating a quantity by sampling, the sample size required, and the resulting confidence guarantee.

Arbitrary Precision Arithmetic in Practice

Implementation concerns for multiprecision arithmetic: memory management, algorithm dispatch, and constant-time requirements.

Engineering Mathematics7 Aug 2026 · 26 min read

Archetypes: Reference Catalogue of Worked Systems

A catalogue of canonical worked linear systems, matrices and transformations, each fully analysed, designed so paired entries isolate a single structural variable.

Arithmetic Functions and Mobius Inversion

Multiplicative arithmetic functions, Dirichlet convolution, the Mobius function and the inversion formula.

Asymptotic Cost of Integer Multiplication

The M(n) abstraction, the hierarchy of multiplication algorithms, and why downstream bounds are quoted in terms of M(n) rather than fixed exponents.

Engineering Mathematics7 Aug 2026 · 4 min read

Asymptotic Notation for Algorithm Analysis

Big-O, Omega, Theta and little-o notation, the conventions that make them precise, and the pitfalls of using them carelessly.

Atkin-Morain Elliptic Curve Primality Proving

ECPP: using complex multiplication to construct curves of known order, avoiding point counting entirely.

Engineering Mathematics7 Aug 2026 · 26 min read

Bases

A basis is a linearly independent spanning set: enough raw material to build every vector, and no more. Standard bases, unique coordinates and verification.

Engineering Mathematics7 Aug 2026 · 27 min read

Bases and Nonsingular Matrices

The columns of a square matrix form a basis of C^m exactly when the matrix is nonsingular, and every vector then has a unique representation in that basis.

Engineering Mathematics7 Aug 2026 · 25 min read

Bases from Row Spaces

Stack a spanning set as the rows of a matrix and row-reduce: the non-zero rows are a basis for the span. The method, its canonical output and an equality test.

Basic Polynomial Arithmetic

Representation of polynomials, addition, multiplication and division costs, and the dense versus sparse choice.

Basic Properties of Polynomial Rings

Degree, leading coefficients, the ring structure of R[X], and when it is an integral domain.

Berlekamp's Factorization Algorithm

Berlekamp's algorithm: the Berlekamp subalgebra, the kernel computation, and splitting by gcds.

Bertrand's Postulate

Bertrand's postulate that a prime always lies between n and 2n, its elementary proof, and its use in algorithm analysis.

Better Smoothness Density Estimates

Refined estimates for smooth number density and how they determine optimal sieve parameters.

Binary Powering and Exponentiation Chains

Square-and-multiply exponentiation, left-to-right and right-to-left variants, windowing, and why exponentiation cost drives primality testing.

Binary Quadratic Forms and the Ideal Correspondence

The dictionary between binary quadratic forms and ideals of a quadratic order, and why the form language is computationally preferable.

Brute-Force Discrete Logarithm Search

Exhaustive search for discrete logarithms, its cost, and its role as the baseline against which other methods are measured.

Buchmann's Sub-exponential Algorithm: Overview

Buchmann's algorithm for class groups and units of arbitrary number fields, its structure, and where its cost concentrates.

Cantor-Zassenhaus Equal Degree Splitting

Splitting a product of irreducibles of equal degree by random elements, the probability analysis, and the characteristic two variant.

Engineering Mathematics7 Aug 2026 · 22 min read

Casting Out Vectors: Reducing a Spanning Set

Keep the vectors indexed by pivot columns and discard the rest: the basis-of-a-span theorem turns any generating set into a linearly independent one with the same span.

Engineering Mathematics7 Aug 2026 · 24 min read

Characterization of Vector Spaces

Every vector space of dimension n is isomorphic to complex n-space, and two finite-dimensional spaces are isomorphic exactly when their dimensions agree.

Chebyshev's Theorem on the Density of Primes

Chebyshev's elementary bounds on the prime counting function, the binomial coefficient argument, and what they establish short of the prime number theorem.

Checking the Condition C_p

The central congruence condition of the Jacobi sum test, what it asserts, and how it is verified in practice.

Chinese Remainder Theorem Algorithms

Reconstructing an integer from residues, Garner's incremental method, and CRT as a strategy for controlling coefficient growth.

Chinese Remaindering and Polynomial Interpolation

The Chinese remainder theorem for polynomials, and Lagrange interpolation as its special case.

Engineering Mathematics7 Aug 2026 · 1 min read

Choosing an Algorithm: Decision Guide

A consolidated decision guide across the main computational tasks in this collection.

Class Group and Unit Computation: the Computational Problem

Why class group and unit computation are a single problem, what the combined algorithm produces, and how the results are verified.

Class Numbers from Analytic Class Number Formulas

Using L-functions and the analytic class number formula to compute or verify class numbers and regulators.

Coefficient Explosion in Hermite Normal Form Computation

Why intermediate entries in normal form computation grow so violently, how to recognise it, and the three standard mitigations.

Engineering Mathematics7 Aug 2026 · 25 min read

Column Space of a Nonsingular Matrix

A square matrix is nonsingular exactly when its column space is all of C^n: proof, the right-inverse construction and the cost of a deficient column space.

Column Space Spanned by Original Columns

Row-reduce, note the pivot columns, then take those columns of the original matrix: a basis for the column space built from genuine columns rather than reduced ones.

Engineering Mathematics7 Aug 2026 · 24 min read

Column Spaces and Systems of Equations

The column space of a matrix is the span of its columns: exactly the set of right-hand sides for which a linear system is consistent, plus the membership test.

Column Vectors: Equality, Addition and Scalar Multiplication

The vector space of column vectors, with entrywise definitions of equality, addition and scalar multiplication, and how one vector equation replaces a system.

Engineering Mathematics7 Aug 2026 · 19 min read

Complex Arithmetic and Vectors

Conjugation of a column vector entry by entry, and the theorems showing it distributes over vector addition and over scalar multiplication with the scalar conjugated.

Complex Multiplication and Class Numbers

The link between curves with complex multiplication and class groups of imaginary quadratic orders, and the Hilbert class polynomial.

Complex Number Arithmetic

Addition, subtraction, multiplication and division of complex numbers, the imaginary unit, reciprocals, and why linear algebra takes its scalars from C.

Composition of Binary Quadratic Forms

Gauss composition as the group law on form classes, its practical computation, and reduction between compositions.

Engineering Mathematics7 Aug 2026 · 4 min read

Computational Algebraic Number Theory: Field Overview

What computational algebraic number theory actually computes, why the problems are hard, and how the subject's algorithms fit together.

Engineering Mathematics7 Aug 2026 · 5 min read

Computational Number Theory and Algebra: Field Overview

How number theory, abstract algebra and algorithm analysis combine into a single computational discipline, and how the KEVOS Mathematics library is organised.

Computational Number Theory: Tools and Libraries

Software for computational number theory and algebra, what each is suited to, and how to choose.

Computing Column Spaces

Four methods for describing the column space of a matrix: raw columns, pivot columns, the transpose-and-reduce route, and expressing the column space as a null space via extende…