---
layout: default
title: Mathematical Landscape
---
# Mathematical Landscape
A map of mathematical domains relevant to this project, grouped by how the concepts relate
to and depend on each other. Pure mathematics only — no implementation, no prioritization,
no versioning.
Groups are ordered by mathematical dependency: later groups build on earlier ones.
---
## Group A — Elementary functions
The scalar functions that everything else is built from.
- Powers and roots: square root, cube root, general power, hypotenuse (√(x²+y²))
- Exponential and logarithm: exponential, natural log, log base 2, log base 10
- Trigonometric: sine, cosine, tangent, and their inverses (arcsine, arccosine, arctangent,
two-argument arctangent)
- Hyperbolic functions: sinh, cosh, tanh, and their inverses
- Rounding and classification: floor, ceiling, round, truncation, fractional part, absolute
value, sign, detection of special values (not-a-number, infinity)
- Constants: π, e, τ, √2, etc.
These operate on single numbers, but every other group depends on them.
---
## Group B — Vectors
The smallest object with linear-algebraic structure: an ordered tuple of numbers with
addition and scalar multiplication.
- Arithmetic: addition, subtraction, scalar multiplication, element-wise multiplication and
division
- Products: dot product (inner product), cross product (three dimensions only)
- Norms: L1 (sum of absolute values), L2 (Euclidean length), L∞ (maximum absolute value),
general Lp norm
- Derived quantities: normalization (unit vector), angle between two vectors, projection of
one vector onto another, distance between two vectors
- Comparisons: element-wise minimum/maximum, clamping to a range
Depends on Group A (norms require square roots; angles require inverse trigonometric
functions).
---
## Group C — Dense matrices
A matrix represents a linear map between vector spaces; this group builds directly on
vectors.
**Basic arithmetic**
- Addition, subtraction, scalar multiplication, element-wise multiplication
- Matrix–vector multiplication, matrix–matrix multiplication
- Transpose, trace
**Structural properties**
- Determinant
- Rank
- Matrix norms: Frobenius norm, induced 1-norm and ∞-norm, spectral norm
- Condition number
**Solving and inverting**
- Solving a linear system of equations
- Matrix inverse (well-defined and tractable mainly for small matrices)
**Decompositions** — each of these is itself a small algorithm with its own numerical
behavior, not a single operation
- LU decomposition, with and without pivoting
- QR decomposition (via Gram-Schmidt orthogonalization or Householder reflections)
- Cholesky decomposition (symmetric positive-definite matrices only; cheaper than LU when
applicable)
- Eigendecomposition
- Singular Value Decomposition (the most general and most expensive; underlies
least-squares fitting, principal component analysis, the pseudo-inverse, and the
condition number)
Depends on Groups A and B.
---
## Group D — Sparse matrices
The same mathematical objects as Group C, in the case where most entries are zero. This is a
representation and access-pattern distinction, not new mathematics: the operations mirror
Group C, but the algorithms to carry them out differ substantially.
**Representations**
- Coordinate list: each non-zero stored as (row, column, value) — simple to construct,
inefficient to compute with
- Compressed row representation — efficient for row-wise access and for multiplying by a
vector
- Compressed column representation — the column-wise counterpart
- Conversion between representations
**Operations**
- Addition, scalar multiplication
- Multiplication by a dense vector
- Multiplication by another sparse matrix
- Multiplication by a dense matrix
- Measures of sparsity (proportion and pattern of non-zero entries)
Depends on Group C conceptually, and on Group A for any numeric reductions involved.
---
## Group E — Krylov subspace methods
For matrices too large to decompose directly, where forming the full matrix or its
decomposition is computationally infeasible. Instead of operating on the whole matrix, a
small subspace is built from repeated matrix–vector products, and the problem is solved
within that subspace.
- **Lanczos iteration** — applies to symmetric matrices; computationally cheap, since it only
needs to retain the last couple of vectors generated. Underlies:
- The Conjugate Gradient method, for solving linear systems with symmetric
positive-definite matrices
- Eigenvalue and eigenvector estimation for symmetric matrices
- **Arnoldi iteration** — applies to general (non-symmetric) matrices; more expensive, since
it requires orthogonalizing against all previously generated vectors. Underlies:
- GMRES, for solving general linear systems
- Eigenvalue and eigenvector estimation for general matrices
- **Power iteration and inverse power iteration** — the simplest eigenvalue estimator,
finding only the dominant eigenvalue; a natural stepping stone before Lanczos or Arnoldi
- **Preconditioning** — a family of techniques for transforming a problem so that iterative
methods converge faster, rather than a single operation
Depends on Group B (vector operations) and on Group C or D as the underlying linear operator
— Krylov methods only require the ability to multiply that operator by a vector, regardless
of whether it is dense or sparse.
---
## Group F — Numerical calculus
The bridge between continuous mathematics and discrete computation.
- **Differentiation**: forward, backward, and central finite differences
- **Integration (quadrature)**: trapezoidal rule, Simpson's rule, Gaussian quadrature
- **Root finding**: bisection method, Newton-Raphson method, secant method
- **Interpolation**: linear interpolation, polynomial interpolation (Lagrange or Newton
form), cubic splines
- **Optimization**: gradient descent, line search, as foundational methods
Depends on Group A (the functions typically being differentiated, integrated, or solved are
built from elementary functions), and on Group B for the multivariate case (gradients are
vectors).
---
## Group G — Dynamical systems
Models of systems that evolve over time according to differential equations, including
their long-term and chaotic behavior.
**Solving ordinary differential equations** (in increasing order of accuracy and cost)
- Euler's method (explicit) — simplest, least accurate
- Runge-Kutta methods (second order, fourth order) — standard general-purpose solvers
- Adaptive step-size methods (e.g. Runge-Kutta-Fehlberg / Dormand-Prince) — adjust the step
size to control local error
- Implicit methods (e.g. backward Euler) — required for stiff systems, where explicit
methods become numerically unstable
**Systems and their analysis**
- Systems of coupled differential equations, not just single equations
- Canonical examples used to study and validate solver behavior: the logistic map, the
Lorenz system, the Rössler system
- Fixed points and their stability
- Lyapunov exponents — quantify sensitivity to initial conditions, the formal signature of
chaotic behavior
- Bifurcation analysis — how qualitative system behavior changes as a parameter varies
- Phase-space trajectories and Poincaré sections, as tools for analyzing long-term behavior
Depends on Group A, on Group F (root finding for fixed points, differentiation for
linearization and stability analysis), on Group B (state vectors), and on Group C
(Jacobian matrices for stability analysis).
---
## Group H — Signal processing
The mathematics of processing discrete, sampled signals over time.
**Filtering**
- Finite impulse response filtering
- Infinite impulse response filtering (cheaper per sample, but can be unstable)
- Standard filter responses: low-pass, high-pass, band-pass, band-stop
- Convolution and correlation (cross-correlation, autocorrelation)
**Transforms**
- The Discrete Fourier Transform, as the conceptual baseline
- The Fast Fourier Transform, as its efficient computational form
- Window functions (Hamming, Hanning, Blackman), used to reduce spectral leakage before
transforming a signal
**Signal analysis**
- Root mean square, peak detection, zero-crossing rate
- Spectral analysis: power spectrum, dominant frequency
- Resampling: decimation (reducing the sample rate) and interpolation (increasing it)
Depends on Group A (filters and transforms are built from trigonometric functions), Group B
(a signal is naturally represented as a vector of samples), and conceptually on Group F
(convolution is the discrete analogue of integration).