A mean or variance summarizes a distribution, but a time series adds two questions: which date's distribution, and how are values at different dates related? A single average of the whole record cannot answer both. A rising level can be hidden by that average, and two sequences with identical histog...
A noisy-looking plot can still contain structure. Consecutive values may cluster on one side of the mean, changes may alternate unusually often, or large residuals may arrive together. A diagnostic test turns one such observation into a defined statistic and compares it with a reference distribution...
Kod bajtowy (ang. bytecode) w Pythonie to pośrednia, niskopoziomowa reprezentacja kodu źródłowego, która jest zrozumiała dla wirtualnej maszyny Pythona (Python Virtual Machine, PVM). Kiedy uruchamiamy skrypt Pythona, interpreter nie wykonuje bezpośrednio kodu źródłowego; zamiast tego, najpierw kompi...
PIP (Python Package Installer) to menedżer pakietów dla języka Python, który ułatwia zarządzanie pakietami z repozytorium PyPI (Python Package Index). PIP pozwala na łatwą instalację, aktualizację i usuwanie pakietów, co jest nieocenione przy rozbudowie projektów i zarządzaniu zależnościami...
Enum (od angielskiego słowa "enumerate" - numerować) to specjalny typ danych w Pythonie, który pozwala na definiowanie uporządkowanych zestawów nazwanych wartości. Wartości te są unikalne i niemodyfikowalne, co czyni je idealnymi do reprezentowania stałych w kodzie...
A partial differential equation (PDE) relates an unknown function of several independent variables to its partial derivatives. PDEs describe fields rather than single trajectories: temperature in space and time, pressure in a fluid, displacement in an elastic body, or the amplitude of a wave...
An ordinary differential equation (ODE) relates an unknown function of one independent variable to one or more of its derivatives. ODEs are the natural language of dynamical systems: they describe how a state changes when its instantaneous rate of change is known...
Heun's method improves Euler's method by using information from both ends of each step. It is a second-order explicit Runge–Kutta method and is also known as the explicit trapezoidal method or improved Euler method...
Picard iteration turns an initial value problem into a sequence of integral approximations. Each iterate is a function obtained by substituting the previous function into the ODE's integral form. This construction is central to the Picard–Lindelof existence-and-uniqueness theorem and explains how fi...
Runge–Kutta methods advance an ODE solution using several carefully chosen slope evaluations inside each step. They achieve higher-order accuracy without requiring derivatives of f; only evaluations of f itself are needed...
Euler's method is the simplest explicit time-stepping method for an initial value problem (IVP). It is useful both as a practical first approximation and as a way to understand how numerical ODE solvers work...
Checksum algorithms are methods for computing a small, fixed-size value from a block of data. They serve as a simple mechanism to detect accidental changes or corruption in data during transmission or storage. While not cryptographically secure, checksums are fundamental to error detection in digita...
Lagrange evaluation is a technique from polynomial interpolation that finds a direct cryptographic application in Shamir's Secret Sharing. It allows a secret to be divided into multiple shares such that only a sufficient number of shares can reconstruct the original secret, while any smaller number ...
Elliptic curve cryptography (ECC) is a public-key cryptographic approach based on the algebraic structure of elliptic curves over finite fields. The security of ECC relies on the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP), which makes it possible to achieve strong security w...
Monte Carlo integration approximates an integral by averaging function values at randomly selected points and applying the appropriate volume or probability weights. The sampling distribution is part of the method: uniform samples use the domain volume, while nonuniform samples require weights that ...
Simpson's rule approximates a definite integral by integrating quadratic interpolants through sampled function values. On each panel it uses two endpoints and their midpoint, fitting a parabola through the three points. A composite rule repeats this construction across pairs of equal subintervals an...
The midpoint rule approximates a definite integral by dividing its interval into smaller pieces and representing each piece by a rectangle. The rectangle's width is the subinterval width, and its height is the function value at that subinterval's midpoint. Adding the signed rectangle areas estimates...
Integration accumulates a quantity over an interval. It can recover total change from a known rate of change, measure area, and express solutions of differential equations. For example, integrating velocity over time gives displacement, including the sign of the motion...
The trapezoidal rule approximates a definite integral by connecting neighboring samples with straight line segments and integrating those segments. When the sampled values are nonnegative, each contribution is the area of a trapezoid; negative values contribute signed area in the same way as the exa...
Matrices organize numbers into rows and columns and represent linear transformations, systems of equations, and collections of data. Their dimensions determine which operations are defined, while their structure determines which numerical methods are appropriate. In applied mathematics and engineeri...
Singular value decomposition (SVD) expresses a matrix as orthogonal or unitary coordinate changes surrounding a rectangular diagonal scaling. It exists for every real or complex matrix, including rectangular and rank-deficient matrices, whereas a diagonal eigenvalue decomposition requires a diagonal...
The QR method computes the eigenvalues of a square matrix through a sequence of similarity transformations. Each iteration factors the current matrix into an orthogonal factor Q and an upper-triangular factor R, then forms the product in the opposite order. The new matrix has the same eigenvalues bu...
The power method estimates an eigenvalue of largest magnitude and its associated eigendirection by repeatedly multiplying a vector by a matrix and normalizing the result. Each multiplication strengthens the contribution from a strictly dominant eigenvalue relative to smaller ones. For the eigenbasis...
Eigenvalue decomposition, also called eigendecomposition, expresses a diagonalizable square matrix in a basis of its eigenvectors. In that basis, the transformation acts independently along each eigenvector by multiplying it by the corresponding eigenvalue. Changing back to the original coordinates ...
The inverse of a square matrix A is denoted by A⁻¹. When it exists, it is the unique matrix whose product with A, on either side, is the identity matrix I. Mathematically, this is expressed as...
LU decomposition, or LU factorization, stores Gaussian elimination as a product of triangular matrices. In the unpivoted form, A = LU, where L is lower triangular and U is upper triangular; with row pivoting, the convention used here is PA = LU. The factors support linear-system solves, determinant ...
Jacobi iteration approximates the solution of a linear system A x = b by repeatedly updating every component from the previous iterate. Each row is solved for the variable on its diagonal, with all other components held at their old values during that sweep. The resulting vector becomes the input to...
Gauss–Seidel is an iterative method for a linear system A x = b, with A ∈ ℝ^(n×n), unknown vector x = (x₁,…,xₙ)ᵀ, and given right-hand side b. It updates one component at a time during a sweep through the equations. Each new component is used immediately in the remaining updates, while components no...
The secant method approximates a root of f(x) = 0 using the two most recent points on the graph. Instead of evaluating the derivative in Newton's formula, it uses the slope of the secant line through those points. The next approximation is the zero of that line. This makes the method useful when an ...
The relaxation method, commonly referred to as the fixed-point iteration method, is an iterative approach used to find solutions (roots) to nonlinear equations of the form f(x) = 0. Instead of directly solving for the root, the method involves rewriting the original equation in the form...
Golden-section search, also called golden-ratio search, locates a minimum or maximum of a unimodal function on a specified interval. It compares function values at two interior points and discards a part of the interval while preserving a minimizer under the assumptions stated below. No derivatives ...
Gradient descent is a first-order method for minimizing a differentiable objective in mathematics, statistics, machine learning, and artificial intelligence. At the current point, the gradient gives the direction of steepest local increase in the Euclidean norm. The method moves in the opposite dire...
Bisection approximates a real root of a continuous function f by repeatedly halving an interval [a,b] whose endpoints have opposite signs. It is a bracketing method: each update preserves an interval known to contain at least one root, rather than relying on a single guess. Its main advantages are a...
Root-finding algorithms aim to solve equations of the form...
The backward difference method estimates a derivative using the function value at the target point x and a preceding point x-h, with h > 0. It is a finite difference method: the slope over a nonzero interval replaces the limiting slope that defines the derivative. This stencil is useful at the right...