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. It is important less as a production ODE solver than as the constructive idea behind a fundamental existence-and-uniqueness theorem...
Runge--Kutta methods advance an ODE solution using several carefully chosen slope evaluations inside each step. They achieve higher-order accuracy without requiring explicit derivatives of $f$ beyond the first derivative already present in the ODE...
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...
Monte Carlo integration is a numerical technique for approximating integrals using randomness. Rather than systematically sampling a function at predetermined points, as done in methods like the trapezoidal rule or Simpson’s rule, Monte Carlo methods rely on random samples drawn from a prescribed do...
Simpson's Rule is a powerful technique in numerical integration, utilized for approximating definite integrals when an exact antiderivative of the function is difficult or impossible to determine analytically. This method enhances the accuracy of integral approximations by modeling the region under ...
The Midpoint Rule is a robust numerical method for approximating definite integrals. It seeks to estimate the area under a curve by partitioning it into a collection of rectangles and then summing the areas of these rectangles. This method is particularly useful when an antiderivative of the functio...
$$\int_{1}^{2} x^2 dx \approx \sum_{i=1}^{10} h \cdot f(1 + 0.1i)$...
The Trapezoidal Rule is a fundamental numerical integration technique employed to approximate definite integrals, especially when an exact antiderivative of the function is difficult or impossible to determine analytically. This method is widely used in various fields such as engineering, physics, a...
Eigenvalues and eigenvectors are foundational concepts in linear algebra, with extensive applications across various domains such as physics, computer graphics, and machine learning. These concepts are instrumental in decomposing complex matrix transformations, thereby simplifying numerical computat...
Matrices are often described as rectangular arrays of numbers organized into rows and columns, and they form the bedrock of numerous processes in numerical methods. People use them for solving systems of linear equations, transforming geometric data, and carrying out many algorithmic tasks that lie ...
Singular Value Decomposition (SVD) is a fundamental matrix decomposition technique widely used in numerous areas of science, engineering, and data analysis. Unlike the Eigenvalue Decomposition (EVD), which is restricted to square and diagonalizable matrices, SVD applies to any rectangular matrix. It...
The QR method is a widely used algorithm in numerical linear algebra for determining the eigenvalues of a given square matrix. Unlike direct methods such as solving the characteristic polynomial, which can be complicated and unstable numerically for large matrices, the QR method leverages iterative ...
The power method is a fundamental iterative algorithm for estimating the eigenvalue of largest magnitude and its associated eigenvector for a given matrix. This technique is particularly appealing when dealing with large and sparse matrices, where direct eigenvalue computations (e.g., via the charac...
Eigenvalue Decomposition (EVD), also known as Eigendecomposition, is a fundamental operation in linear algebra that breaks down a square matrix into a simpler form defined by its eigenvalues and eigenvectors. This decomposition provides deep insights into the properties and structure of a matrix, en...
The inverse of a matrix A is denoted as A^-1. It is a unique matrix such that when it is multiplied by the original matrix A, the result is the identity matrix I. Mathematically, this is expressed as...
A linear system of equations is a collection of one or more linear equations involving the same set of variables. Such systems arise in diverse areas such as engineering, economics, physics, and computer science. The overarching goal is to find values of the variables that simultaneously satisfy all...
LU Decomposition (or LU Factorization) is a powerful and widely used technique in numerical linear algebra for solving systems of linear equations, computing inverses, and determining determinants. The core idea is to factorize a given square matrix $A$ into the product of a lower-triangular matrix ...
The Jacobi method is a classical iterative algorithm used to approximate the solution of a system of linear equations $A\mathbf{x} = \mathbf{b}$. Instead of attempting to solve the system directly using methods such as Gaussian elimination, the Jacobi method iteratively refines an initial guess for ...
The Gauss-Seidel method is a classical iterative method for solving systems of linear equations of the form $A\mathbf{x} = \mathbf{b}$, where $A$ is an $n \times n$ matrix, $\mathbf{x}$ is the vector of unknowns $(x_1, x_2, \ldots, x_n)$, and $\mathbf{b}$ is a known vector. Unlike direct methods suc...
Gaussian elimination is a fundamental algorithmic procedure in linear algebra used to solve systems of linear equations, find matrix inverses, and determine the rank of matrices. The procedure systematically applies elementary row operations to transform a given matrix into an upper-triangular form ...
The Secant Method is a root-finding algorithm used in numerical analysis to approximate the zeros of a given function $f(x)$. It can be regarded as a derivative-free variant of Newton's method. Instead of computing the derivative $f'(x)$ at each iteration (as done in Newton’s method), it approximate...
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...
Newton's method (or the Newton-Raphson method) is a powerful root-finding algorithm that exploits both the value of a function and its first derivative to rapidly refine approximations to its roots. Unlike bracketing methods that work by enclosing a root between two points, Newton's method is an ope...
The Golden Ratio Search is a technique employed for locating the extremum (minimum or maximum) of a unimodal function over a given interval. Unlike gradient-based or derivative-requiring methods, this approach uses only function evaluations, making it broadly applicable even when derivatives are dif...
Gradient Descent is a fundamental first-order optimization algorithm widely used in mathematics, statistics, machine learning, and artificial intelligence. Its principal aim is to find the minimum of a given differentiable function $f(x)$. Instead of searching blindly, it uses gradient information —...
The bisection method is a classical root-finding technique used extensively in numerical analysis to locate a root of a continuous function $f(x)$ within a specified interval $[a, b]$. It belongs to the family of bracketing methods, which use intervals known to contain a root and systematically redu...
Root-finding algorithms aim to solve equations of the form...
The Taylor series is a fundamental tool in calculus and mathematical analysis, offering a powerful way to represent and approximate functions. By expanding a function around a specific point, known as the "center" or "point of expansion," we can express it as an infinite sum of polynomial terms deri...
Differentiation is a cornerstone concept in calculus, fundamental to understanding how quantities change in relation to one another. At its core, differentiation is used to determine the rate at which a particular quantity is changing at a specific point. This rate of change is quantitatively expres...
The central‐difference method is a finite‐difference scheme for estimating derivatives that combines forward and backward differences via Taylor‐series expansions. By evaluating the function at points symmetrically placed around the target, it cancels out many of the lower‐order error terms, yieldin...
Thin-plate spline (TPS) interpolation is a smooth method for fitting a surface through scattered points in two or more dimensions...
Interpolation constructs a function that passes exactly through a set of known data points. If the data are...
Gaussian radial basis function (RBF) interpolation builds a smooth interpolant from Gaussian functions centered at the data sites...
A cubic spline interpolates data with a sequence of cubic polynomials rather than one high-degree polynomial across the entire interval...
Least squares fits a model to data by minimizing the sum of squared residuals...
Linear interpolation estimates a value between two known points by assuming the function is a straight line over that interval...
Newton interpolation represents the same unique polynomial obtained by Lagrange interpolation, but in a form that is easier to build incrementally...
Regression models the relationship between one or more predictors and a response variable. Unlike interpolation, regression generally does not try to pass exactly through every observation. Instead, it chooses model parameters that balance data fit with a chosen model structure...
Lagrange interpolation constructs the unique polynomial of degree at most $n$ that passes through $n+1$ distinct data points...
The backward difference method is a finite difference technique employed to approximate the derivatives of functions. Unlike the forward difference method, which uses information from points ahead of the target point, the backward difference method relies on function values from points preceding the...
The forward difference method is a fundamental finite difference technique utilized for approximating the derivatives of functions. Unlike the central and backward difference methods, which use information from both sides or preceding points, respectively, the forward difference method relies solely...