"The study of algorithms that use numerical approximation for the problems of mathematical analysis."
The study of algorithms for solving mathematical problems, often involving numerical approximations.
Mathematical Fundamentals: This includes topics such as algebra, calculus, linear algebra, and differential equations, which are the foundation for numerical analysis.
Approximation Theory: This involves approximating complicated functions by simpler functions. It covers topics such as interpolation, curve fitting, and polynomial approximation.
Numerical Methods: These are algorithms used to solve mathematical problems. They include numerical integration, numerical differentiation, root-finding, and numerical linear algebra.
Error Analysis: This studies the accuracy and precision of numerical methods. It covers topics such as rounding errors, truncation errors, and convergence analysis.
Optimization: This studies techniques for finding the minimum or maximum of a function. It includes topics such as unconstrained and constrained optimization, gradient descent, and quadratic programming.
Monte Carlo Methods: This involves simulating random events to obtain a numerical solution to a problem. It covers topics such as simulation of random variables, random walks, and Markov chain Monte Carlo.
Partial Differential Equations: This studies solving partial differential equations using numerical methods. It covers topics such as finite difference methods, finite element methods, and spectral methods.
High-Performance Computing: This studies techniques for implementing computationally intensive algorithms on modern computer architectures. It includes topics such as multithreading, parallel computing, and GPU computing.
Data Analysis: This studies techniques for analyzing and visualizing data. It includes topics such as statistical analysis, data visualization, and machine learning.
Applications: This studies applying numerical methods to real-world problems. It includes topics such as financial modeling, computational physics, and computational biology.
Numerical Linear Algebra: Solves the systems of linear equations that arise in mathematical models by using algorithms that involve matrix manipulations.
Approximation Theory: Develops methods to approximate the solutions to equations or functions that are difficult or impossible to solve analytically.
Numerical Integration: Approximates the definite integral of a function over an interval, using techniques like the Trapezoidal rule or Simpson's rule.
Ordinary Differential Equations (ODEs): Studies numerical methods for solving ODEs that arise in physical, engineering, and biological applications.
Partial Differential Equations (PDEs): Focuses on numerical methods for solving PDEs that model a range of phenomena like heat transfer, fluid flow, and elasticity.
Optimization: Solves optimization problems where the goal is to minimize or maximize an objective function subject to constraints using numerical methods.
Monte Carlo Methods: Simulates stochastic processes by generating a large number of random sample values to estimate an unknown quantity or probability.
Numerical Optimization: Develops algorithms to optimize a function, usually a function of several variables, over its domain.
Finite Element Method: Computes numerical solutions to partial differential equations by discretizing the domain into a finite number of subdomains and solving them separately.
Numerical Methods for Scientific Computing: Develops algorithms that enable the solution of complex mathematical models in science and engineering.
"Numerical analysis uses numerical approximation, while symbolic manipulations involve exact solutions."
"All fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business, and even the arts."
"Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering."
"Examples of numerical analysis include ordinary differential equations in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains in medicine and biology."
"Before modern computers, numerical methods often relied on hand interpolation formulas, using data from large printed tables."
"Since the mid 20th century, computers calculate the required functions instead."
"Many of the same formulas continue to be used in software algorithms."
"The numerical point of view goes back to the earliest mathematical writings."
"A tablet from the Yale Babylonian Collection (YBC 7289) gives a sexagesimal numerical approximation of the square root of 2."
"Approximate solutions within specified error bounds are used."
"Predicting the motions of planets, stars, and galaxies through ordinary differential equations."
"Numerical linear algebra is used in data analysis."
"Stochastic differential equations and Markov chains are used for simulating living cells."
"Numerical analysis provides approximate solutions applicable only to real-world measurements."
"Numerical analysis gives approximate solutions instead of exact symbolic answers."
"Hand interpolation formulas were used before computers to obtain numerical approximations."
"Computing power has enabled the use of more complex numerical analysis, providing detailed mathematical models."
"Numerical analysis has seen growth in all fields, including the life and social sciences, medicine, business, and the arts."
"Complex numerical analysis models provide detailed and realistic mathematical models in science and engineering."