"The study of algorithms that use numerical approximation for the problems of mathematical analysis."
Study of algorithms used to solve mathematical problems numerically.
Linear algebra: The study of linear equations and their solutions, vector spaces, matrices, eigenvectors and eigenvalues, and linear transformations.
Calculus: The study of limits, derivatives, integrals, infinite series, and differential equations.
Numerical methods: Algorithms and techniques used to solve mathematical problems numerically, including root-finding methods, interpolation, numerical integration, and differential equation solvers.
Statistical analysis: The application of statistical methods to analyze and interpret data, including hypothesis testing, regression analysis, and Bayesian statistics.
Optimization: The study of algorithms and techniques used to minimize or maximize a function subject to constraints, including linear programming, nonlinear programming, and dynamic programming.
Numerical linear algebra: The study of algorithms and techniques used to solve linear algebra problems numerically, including matrix factorizations, iterative methods, and sparse matrix techniques.
Partial differential equations: The study of differential equations involving partial derivatives, including numerical techniques for solving them, such as finite difference methods and numerical methods for solving nonlinear PDEs.
Monte Carlo methods: A class of computational algorithms that rely on repeated random sampling to obtain numerical solutions to problems, including simulation and optimization problems.
Computational fluid dynamics: The study of numerical techniques used to solve fluid mechanics problems, including computational methods for solving Navier-Stokes equations.
Machine learning: The study of algorithms and techniques used to train computers to recognize and predict patterns in data, including neural networks, decision trees, and support vector machines.
Finite Difference Method: In this method, differential equations are solved by approximating derivatives with finite differences. It is commonly used to solve partial differential equations.
Finite Element Method: This method is used to solve a wide range of problems from structural engineering to fluid mechanics. It involves dividing the physical domain into smaller, finite elements which are then analyzed using numerical methods.
Monte Carlo Method: This method uses random sampling to solve problems that would otherwise be difficult or impossible to solve analytically. It is used in physics, finance, and many other fields.
Spectral Method: This method is a numerical technique for solving differential equations in which the solution is represented as a sum of basis functions that are carefully chosen to satisfy boundary conditions.
Boundary Element Method: This method is used to solve boundary value problems by dividing the boundary into elements and using numerical methods to solve the equations.
Finite Volume Method: In this method, the physical domain is divided into smaller volumes, and the equations governing the system are integrated over these volumes.
Runge-Kutta Method: This numerical method is commonly used to solve ordinary differential equations. It involves a set of formulas to approximate the solution iteratively over small time steps.
Gauss-Seidel Method: This iterative method is used to solve linear systems of equations by computing the unknowns one at a time in a systematic manner.
Newton-Raphson Method: This method is used to find the roots of a function by using a series of approximations based on the previous iteration.
Fast Fourier Transform: This is a numerical algorithm used to efficiently compute the discrete Fourier transform of a sequence. It is widely used in signal processing, audio and image compression, and other fields.
"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."