Numerical analysis

Numerical analysis is a branch of mathematics that uses numerical approximation techniques to solve mathematical problems that are difficult or impossible to solve analytically. It includes the study of algorithms and computational methods for problems involving continuous mathematics, such as those in calculus and linear algebra.

History

The history of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago. Many great mathematicians of the past were preoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Gauss elimination, and Euler's method.

Basic concepts

Numerical analysis is concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.

Error analysis

In numerical analysis, error analysis comprises both the analysis of the rounding errors inherent in an approximation method, and the propagation of these errors in the computation.

Interpolation, extrapolation, and regression

Interpolation involves estimating a function at a point by using nearby known values of the function. Extrapolation is similar, but it involves estimating values outside the known range. Regression involves fitting a function to a set of data.

Numerical differentiation and integration

Numerical differentiation involves approximating the derivative of a function, while numerical integration involves approximating the integral of a function.

Numerical solution of differential equations

Numerical analysis also involves the development and analysis of algorithms for solving differential equations, both ordinary differential equations and partial differential equations.

Applications

Numerical analysis is used in a wide range of fields, from computational physics and computational chemistry to statistics, economics, and engineering. It is also used in computer graphics, and in the development of algorithms for numerical approximation in computer algebra systems.

See also

• Computational mathematics
• Scientific computing
• Computational science
• Mathematical software