Numerical Analysis is an area of mathematics that focuses on finding approximate solutions to problems that cannot be solved exactly. It involves the use of algorithms and theoretical tools to solve problems that involve the manipulation of numbers. Numerical analysis has a wide range of applications in computer science, ranging from studying complex models such as simulations of game theory to finding solutions to linear and nonlinear systems of equations.
Numerical analysis is usually divided into two main classes of techniques, approximation techniques and exact techniques. Approximation techniques involve using approximate methods to get a general idea of the solution, while exact techniques involve a more detailed process of finding exact solutions. Approximation techniques are often used to solve problems that are too complex to solve using exact techniques.
The most common method of numerical analysis is numerical integration. This involves using a numerical technique to calculate the area under a given curve. Other numerical methods include solving linear and nonlinear systems of equations, finding the roots of functions, optimization techniques, and even interpolation and extrapolation.
Numerical analysis has been used to solve problems for many decades. Early examples include numerical integration methods developed by Isaac Newton and Gottfried Leibniz in the 17th century. In the 19th century, numerical analysis saw a dramatic surge in popularity with the development of Monte Carlo simulations, Gaussian quadrature methods, and numerical linear algebra. With the growth of computer science and computing power, numerical analysis has become a critically important area of research.
Today, numerical analysis is used in a variety of fields. Some of the most common applications include: statistical analysis, image processing, and weather prediction. Numerical analysis is also used extensively in the field of finance to develop mathematical models of investment decisions. Numerical analysis is also used for solving complex problems in physics, engineering, and the sciences.
