Numerical analysis
Numerical Analysis is a branch of applied mathematics that involves designing methods that solve complex problems using accurate numerical approximation. The goal of numerical analysis is to determine if the inaccuracy at every stage is within permissible boundaries, not merely the numerical output of such a procedure.
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What is Numerical Analysis?
Numerical analysis studies algorithms that employ numerical approximation rather than symbolic operations to solve mathematical analysis issues as distinguished from discrete mathematics. Numerical methods are tools designed under Numerical Analysis for approximating solutions. The analysis entails developing techniques that provide approximate, precise numeric answers, which are essential when calculating the accurate solutions is impractical or highly expensive. The numerical analysis also involves classifying these approaches' convergence, precision, steadiness, and computing issues. Numerical analysis is used in all sectors of engineering and physical sciences, as well as in life and social sciences, health, commerce, and even the arts.
History
Modern numerical analysis may be traced back to the 1947 publication "Numerical Inverting of Matrices of High Order" by John von Neumann and Herman Goldstine. Even though numerical analysis has a more extensive background, "contemporary" numerical analysis, as defined above, is distinguished by the confluence of the programmed electronic computer, mathematical interpretation, and the possibility and necessity to answer vast complex questions and tasks in situations.
Examples of Numerical Methods
Newton-Raphson method
Some computations are impossible to solve using algebra or other computational formulas. Numerical approaches are required for this. One such approach is Newton-Raphson's method, which enables us to determine the solution using linear approximation.
Simpson's Rule
Simpson's Rule is a mathematical formula for approximating the numerical value of integral of a function between two intervals, by using the concept of area under the function curve for estimation.
Trapezoidal Rule
Trapezoidal Rule is another numerical method for determining the numerical value of a direct integral.
Uses and Applications
- Numerical Methods are used to solve complex mathematical problems that come across in all fields of engineering.
- Various numerical methods can be used to solve equations with variables, find minimum or maximum of a function, estimate the value of a definite integral, and solve differential equations; which are all employed to devise solutions for complex mathematical problems in engineering applications.
- Advances in computing has enabled complex numerical analysis to be employed for preparing models with high accuracy and detail in science and engineering.