Numerical Analysis

Numbers play an important role in solving different problems. Whenever we want to make a rational decision in the stock market or something that is very unpredictable, we try to analyze the past numbers to get a clear picture of stocks and the variables. This analysis is possible because there is a specialized branch of mathematics that is dedicated to the analysis of these numerical methods and algorithms. This branch is called numerical analysis. 

Numerical analysis comes into the picture in almost every field where there is some sort of prediction about the future. Understanding this branch, therefore, becomes important to make these algorithms more accurate. So in this article, we will understand in detail numerical computation and different types of numerical methods. 

Introduction to numerical analysis: 

Numerical Analysis refers to the process of in-depth analysis of algorithms and natural approximations. It is considered a part of both computer science and mathematical sciences. It is used in several disciplines like engineering, medicine, social science, etc. Because this branch is concerned with understanding, creating, and implementing the algorithms for common problems. 

Some commonly used numerical computation methods include differential equations (that help predict planetary motion), linear algebra, etc. 

Applications of Numerical Analysis: 

1. The major benefit of numerical analysis is that the scientific models which were earlier difficult to make became comparatively easier.   
2. It helps us in coming up with formulas and solutions that can give us near to accurate answers for various problems. 
3. Numerical weather predictions have become easier with advanced numerical analysis. 
4. We can also better understand how human activities are influencing climatic conditions with the help of numerical computation because several variables will get easily adjusted in numerical equations in the form of alphabets. 
5. Through numerical analysis, scientists also try to calculate spacecraft trajectories. 
6. Insurance companies and hedge fund companies (that have to deal with future pricing) also use this information for analyzing information about participants. 
7. Use by airlines: Airline companies use numerical analysis for understanding fuel needs, ticket pricing, and other such mathematical decisions. 
8. Even famous car companies rely on numerical analysis for making their car safer by checking crash possibilities.  
9. In Engineering based on computer science, several numerical analyses are used. Mixed systems in which two different kinds of mechanical systems are used require numerical analysis. 
10. Even in the production and manufacturing sector, numerical analysis is useful for deciding the apt location, the amount of inventory, and other such decisions. 

Types of numerical methods: 

The numerical analysis takes the help of geometry, algebra, calculus, and other mathematical tools in which the variable is not constant. Following are types of numerical methods: 

1. Linear and nonlinear equations: Linear equations which are usually in the form of px=q are found in almost all areas. We use it to calculate demand and supply, time and speed, and much more. Similarly, nonlinear equations in which our sole goal is to find roots of x are also helpful in several fields like aeronautics. 

2. Integration and differentiation: Almost all of us have solved mathematical problems involving integration and differentiation. These two equations also form an important part of numerical analysis. These are further divided into partial differential equations and ordinary differential equations. 

3. Approximation theory (Use of nearby problems): 

Approximations are also used widely for computing numerical functions. The method is based on simple mathematical operators (addition, subtraction, multiplication, and division) but helps solve complex polynomial problems. 

One of the widely used methods for numerical computation through approximations is the use of nearby problems. It is used when we face a situation where direct solvation is difficult. So, we use a nearby problem in place of that situation and then solve it. This makes the problem easy to solve. 

An example of this method is the interpolation technique which is used to replace difficult numerical integration situations. 

4. Vectors and functional analysis: Sometimes vectors and functional analysis are also used for computing Numerical problems.

Besides these simple methods, certain complex theorems and mathematical rules are also used for Numerical analysis. Examples include the Euler method, Taylor Series method, Shooting method, RK-2 and RK-4 method, trapezoidal rule, etc. Out of all these methods, the finite element approach is used for computer automated calculation and analysis. The method is simple. All the differential equations are first converted into simple equations and then the computer can easily solve them. 

Conclusion:

In this article, we looked at one of the most important branches of mathematics used for approximations and calculations, called Numerical Analysis. From the introduction of Numerical Analysis to its application, we understood how it plays a major role in scientific, engineering, and other disciplines. We also looked at different types of Numerical computation methods like linear equations, Integration, approximations, etc. So you can use them according to the situation and calculate the possibilities of a real-life problem.