# What is Numerical Methods

##### Numerical Methods: Understanding the Basics

Numerical methods refer to the various algorithms and techniques used in computing that approximate mathematical models or solve mathematical problems. They play a crucial role in scientific and engineering applications that involve complex computations, large volumes of data, and simulation of physical processes that are difficult or impossible to solve analytically. Numerical methods are used in diverse fields such as physics, biology, finance, and computer science to address problems that require numerical solutions.

In this article, we will take a closer look at the fundamental concepts of numerical methods and some of the most commonly used techniques.

###### Why Use Numerical Methods?

Unlike analytical methods, which rely on solving mathematical equations using algebraic or calculus techniques, numerical methods use algorithms to approximate solutions to mathematical problems. Analytical solutions are usually only possible for simple or idealized systems, while numerical methods can be used for more complex and realistic models. In addition, numerical methods are often much faster than analytical methods and can handle larger amounts of data and more complex models.

###### Numerical Techniques

There are dozens of numerical techniques, each with its own approach to solving mathematical problems. In this section, we will discuss some of the most commonly used numerical techniques in scientific and engineering applications.

• Root Finding: Root finding is a technique used to find the roots (or zeros) of a function. The roots of a function are the values of the independent variable that make the function equal to zero. One of the most well-known root-finding techniques is the bisection method, which involves iteratively cutting an interval in half until a root is found. Other methods include Newton-Raphson method and Brent's method.
• Interpolation: Interpolation is a method used to estimate the value of a function based on known data points. Interpolation is often used to find intermediate values of a function that are not explicitly known. Some of the most commonly used interpolation methods include linear interpolation, polynomial interpolation, and spline interpolation.
• Integration: Integration is the process of finding the area under a curve. Numerical integration methods are used when an analytical method of integration is not feasible. Some of the most commonly used numerical integration methods include the trapezoidal rule, Simpson's rule, and Gaussian quadrature.
• Differential Equations: Differential equations are equations that describe how a system changes over time. Numerical methods are used to solve differential equations when analytical solutions are not feasible. Some of the most commonly used numerical methods for differential equations include Euler's method, Runge-Kutta methods, and finite element methods.
• Optimization: Optimization methods are used to find the minimum or maximum value of a function. This is often useful in engineering and financial applications where a certain objective must be achieved. Some of the most commonly used optimization methods include Newton's method, gradient descent, and simulated annealing.
###### Accuracy and Error

When using numerical methods, it is important to consider the accuracy and error of the solutions. Numerical methods always involve some degree of error due to the fact that they are approximating and not exact solutions. There are two main sources of error in numerical methods: rounding error and truncation error.

• Rounding Error: Rounding error occurs when numbers are rounded off during computations. This error can accumulate over many computations, resulting in a significant loss of accuracy.
• Truncation Error: Truncation error occurs when an approximation is used instead of the exact solution. This error can be reduced by using more accurate numerical methods or by decreasing the step size in the approximation.

It is also important to consider the order of accuracy of a numerical method. The order of accuracy refers to the rate at which the error decreases as the step size is decreased. Higher order methods are generally more accurate but can also be more computationally demanding.

###### Conclusion

Numerical methods are essential in scientific and engineering applications that require complex computations and simulations of physical processes. Understanding the basics of numerical techniques and the sources of error is important in utilizing numerical methods correctly and effectively. With the advancements in computing power, numerical methods will continue to play a significant role in solving complex problems in a wide range of applications.