Why I love Numerical Analysis

Importance of Numerical Analysis

Here I am going to write about something I love, and that is ‘Numerical Analysis’. It is a branch of applied mathematics that uses various methods (algorithms) to find approximate solutions of complex mathematical problems. In modern science and engineering- there are many problems where the equations are so complex that analytical or exact solutions are not possible. Truly speaking, such real-world problems can rarely be expressed in completely perfect mathematical formulas. When such situation occurs, or when an analytical solution is very difficult to obtain, numerical analysis is then the only hope. Numerical methods fill that specific gap and provide solutions with a desired accuracy. From calculating square roots on a calculator to modelling climate change and planning spacecraft trajectory, these methods are essential. The greatest strength of these methods is their universality- which means, the same method can be applied to all fields.

Therefore, numerical methods are not just for mathematicians or engineers- they are increasingly important in other fields too (for example, economics, finance, environmental science, and even in sociology). If we think about how pandemics spread, how bacteria evolve, how the planets move, how airplane wing’s function, how climate change impacts, then we will have to deal with Dynamical Systems (another big branch of mathematics). To solve those complex systems, what we need is nothing but numerical methods, which connect theories, computations and applications.

From an educational perspective, a numerical analysis course develops logic and problem-solving skills in students. Some topics that we should learn very carefully are root finding methods, interpolation, least squares, systems of equations, numerical differentiation and integration, ODEs (IVP & BVP), PDEs, approximation theory, eigenvalue problems, optimization methods, and continuation methods. While learning these methods or algorithms, we can use any programming language such as MATLAB, Python, Fortran etc. to write the associated codes. There are also some advanced and specialized software tools, which we can use while dealing with computationally extensive projects.

Below is an animation of the motion of a double pendulum, which is described by a set of ODEs. To find the solution of this system, I have applied numerical techniques. This is just an example how numerical methods can play roles in solving real-world problems, and that's the main reason why I love Numerical Analysis.

(I always describe this animation with FUN 😊 by saying: although red and blue started their journey together, they got separated at some point and their lives became very very chaotic at the end 😗)

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(written by Md. Azmir Ibne Islam)

---- See some more Dynamical Systems related Animations & Maths, where numerical techniques have been used.