# Articles

## “Numerical Method”-in Science & Technology

By Sanjenbam Jugeshwor Singh

Numerical algorithms are at least as old as the Egyptian RHIND PAPYRUSC (1650 BC), which describes a root- finding method for solving a simple equation. Ancient Greek mathematicians made many further advancement in numerical methods. In particular, EUDOXUS of Cnidus (400-350 BC) created and Archimedes (285-212/211 BC) perfected the method of exhaustion for calculating lengths, areas, volumes of geometrical figures. When used as a method to find approximations, it is in much the spirit of modern numerical integration; and it was an important precursor to the development of Calculus by Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716).

Calculus, in particular, led to accurate mathematical models for physical reality, first in physical Science and eventually in the other sciences, Engineering, medicine and business. These mathematical models are usually too complicated to be solved explicitly and effort to obtain approximate, but highly useful solutions gave a major impetus to numerical Analysis. Another important aspect of the development of numerical methods was the creation of Logarithms by Scottish mathematician John Napier (around 1614) and others. Logarithms replaced tedious multiplication and division (often involving many digits of accuracy) with simple addition and subtraction after converting to the original values to their corresponding logarithms through special tables. Mechanization of this process spurred the English inventor Charles Babbage (1791-1871) to build the first computer. Newton created a number of numerical methods for solving a variety of problems and his name still attached to many generalization of his original ideas. Of particular note is his work on finding roots(solutions) for general functions and finding a polynomial equation that best fit a set of data( polynomial interpolation).Following Newton ,many of the mathematical giants of 18th and 19th centuries made major contributions to numerical analysis. Foremost among these were the Swiss Leonhard Euler (1707-1783).the French Joseph-Louis Lagrange (1736-1813) and the German Carl Friedrich Gauss (1777-1855).

Now a days, numerical method is a popular subject and the basis of all branches of Science & Technologies. This is because, real life problems can be solved using computational techniques based on numerical methods. The subject matter of numerical method is often described as that part of mathematics which is constructive. In fact, the study of numerical methods involves something more than the mere use of the constructive result of mathematics for computation. The ability to differentiate the abstract numbers of the mathematics and the numbers which a computer has to work with adds to the usefulness of numerical methods. For better utilization of the computing power of a computing tool, we need better computing techniques for approximations and numerical analysis.

The mathematical value of a constant (say,π) is unique, though there may be many approximate values. The problem of choosing the right approximate value is easy: we should choose that value which is in some sense closest to the true value. But how do we ensure this? This is the first problem of a numerical analyst, who live and work in the world of finite precision and approximate numbers. Numerical method, clearly describes the scope of the work of a numerical analyst. This world is made more complicated by another kind of exact mathematical result which are constructive and hence come within the purview of numerical methods. We cannot expect the rules of mathematics to be valid while working with numerical methods and hence this subject becomes different from mathematics, requiring new enquiries and now insights into new problems that arises. The area is challenging because we tend to think that we were working according to the rules of mathematics and so our answers and results must be correct, while in reality at almost every step of numerical work, we do deviate from established mathematical rules. The problem is that in spite of such deviations, we have to ensure a result which in some sense, is close to true value. This is achieved using numerical analysis which examines the numerical method to find strategies and measures to ensure an acceptable approximation to the true solution of the problem.

The subject of numerical method is not the blind uncritical use of the constructive methods of mathematics. It is to examine these methods when implemented with finite precision arithmetic and then use them cautiously with the added information gained by such analysis .Available methods should also be examined with reference to their applicability to a given problem. Because the methods were developed by mathematicians with infinite precision in view, their uncritical use with finite precision may produce wrong result.

Following the development by Newton of his basic laws of physics, many mathematician and physicists applied these laws to obtain mathematical model for solid and fluid mechanics. Civil and mechanical engineers still base their models on this work and numerical analysis is one of their basic tools .In 19th century ,phenomena involving heat, electricity and magnetism were successfully modeled and in the 20th century ,relativistic mechanics, quantum mechanics and other theoretical constructs were created to extend and improve the applicability of earlier ideas. Numerical analysis is widely applied in CAE (Computer aided Engineering), atmospheric modeling to improve weather forecasts, in the development of computer software & optimization technique for modern businesses etc. So without numerical methods or numerical analysis the fast growing technology with high precision would not have been possible. Therefore Engineering & science students even business tycoon need to know this subject precisely.

(Writer can be reached to: [email protected])

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