Introduction to Numerical Techniques/ Analysis

Introduction to Numerical Techniques/ Analysis

Introduction to Numerical Techniques/ Analysis – Numerical techniques are the study of algorithms that are used for numerical approximation for the problems of mathematical analysis. A numerical method is complete and defines a set of methods for the solution of a problem, together with measurable error estimates. The study and implementation of such methods is the field of numerical analysis.

There is a trick that lets you get nearer and nearer to an exact answer is “numerical methods”. Numerical methods find out the solutions close to the answer without ever knowing what that answer is. As such, an important part of every numerical method is proof that it works.

Numerical analysis discovers the application in all fields of engineering and the physical sciences like physics, astronomy, and geology, and now in 21st-century life, social sciences as well as medicine, and business. The current growth in computing power has allowed the use of more complex numerical methods and provides detailed and practically mathematical models in science and as well as engineering.

The field of numerical techniques predates the invention of modern computers by many centuries. Linear interpolation was already used even more than 2000 years ago. Several great mathematicians of the past were already occupied by numerical analysis, as it is clear from the names of important algorithms like Newton’s method, Lagrange interpolation polynomial, Gaussian elimination, or Euler’s method.

To speed up computations by hand, large books were built with formulas and tables of data such as interpolation points and function coefficients. By using these tables, and frequently calculating out to 16 decimal places or more some functions, one will look up values to plug into the formulas given and reach good numerical estimates of some functions. The field in canonical work is the NIST publication which is edited by Abramowitz and Stegun, a 1000-plus page book with a very large number of commonly used formulas and functions and their values at many points. The function values are no longer useful when a computer is available, but the large listing of formulas can still be very useful.

The mechanical calculator was also spread as a tool for hand computation. These calculators are involved in electronic computers in the 1940s, and it was found that these computers were very useful for administrative purposes. Basically, the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done.

Why is Numerical Techniques/ Analysis needed?

The analytics method may not exit. It means there may be no formula available in numerical methods.

– Data available does not admit applicability of the direct analytic method

-Analytic methods exist but there are quite time-consuming due to the huge data/complex functions involved.

EXAMPLE

Quadratic equation: ax²+bx+c=0

Here the quadratic equation with real or complex coefficients has two solutions, called roots. From this there are two solutions that may or may not be distinct, and they may or may not be real.

Quadratic equation: X²-5x-6=0

From the above equation, the root is x-2 and x=3.

Quadratic equation: 4x²+10x+34=0

From the above equation if we find the root then we put the quadratic formula:

Here, a=4, b=10,c-34 then put these values in the above equation.

There’s a solution by for 3rd-degree polynomials and another for 4th-degree polynomials but they are very hard. There never be a solution for the 5th or higher degree polynomials.

Applications of Numerical Techniques:

  • Basically used in computer science for root algorithm.
  • It is used to identify profit and loss in the company.
  • Used for Multidimensional root finding.
  • Solving practical technical problems using scientific and mathematical tools.
  • Network Simulation
  • Train and Traffic signal
  • Weather prediction
  • Build up an algorithm.

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