**Introduction to Numerical Techniques/ Analysis**

Introduction to Numerical Techniques/Analysis – Numerical techniques are the study of algorithms for numerical approximation for mathematical analysis problems. A comprehensive numerical method defines a set of approaches to problem-solving along with quantifiable estimates of error. The study and use of such methods are part of the field of numerical analysis.

A method is known as the “numerical method” enables you to approach an exact solution more and more closely. Numerical methods determine solutions that are close to the true answer without ever knowing the true answer. Therefore, a key element is providing proof that a numerical method works.

The numerical analysis finds use in the social sciences, as well as in business, medicine, and other facets of contemporary life, in addition to the physical sciences like physics, astronomy, and geology. The current increase in computing power has enabled the use of more sophisticated numerical techniques as well as the accessibility of extensive and practical mathematical models in science and engineering.

Numerical techniques existed for many centuries before the development of modern computers. Even more than 2000 years ago, linear interpolation was already in use. Numerical analysis was a field that attracted the attention of many great mathematicians in the past, as is evident from the names of crucial algorithms like Newton’s method, Lagrange interpolation polynomial, Gaussian elimination, or Euler’s method.

To facilitate calculations performed manually, large books with formulas and data tables, including interpolation points and function coefficients, were created. By using these tables and frequently computing some functions out to 16 decimal places or more, one can look up values to enter into the formulas and obtain precise numerical estimates of some functions. The field’s standard work is the NIST publication, edited by Abramowitz and Stegun. There are many frequently used formulas and functions in this more than 1,000-page book, along with their values at various points. The function values are no longer useful when a computer is available, but the extensive list of formulas can still be very useful.

The mechanical calculator became well-liked as a tool for performing calculations by hand. These calculators were incorporated into electronic computers in the 1940s, and it was found that these computers were extremely useful for administrative tasks. The advent of the computer essentially changed the field of numerical analysis because it made it possible to perform calculations that were longer and more complex.

**Why is Numerical Techniques/ Analysis needed?**

The analytics method might not terminate. It implies that there might not be a numerical method formula.

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

-There are analytical methods, but they take a lot of time because there is a lot of large data and complicated functions.

EXAMPLE

**Quadratic equation: ax²+bx+c=0**

Here the quadratic equation with real or complex coefficients has two solutions, called roots. This leads to two possible outcomes, both of which may or may not be distinct and 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 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 for root algorithms in computer science.
- It is used to determine the company’s profit and loss.
- used to find multidimensional roots.
- using scientific and mathematical methods to solve real-world technical issues.
- Simulation of a network
- both traffic and train signals
- weather forecast
- Create an algorithm.

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