Introduction to Numerical Methods

In introduction to numerical methods, the definition, need and applications of Numerical methods are explained. The word “Numerical Method” is synonymous to “Numerical Analysis”, “Computational Methods”.

What is Numerical Method?

  • In reality “Solving a math problem” generally involves findingananswer rather thanexactanswer.
  • Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis.
  • A numerical method is a complete and definite set of procedures for the solution of a problem, together with computable error estimates. The study and implementation of such methods is the field of numerical analysis/ numerical methods.
  • A trick that lets you get closer and closer to an exact answer is a “numerical method”.
  • Numerical methods find solutionscloseto the answer without ever knowing what that answer is. As such, an important part of every numerical method is a proof that it works.

Need of Numerical Methods

Numerical analysis is necessary in following conditions.

  • Analytic method may not exist.
  • Data available does not admit applicability of direct analytic method.
  • Analytic method exists but are quite time consuming due to huge data/complex functions involved.

All of the above needs are explained in details as follows.

Analytic Method May Not Exist

To solve any mathematics problem generally two approaches are used. The first is, analytical method and second is numerical method. Generally, analytical approach is preferred because they are intuitive, simple, and based on how activists approach everyday problems.

The Analytical Method is a generic process combining the power of the Scientific Method with the use of formal process to solve any type of problem. It has following key steps:

  1. Identify the problem to solve.
  2. Choose an appropriate process.
  3. Implement the solution.

Let’s consider an example of Quadratic equation of type ax2+bx+c=0

A quadratic equation with real or complex coefficients has two solutions, calledroots. These two solutions may or may not be distinct, and they may or may not be real.

Now, try to find the roots of following equation given by

x2-5x+6=0

It is very easy to find the roots. You may have found the roots as x= 3 and x=2. Now, again try to find the roots for following equation.

4x2+10x+34=0

To find the roots of such quadratic equations following Quadratic formula is used.

{x_{1,2}} = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}

Where, a=4, b=10 and c=34.

There is a solution by for 3rd degree polynomials and another for 4th degree polynomials but they are very hard. However, there can never be a solution by for 5th or higher degree polynomials.

5x5+3x4-4x3+45x2+10x-20=0

Also, analytical methods are not available to find roots for transcendental equations such as

x2+ cos x + 10 = 0

2x – log 7 + 10 = 0

Data Available Does Not Admit Applicability of Direct Analytic Method

Sometimes the direct analytical method for the solution of a problem does not available or analytical method can not be applied in such cases. The graphical method may exist but again accuracy may be less.

introduction to numerical methods

If in above graph you want to find the value which lies between any two points in the data then this graph can be a solution but plotting graph for every data and find the solution is not convenient especially when a huge amount of data is available.

Suppose we have the population of a city for year 1991, 1995, 1999, 2006 and 2011. Now suppose if we need the population figure for example 1996 and 2008, it would be impossible to conduct census for these years. The only alternative is to make use of numerical methods.

Analytic Method Exists but are Quite Time Consuming

Let’s take the example of 4th degree polynomials (Quartic equation). The equation is having the form

ax4 + bx3 + cx2 + dx + e = 0

The solution of this type of equation can be given by using following formula.

Introduction to numerical methods

This formula is not easy to remember because a lot of complex terms are involved in it. Also, its solution is time consuming.

Sometimes huge data is available for a system and an analytical method can also be applied to the data for solution. But due to huge data the solution for such problems is time consuming.

Numerical method can provide quick solution for such problems. Computer programing make this task easier.

Applications

  • Used for Multidimensional root finding.
  • Widely used in statistics for prediction of growth, production etc.
  • Usually used in computer science for root algorithm.
  • It is used to determine profit and loss in the company.
  • The formal academic area of numerical analysis ranges from quite theoretical mathematical studies to computer science issues.
  • To find solution of problems in the field of natural sciences, social sciences, engineering, medicine, and business.
  • Subtle numerical analysis is required to implement detailed models of the world. For example, ordinary differential equations appear in celestial mechanics (predicting the motions of planets, stars and galaxies); numerical linear algebra is important for data analysis; in simulating living cells for medicine and biology.
  • Network Simulation
  • Weather prediction

In other words, Numerical methods are used almost in every field.

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