Numerical methods are capable of handling large systems of equations, non-linearities, and complicated geometrics. The main aim of numerical methods is to provide constructive methods for obtaining answers to such problems in a numerical form with the following characteristics:
- Numerical stability
- Rate of convergence
The numerical method plays a very important role in solving real-life mathematical, physical, and engineering problems. In the field of science and engineering complex, mathematical problems are solved using methods and algorithms of numerical methods. This has helped both the field to tackle the complex numerical in a more relevant and accurate way to obtain the desired solution.
Let’s come into the actual topic, about errors in numerical methods.
Deviations from exact or true values are called
errors. They are an integral part of any numerical calculations.
An error represents inaccuracy and imprecision(not exact) of a numerical calculation or computation.
In this blog, we will discuss three main types of errors along with their sub-types:
- Inherent error(Input error)
- Numerical error(Procedural error)
Types of Error in Numerical Methods
1. Inherent error
Inherent error is an error found in a program that causes it to fail regardless of what the user does and is commonly unavoidable. This error requires the programmer to modify the code to correct the issue. This is usually caused by errors in data before processing.
It is further divided into two types: Data error and Conversion error.
a) Data error
Data error arises when data for a problem are obtained by some experimental means and therefore are of limited accuracy.
b) Conversion error
Conversion error arises due to the limitation of computers that store data.
2. Numerical error
Numerical errors are introduced during the process of implementation of the numerical method. It also has two types: Round-off error and Truncation error.
a) Round-off error
Round-off error arises when a fixed number of digits are used to represent the exact number. Such numbers need to be rounded off to some near approximation which is dependent on the word size used to represent the numbers of the device.
b) Truncation error
Truncation error is raised by using approximate results or by replacing an infinite process with a finite one. It is the type of algorithm error i.e. procedural error which we can predict the extent of the error that occurs in the method.
Further, have knowledge of these errors:
Absolute, Relative, and Percentage errors
Absolute error is the magnitude of the difference between the true value of a quantity(Xt) and the approximate value(Xa).
Absolute error(Ea) = |Xt – Xa|
The relative error is the ratio of the magnitude of the absolute error to the true value of a quantity(Xt).
Relative error(Er) = |(Xt – Xa) / Xt |
When the relative error is expressed in percentage, it is called the percentage error.
Percentage error(Ep) = |(Xt – Xa) / Xt | * 100%
Numerical methods are extremely powerful problem-solving tools that can be used to generate reliable solutions to mathematical problems. While calculating those big problems, there could be chances of an error occurring. So we should be aware of different types of errors in numerical methods.
If you have any queries regarding the topic, don’t forget to leave a comment below. Thank you 🙂
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