When we want to formulate a problem and want to implement it on computer then various errors occur at various stages. The various types and sources of errors are discussed in the following article.
Numerical analysis deals with developing methods, called numerical methods, to approximate a solution of a given Mathematical problem.
The approximate solution obtained by this method will involve an error which is precisely the difference between the exact solution and the approximate solution.
Error = Exact value – approximate value
Inherent Errors
These types of error occur due to simplified assumptions made during mathematical problem. These also occur when the data is obtained from certain physical measurement of the parameters of the proposed problem.
There are three types of inherent errors.
- Modelling error
- Data error
- Blunder
The sources and causes of above errors are explained below.
Modelling Errors
A mathematical model is built to represent a physical process or a phenomenon. When a mathematical model being formulated is not exact/accurate when compared to the underlying physical process, errors can occur in the resulting solution.
Models often require many simplifying assumptions. The physical process being modelled may be overly complex in some cases.
In such situations, it will be impossible/impractical to create an exact mathematical model. Hence, the resultant model may be a simplified version of the underlying physical process.
Example:
While calculating the force acting on a freely falling body, we may assume that the air resistance is linearly proportional to velocity of the falling body. This simplification will have its impact on the accuracy of the result.
To reduce this error, one can enhance the model, thus adding to the model complexity. Increase in the model complexity will make the model more difficult to solve and the numerical process to consume more computing resources.
On the other hand, oversimplifying the model will make the numeric process simpler but will produce results with unacceptable levels of accuracy.
It is therefore needed to strike a balance between the level of accuracy required for the results and the complexity level of the model. A model must be enhanced only upto a level of complexity that reduces the error to an acceptable level.
Data Errors
Data errors arise when data to be input into a model are acquired using experimental methods. These are also called empirical errors.
Such errors occur mostly due to the limitations or errors in the instrumentation.
Example:
If the temperature of the room is to be recorded after every 10 min but electronic thermometer gives reading correct only for an hour then after an hour data recorded will be wrong.
Hence, to reduce such errors, it is more important to improve the accuracy of the data being read than improving the precision of arithmetic operations.
Human Errors or Blunders
These are errors introduced due to human imperfections or mistakes. Human errors can occur at any stage of the problem-solving cycle.
Some common types of human errors are:
- Mistakes in Data input such as misprints, giving values column-wise instead of a row wise to a matrix, forgetting a negative sign, etc.
- Wrong guess of initial values.
- Lack of understanding of the problem (physical system).
- Overlooking of some basic assumptions required for formulating the model or making wrong assumptions (Modelling Error).
- Errors in deriving the mathematical equation or using a model that does not describe adequately the physical system under study (Modelling error).
- Selecting a wrong numerical method for solving a mathematical model.
- Selecting a wrong algorithm for implementing a numerical method.
- Programming mistakes.
All these mistakes can be avoided through a reasonable understanding of the problem and the numerical solution method, use of good programming techniques and tools, effective code reviews and testing.
Types and Sources of Errors in Numerical Analysis
Following diagram represents the types and sources of errors in numerical analysis or numerical methods.
When we want to implement any engineering problem in computer at that time already inherent errors may exist in the problem. Various reasons are explained in the above section.
Due to use of computer in numerical methods again some errors occur. They are called “Numerical Errors”.
Numerical Errors
Numerical errors can arise during the process of implementation of numerical method. Hence these are also called procedural errors.
The study of numerical methods is incomplete if we don’t develop algorithms and implement the algorithms as computer codes.
The outcome of the computer code is a set of numerical values to the approximate solution obtained using a numerical method.
Such a set of numerical values is called the numerical solution to the given Mathematical problem.
During the process of computation, the computer introduces new errors, called
- Round-off error
- Truncation error.
Round-off Errors
Round-off error occurs because of the computing device’s inability to deal with certain numbers.
1/3 = 0.3333333…
1/6 = 0.166666…
π(pi) = 3.141592653589793238…
Such numbers need to be rounded off to some near approximation which is dependent on the word size used to represent numbers of the device.
When the number of bits required for representing a number are less than the number is usually rounded to fit the available number of bits. This is done either by chopping or by symmetric rounding.
- Chopping error
- Symmetric Round-off Error
Chopping Error
In chopping error, digits that are beyond the storage capacity of the computer are dropped.
If the word length of the computer is 4 digits, then a number like 32.5472 will be stored as 32.54. Digits 7 and 2 will be dropped.
Now to evaluate the error due to chopping let us consider the normalized representation of the given number i.e., 32.5472 – 32.54 = 0.0072
Symmetric Round-off Error
In a symmetric round-off, the last retained significant digit is rounded by 1, if the first digit being discarded is greater than or equal to 5. If it is less than 5, the last retained digit is unchanged.
In the previous example, 32.5472 will become 32.55 because 7 is greater than 5.
If the original number was 32.5432, then it will be stored as 32.54.
Symmetric round-off error illustration
Number A | Number B | Subtraction C=A-B | D= C *1000 | D+248.3724 | |
Exact value | 1.7612 | 1.4565 | 0.3047 | 304.7 | 553.0724 |
Rounded Value | 1.76 | 1.46 | 0.3 | 300 | 548.37 |
Error | 0.0012 | 0.0035 | 0.0047 | 4.7 | 4.7024 |
Truncation Errors
In a numerical computation round-off errors are introduced at every stage of computation. Hence though an individual round-off error due to a given number at a given numerical step may be small but the cumulative effect can be significant.
The word ‘Truncate’ means ‘to shorten’.
Truncation error refers to an error in a method, which occurs because some number/series of steps (finite or infinite) is truncated (shortened) to a fewer number. Such errors are essentially algorithmic errors and we can predict the extent of the error that will occur in the method.
For example, if we approximate the exponential function by the first four non-zero term of its Maclaurin series (special case of Taylor series) as
Truncation Error
The resulting error is a truncation error. It is present even with infinite-precision arithmetic, because it is caused by truncation of the infinite Taylor series to form the algorithm.
Causes of Truncation Error
- Limiting infinite number of terms in a series to finite number of terms.
- Taking finite step size instead of infinitesimal step size (Example: Numerical integration and numerical differentiation).
- Limiting infinite number of iterations to a limited number.
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