ERRORS IN NUMERICAL COMPUTATIONS
PRESENTED BY
LT. SONIA MAHINDRU
ASSTT. PROF. IN COMPUTER SCIENCE
ERRORS
The numerical error is the difference between the exact solution and the approximate solution.
The numbers that arise in technical applications are better described as “exact numbers” because there is not the sort of uncertainty in their values. They are the result of counting discrete items. For example, one dozen eggs is exactly 12 eggs, never 12.1 eggs or 11.9 eggs
EXACT NUMBERS
APPROXIMATE NUMBERS
Approximate number is defined as a number approximated to the exact number and there is always a difference between the exact and approximate numbers. For example, are exact numbers as they do not need any approximation. But, , are approximate numbers as they cannot be expressed exactly by a finite digits. They can be written as etc. which are only approximations to the true values.
• Blunders (Gross Errors) These errors also called humans errors, and are caused by humans mistakes and oversight and can be minimised by taking care during scientific investigations. These errors will add to the total error of the underlying problem and can significantly affect the accuracy of solution.
• Modelling Errors These errors arise during the modelling process when scientists ignore effecting factors in the model to simplify the problem. Also, these errors known as formulation errors.
Sources of Error in Numerical Computations
• Discretisation Errors Computers represent a function of continuous variable by a number of discrete values. Also, scientists approximate and replace complex continuous problems by discrete ones and this results in discretisation errors.
TYPES OF ERRORS
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.
Some examples of model simplification, resulting in errors:
While calculating the force acting on a freely falling body, we may assume that the drag coefficient (air resistance) is linearly proportional to velocity of the falling body. This simplification will have its impact on the accuracy of the result.
While evaluating disease control programs, mainly epidemiological factors are included in the model, while others like social factors may be left out to make the model less complex.
Inherent Errors / Input Errors
Errors that are present in the data that are input to the model are inherent errors. They are also called input errors.
They are classified into two – Data Errors and Conversion Errors.
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. A voltage reading can be accurate only upto the accuracy of the voltmeter. Similarly, the accuracy of distance measurement is limited by the accuracy of the instrument used to measure distance.
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.
Conversion Errors
Conversion errors arise due to the limitation of computers to store exact decimal data. Hence, these are called representational errors. In a floating point representation, a computer can only retain a limited number of digits. Digits that are not retained causes a round-off error.
Numerical Errors
Errors can arise during the process of implementation of numerical method. Hence these are also called procedural errors. They are classified into two – Round-off errors and Truncation errors.
The total numerical error in a process can be calculated as the sum of round-off errors and truncation errors in the process. Considering these factors, suitable techniques can be employed during implementation of a numerical method to reduce the total numerical error.
Round off error
Round off errors occur because computers have limited capacity to store exact numbers. These errors can have cumulative effect in a numerical process. When an exact number is stored round-off error arises once. When repeated arithmetic operations are performed, round off error may occur in each operation and these errors add up. Even though the initial round off error is insignificant, after repeated arithmetic operations, the total round-off error may become significant due to cumulative effect.
Round-off errors can be categorised into two :-
Chopping
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 10.6872 will be stored as 10.68. Digits 7 and 2 will be dropped.
Symmetric Round Off
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 above example, 10.6872 will become 10.69 because 7 is greater than 5. If the original number was 10.6842, then it will be stored as 10.68.
Truncation Error
Truncation errors occur when an exact mathematical procedure is approximated. When a numerical process is truncated after a finite number of iterations for computational simplicity, truncation error arises.
This is often the case while an infinite series is to be added up to arrive at the exact result, but we are forced to truncate the process after attaining an acceptable level of accuracy due to practicality issues.
Calculation of Sin of a value, exponential function etc. are infinite series to be added up to arrive at the result. Because of computational limitations, we normally truncate the numerical process after certain number of terms, resulting in truncation error.
Human Errors
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 errors are :
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
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
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