Lecture No 2: Error Analysis

Numerical methods provide estimates that are very close to an exact analytical solution, obviously an error is introduced into the computation therefore it is important to understand the errors. Numerical computation can also be defined as a vehicle to study errors in computations.

             

 Error

Error is the difference between the actual value and the computed value. Let x be the actual value of a certain problem and  be its approximate value then mathematically error can be written as

                

Let x=2 be the actual value and be its approximate solution then an error is 0.01

 

Three Important steps in Error Analysis

 

It is the study and evaluation of errors. An error analysis generally consists of the following three steps:

1. ·         To identify the errors, secondly
   
2. ·         To quantify the errors, and then the lastly
   
3. ·         To minimize as per our need.

 

SOURCES OF ERROR:

A numerical method for solving a given problem will, in general, involve an error of one or several types. Although different sources initiate the error, they all cause the same effect: diversion from the exact answer. Some errors are small and may be neglected, while others may be devastating if overlooked. In all cases, error analysis must accompany the computational scheme, whenever possible.

The main sources of error are as follows:

 Gross errors, Rounding errors, and Truncation errors.

1. Gross Errors

Gross errors are either caused by human mistakes or by the computer. A few examples of these errors are as follows:

i.             Misreading or misquoting the figures, particularly in the interchanges of adjacent. digits,

ii.            Use of inaccurate mathematical formula (algorithm) to solve a particular problem, and

iii.           Use of inaccurate data.

Although gross errors are not directly concerned with most of the numerical methods discussed, they can sometimes have a great impact on the success of modeling efforts. Thus, they always are kept in mind when applying numerical techniques in the context of real-world problems. These errors are not very serious and can be avoided if enough care is taken in using proper numerical analysis techniques. We shall primarily concern ourselves with the latter types of errors.

2. Rounding Errors

When a numerical method is actually run on a digital computer after transcription to a computer program form a kind of error called round-off error is introduced.

The error introduced by rounding-off numbers to a limited number of decimal places is called the rounding error. For example, it would be impracticable to mention the distance between two points on the earth as 15.2967 meters. It would be more reasonable if it were to be rounded to the nearest whole number, i.e. 15 meters, Thus, the error introduced by rounding is 0.2967 meters. Another example is the value of n =3.1415926353 which may be meaningfully rounded-off to 3. 1416 or 3.142.

Rounding-off errors play an important role in numerical analysis. In order to obtain a

smaller error as a result of rounding-off, we may apply the following rules when performing

manual calculations (these rules are not normally applied when performing extensive computer calculations).

3. Truncation error:

It is defined as the replacement of one series with another with fewer terms.

Example:

• ·    Binomial expression.
• ·         Infinite geometric series, 
•       Taylor’s or McLaren’s series.