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Round off error/ Truncation error

Introduction

We define error as a quantity which satisfy the identity
True value = Approximate value + error
XT = XA + error

If error in approximation is considerably small then we say that XA is good approximation to X.
Error can be positive or negative. We are interested in absolute value of error.
|error| = |XT - XA|
|error| = |True value - Approximate value|

When the true value is very large or very small, we prefer to study the error by comparing it with the true value. This is known as relative error. We define this error as,
Relative error = |(True Value - Approximate Value)/(True Value)|

If the true value is not available. We replace the true value by the computed approximate value in the definition of the relative error.

In the numerical calculations, we encounter mainly two types of errors
  1. Round-off error
  2. Truncation error

Round-off Error

Sometimes we indicate number with decimal digits followed by dots. The line of dots indicates that the digits continue and we are not able to write all of them. That is, these numbers cannot be represented exactly by a terminating decimal expansion. Whenever we use such numbers in calculations we have to decide how many digits we are going to take into account.

Therefore in general whenever we want to use only a certain number of digits after the decimal point. Then it is always better to use the value rounded-off to that many digits because in this case the error is usually small. The error involved in a process where we use rounding off method is called round-off error.

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