Numerical Analysis –MTH603 VU
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Errors in Computations
Numerically, computed solutions are subject to certain errors. It may be fruitful to
identify the error sources and their growth while classifying the errors in numerical
computation. These are
Inherent errors,
Local round-off errors
Local truncation errors
Inherent errors
It is that quantity of error which is present in the statement of the problem itself, before
finding its solution. It arises due to the simplified assumptions made in the mathematical
modeling of a problem. It can also arise when the data is obtained from certain physical
measurements of the parameters of the problem.
Local round-off errors
Every computer has a finite word length and therefore it is possible to store only a fixed
number of digits of a given input number. Since computers store information in binary
form, storing an exact decimal number in its binary form into the computer memory gives
an error. This error is computer dependent.
At the end of computation of a particular problem, the final results in the computer,
which is obviously in binary form, should be converted into decimal form-a form
understandable to the user-before their print out. Therefore, an additional error is
committed at this stage too.
This error is called local round-off error.
10 2
(0.7625) (0.110000110011)
=
If a particular computer system has a word length of 12 bits only, then the decimal
number 0.7625 is stored in the computer memory in binary form as 0.110000110011.
However, it is equivalent to 0.76245.
Thus, in storing the number 0.7625, we have committed an error equal to 0.00005, which
is the round-off error; inherent with the computer system considered.
Thus, we define the error as
Error = True value – Computed value
Absolute error, denoted by |Error|,
While, the relative error is defined as
Relative error Error
True value
=
Local truncation error
It is generally easier to expand a function into a power series using Taylor series
expansion and evaluate it by retaining the first few terms. For example, we may
approximate the function f (x) = cos x by the series
24 2
cos 1 ( 1)
2! 4! (2 )!
n
n
xx x
xn
=− + − +− +""
If we use only the first three terms to compute cos x for a given x, we get an approximate
answer. Here, the error is due to truncating the series. Suppose, we retain the first n
terms, the truncation error (TE) is given by
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