Numerical Analysis By Shanker Rao

4 NUMERICAL ANALYSIS
F
HG
involves an analytical error.
2 3
1 − x x x
+ − 0
2 3KJ − x
! !
= ,
I
The magnitude of the error in the value of the function due to cutting (truncation) of its series
is equal to the sum of all the discarded terms. It may be large and may even exceed the sum of the
terms retained, thus making the calculated result meaningless.
(iii) Round-off errors. When depicting even rational numbers in decimal system or some other
positional system, there may be an infinity of digits to the right of the decimal point, and it may not
be possible for us to use an infinity of digits, in a computational problem. Therefore it is obvious
that we can only use a finite number of digits in our computations. This is the source of the socalled
rounding errors. Each of the FORTRAN Operations +, –, *, /, **, is subject to possible roundoff
error.
To denote the cumulative effect of round-off error in the computation of a solution to a given
computational problem, we use the computational error and the computational error can be made
arbitrarily small by carrying all the calculations to a sufficiently high degree of precision.
Definition 2 By the error of an approximate number we mean the difference between the exact
number X, and the given approximate number x.
It is denoted by E (or by ∆)
E= ∆ =X – x.
Note An exact number may be regarded as an approximate number with error zero.
Definition 3 The absolute error of an approximate number x is the absolute value of the difference
between the corresponding exact number X and the number x. It is denoted by E A
. Thus
E A = X − x
Definition 4 The limiting error of an approximate number denoted by ∆x is any number not less
than the absolute error of that number.
Note From the definition we have
E A = X − x ≤ ∆x.
Therefore X lies within the range
x – ∆x ≤ X ≤ x + ∆x
Thus we can write X=x±∆x
Definition 5 The relative error of an approximate number x is the ratio of the absolute error of
the number to the absolute value of the corresponding exact number X, where bX ≠ 0g. It is denoted
by E R
(or by δ)
E
R
E A
= δ = .
X