Numerical Analysis By Shanker Rao

10 NUMERICAL ANALYSIS
First two significant
digits 2 3 4
55, …, 59 0.8 0.08 0.008
60, …, 69 0.7 0.07 0.007
70, …, 79 0.6 0.06 0.006
80, …, 99 0.5 0.05 0.005
1.6 GENERAL ERROR FORMULA
Let u be a function of several independent quantities x 1
, x 2
, …, x n
which are subject to errors of
magnitudes ∆x 1
, ..., ∆x n
respectively. If ∆u denotes the error in u then
b
= 1 2
u f x , x ,..., x n
b
g
u + ∆u = f x + ∆x , x + ∆x ,..., x + ∆x
.
1 1 2
Using Taylor’s theorem for a function of several variables and expanding the right hand side
we get
b g
∂
2
f ∂f
∂f
u + ∆u = f x1, x2, ..., xn
+ ∆x1 + ∆x2
+ ... + ∂xn
+
∂x
∂x
∂x
terms involving b∆x i g 2 , etc.,
n
n
n
g
1 2
∂f
∂f
∂f
u + ∆u = u + + ∆x1 ∆x2
+ L + ∆x n +
∂x
∂x
∂x
terms involving b∆x i g 2 , etc.
1
2
The errors ∆x 1 , ∆x 2 ,..., ∆x n , are very small quantities. Therefore, neglecting the squares and
higher powers of ∆x i , we can write
The relative error in u is
Formula (2) is called general error formula.
E
R
f f
x
∆u
≈ ∂ ∆x
+ ∂ ∆x
+ + ∂ 1
2 ... ∆xn. ∂x
∂x
∂x
(1)
1
F
HG
2
∆u
1 ∂u
∂u
∂u
= = ∆x1 + ∆x2
+ ... + ∆xn
. (2)
u u ∂x
∂x
∂x
1 2
n
n
n
I
KJ
F
HG
Q ∂ f
∂x
i
∂u
=
∂x
i
I
KJ
(i = 1, 2, ..., n)