Numerical Analysis By Shanker Rao

6 NUMERICAL ANALYSIS
X
1 − +
− x = 003 . < (
2 10 ) 1 3 1 ,
b g
i.e., E A = 003 . 1
<
2 01 . .
Note
Theorem
1. All the indicated significant digits in mathematical tables are correct.
2. Sometimes it may be convenient to say that the number x is the approximation to an exact number
X to n correct digits. In the broad sense this means that the absolute error E A
does not exceed
one unit in the nth significant digit of the approximate number.
If a positive number x has n correct digits in the narrow sense, the relative error E R
of
this number does not exceed
E
R
Proof
1 1
≤
α 10
m
n
F H G I K J − 1
Let
F
n 1
1 I
HG
10 K J −
divided by the first significant digit of the given number or
, where α m
is first significant digit of number x.
x= m
m
( where α ≥1)
m
m–
1
m–
1
m–n+
1
m–n+ 1
α 10 + α 10 + ... + α 10 + ...,
denote an approximate value of the exact number X and let it be correct to n digits.
Then by definition we have
E = X − x ≤
A
1
bg ,
m− n+
2 10 1
1
Therefore X ≤ x – ( ) .
m− n+
2 10 1
If x is replaced by a definitely smaller number α m
10 m we get
X ≥ α 10
m
1
−
2 10 1 ,
m m− n+
F
HG
1 m
⇒ X ≥ −
n−
2 10 2 1
α m ,
10 1
bgb
1 m
∴ X ≥ m −
2 10 2α 1 .
g
I K J