Numerical Analysis
Grinshpan
AN EXAMPLE ON CHOPPING AND ROUNDING
This example highlights a possible effect of rounding on approximation.
It is based on the difference between round-to-zero and round-to-nearest modes.
Consider a five-digit binary computer . in round-to-zero (chopping) mode.
The representation of
√
x = 2 = (1.0110101000001001111...)2
on this device is
x̃ = (1.0110)2 .
Note that x̃ = 1.375. The error of this representation,
err(x̃) = (0.00001010000100111...)2 ,
obviously exceeds 2−5 .
Out of two numbers, a = 1.4375 and b = 1.4142, approximating
smaller error:
√
√
| 2 − b| < | 2 − a|.
√
2, b clearly has a
However, the machine representations ã and b̃ of a and b satisfy the reverse inequality.
Indeed, it is not hard to check that (work out the details!)
a = ã = (1.0111)2
and that
as ã −
√
2 < 2−5 .
and b̃ = x̃ = (1.0110)2 ,
√
√
| 2 − ã| < | 2 − b̃|,
So, perhaps, a is a better approximation.
Can you give an example of x, a, b such that |x − b| < |x − a| and |x̃ − ã| < |x̃ − b̃| ?