Tags: Matlab, Numerical Analysis,floating number system,
The computer systematic calculation is varied from real world computation. It used oating number system, which is the form
:d1:::dt e (1)
is the xed base, such as based 2(1010), base 10(124), and base 16(2A) which is an integer. t is the number of digits; e is the exponent.
It is crucial to have a general standard that could represented dierent computer system. The IEEE Standard could demonstrated one computer system as following:
F(; t; m;M) (2)
While is the xed base of the computer, and it is mostly 2. t is the number of digits, also length of mantissa, which is also a integer. m is the smallest exponent the oating number is going to move, so as the M is the largest exponent that oating number is going to move.
Because there is only nite bit in the computer system, some numbers will be truncated or rounded. The computers cannot represent all numbers so round-o errors cannot be avoided.
The dierent arithmetic hardware or compiler used by computer have dierent digits. The results could be dierent by same hardware dierent compiler and by same compiler but dierent hardware. We will dig deeper into the computer system and see how it make impacts of our computational result.
First of all, we could nd a smallest oat-point number which does not move. That oating number is expressed as following form.
:d1:::dt e =d1+ d22 + ::: +dtt (3)
We have innite number in the real-world problem, but there are only nite bites that could represent the numbers in the binary computer system. Since it is not possible to represent all the numbers in the computer system, we might meet some errors which include truncation error and round-o error.
For example, we assume a IEEE standard is F = (2; 25;64; 63),
In the numerical analysis, we would like to control and assess the errors that incurred in the math-ematical computation process, such as truncation errors and round o errors. Truncation error canbe caused by inexact evaluation of mathematical operators, and round o errors can be made by machine imprecision. The computer uses oating point system, and every number is stored in a programmer of nite digits.
Thus, not all numbers are able to be represented in the oating system. If a number x is not in the oating number system, then it must be rounded to xr. Moreover, we noticed that the laws of arithmetic operations are not exact in some cases. For instance, 1 + e <= 1 and e is positive. The magnitude of round o error happened at one stage could increase as each iteration magnify the errors by a factor.