preciseness of the calculation

Hi I have two rows of data, consists of 3 numbers in each row. which are for example a,b and c. a and c are constant in each row, but b is changed. I sum the numbers in each row, however the answers are the same even though b is different. here is the numbers: a= 9237639,3400267 b=184,279078050855 c=23,3803295791066 a+b+c= 19237846,9994343 a=19237639,3400267 b=184,279078052863 c=23,3803295791066 a+b+c= 19237846,9994343

why these two summations are different?

Answer 1

[Jan]

This is an effect of the limit internal precision. DOUBLEs are represented by 64 bits such that the mantissa has a precision of 53 bits. This is about 16 digits. If you add numbers of different size, small differences can dissappear. See:

a = 1.0E17 + 1 - 1.0E17
% >> 0

although 1 would be mathematically correct. But the intermediate result of the first two summands cannot be represented accurately.

Using the standard IEEE arithmetics, there must be a difference between "mathematically" and "numerically". See also: eps and FAQ 6.1.

BTW. Reading the FAQ completely is always a good idea.