MTH3051 Introduction to Computational Mathematics - User Web ...

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School of Mathematical Sciences Monash University 0 . 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 We humans often write numbers like 0.0001234 or 675.123. This is not how the computer stores them. It will always shuffle the decimal point left or right to put the number in the form shown above. This process is called normalisation and it is applied after every computation. Numbers such as 0.0001234 and 675.123 are known as un-normalised numbers. For the time being we will work only with positive numbers (it saves writing ± with every number). 3.6 Examples 0 . 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 0 . 2 5 6 1 2 3 4 0 0 0 0 0 0 0 0 16-Feb-2014 30
School of Mathematical Sciences Monash University 0 . 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 From these examples we can see that every number x can be written in the form x = ( a + 10 −N b ) × 10 m with both a and b simple numbers in the range 0.1 to 0.99999 · · · In this representation for x we have ˜x = a × 10 m E R (x) = b × 10 m × 10 −N (Note : a has exactly N decimal digits while b has an infinite number of digits.) In most of what we do with E R (x) we will not be too concerned with the exact value of b. Thus as both x and b × 10 m carry a factor of 10 m we write Here is a very important question E R (x) = 10 −N O (x) (1) How do round off errors propagate through a series of calculations? We might be tempted to say that at the end of a series of calculations, the round off error in the answer will be E R (y) = 10 −N O (y) on a n N− digit computer. This is not always true (the above only applies when storing an exact number – it makes no account for the computations that may have preceded this number). As an example, suppose you calculated y = f(x) for some given function f(x). This will entail at least three sources of round off error. First, there will be a round-off error in x (due to its own prior history). Second, there will be a round-off error in computing f and third there may be an error in storing the computed value of f (do we round up or round down?). All three errors errors will combine and the best we can expect will be with M ≤ N. E R (y) = 10 −M O (y) 16-Feb-2014 31
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