By Guderley K. G., Keller C. L.
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Extra resources for A Basic Theorem in the Computation of Ellipsoidal Error Bounds
Suppose that we are dealing with a 32-bit word and deal with single precision real number. This means that the precision is at the 6-7 decimal places. Thus, we cannot represent all decimal numbers with an exact binary representation in a computer. A typical example is , whereas has an exact binary representation even with single precision. 25) and if we see that there is a potential for an increased error in Å . This is because we are subtracting two numbers of equal size and what remains is only the least significant part of these numbers.
A better approach in case of a quadratic expression for Ü is to use a 3-step formula where we evaluate the derivative on both sides of a chosen point Ü¼ using the above forward and backward two-step formulae and taking the average afterward. 9) and we see now that the dominating error goes like ¾ if we truncate at the scond derivative. We call the term ¾ ¼¼¼ the truncation error. It is the error that arises because at some stage in the derivation, a Taylor series has been truncated. As we will see below, truncation errors and roundoff errors play an equally important role in the numerical determination of derivatives.
The È average value for this example is È Since ¾ Ü , it is easy to see that computing Ü Ü Ü can give rise to very large numbers with possible loss of precision when we perform the subtraction. To see this, consider the case where . Then we have5 ¿ ¼ ¼ ¼ ¾¼½¿ ½¼¼¼ ¿ ¼¼ while the exact answer is ´µ Ü¾ ÜÜ ½¼¼¿ ¾ Ü¾ ÜÜ ½¼¼¼ ´µ You can even check this by calculating it by hand. The second algorithm computes first the difference between Ü and the average value. The difference gets thereafter squared.