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**Extra resources for Classical and modern numerical analysis : theory, methods and practice**

**Example text**

Rounding error. This is the error made due to the ﬁnite number of digits available on a computer. 2 on page 4), if f ∈ C 2 [a, b], then f (x) = 1 f (x + h) − f (x) + h h and 1 h x+h f (t)(x + h − t)dt x x+h f (t)(x + h − t)dt ≤ ch. x Thus, f (x) ≈ (f (x + h) − f (x))/h, and the error is O(h). We will call this the method error or truncation error , as opposed to roundoﬀ errors due to using machine approximations. 3 Now consider f (x) = ln x and approximate f (3) ≈ ln(3+h)−ln for h small h using a calculator having 11 digits.

15. Let f (x) = ln(x + 1) − ln(x) . 2 (a) Use four-digit decimal arithmetic with rounding to evaluate f(1000). 32 Classical and Modern Numerical Analysis (b) Rewrite f (x) in a form that avoids the loss of signiﬁcant digits and evaluate f (x) for x = 1000 once again. (c) Compare the relative errors for the answers obtained in (a) and (b). 16. Assume that x∗ and y ∗ are approximations to x and y with relative errors rx and ry , respectively, and that |rx |, |ry | < R. Assume further that x = y. How small must R be in order to ensure that x∗ = y ∗ ?

12345666 · · · × 107 . 12345 × 107 (chopped). 12346 × 107 (rounded). 2 On most modern machines, four rounding modes are actually chosen. 2, starting on page 17. 2 for an example with β = 10 and t = 1. 2) × 101 , . . 1 × 102 . 2: An example ﬂoating point system: β = 10, t = 1, and m = 1. We now have the following error bound. 6 |x − ﬂ(x)| ≤ 1 |x|β 1−t p, 2 where p = 1 for rounding and p = 2 for chopping. α1 α2 · · · αt · · · )β m , we have β m−1 ≤ |x| ≤ β m . In the interval [β m−1 , β m ], the ﬂoating point numbers are evenly spaced with separation β m−t .