Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

92 CHAPTER 7. APPROXIMATING DERIVATIVES
PSfrag replacements
h
φ(h)
1.0 0.39269908169872408
0.1 0.33395069677431943
0.01 0.33333950618106845
0.001 0.33333339506169679
0.0001 0.33333333395058062
1 × 10 −5 0.33333333334106813
1 × 10 −6 0.33333333332441484
1 × 10 −7 0.33333333315788138
1 × 10 −8 0.33333332760676626
1 × 10 −9 0.33333336091345694
1 × 10 −10 0.333333360913457
1 × 10 −11 0.333333360913457
1 × 10 −12 0.33339997429493451
1 × 10 −13 0.33306690738754696
1 × 10 −14 0.33306690738754696
1 × 10 −15 0.33306690738754691
1 × 10 −16 0
The data are illustrated in Figure 7.1. Notice that φ(h) gives at most 10 decimal places of
accuracy, then begins to deteriorate; Note however, we get 13 decimal places from D(2, 2). ⊣
1
total error
0.01
0.0001
1e-06
1e-08
1e-10
1e-12
1e-16 1e-14 1e-12 1e-10 1e-08 1e-06 0.0001 0.01 1
Figure 7.1: The total error for the centered difference approximation to f ′ ( √ 2) is shown versus
h. The total error is the sum of a truncation term which decreases as h decreases, and a roundoff
term which increases. The optimal h value is around 1 × 10 −5 . Note that Richardson’s D(2, 2)
approximation with h = 0.01 gives much better results than this optimal h.