Roughly speaking the error can be estimated by comparing two approxima tions

# Roughly speaking the error can be estimated by

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Roughly speaking, the error can be estimated by comparing two approxima- tions obtained with different h . Consider (1.27). If we halve h we get I [ f ] = T h/ 2 [ f ] + 1 4 C 2 h 2 + R ( h/ 2) . (1.29) Subtracting (1.29) from (1.27) we get C 2 h 2 = 4 3 ( T h/ 2 [ f ] - T h [ f ] ) + 4 3 ( R ( h/ 2) - R ( h )) . (1.30) The last term on the right hand side is o ( h 2 ). Hence, for h sufficiently small, we have C 2 h 2 4 3 ( T h/ 2 [ f ] - T h [ f ] ) (1.31) and this could provide a good, computable estimate for the error, i.e. E h [ f ] 4 3 ( T h/ 2 [ f ] - T h [ f ] ) . (1.32) The key here is that h has to be sufficiently small to make the asymptotic approximation (1.31) valid. We can check this by working backwards. If h is sufficiently small, then evaluating (1.31) at h/ 2 we get C 2 h 2 2 4 3 ( T h/ 4 [ f ] - T h/ 2 [ f ] ) (1.33)
1.2. AN ILLUSTRATIVE EXAMPLE 11 and consequently the ratio q ( h ) = T h/ 2 [ f ] - T h [ f ] T h/ 4 [ f ] - T h/ 2 [ f ] (1.34) should be approximately 4. Thus, q ( h ) offers a reliable, computable indicator of whether or not h is sufficiently small for (1.32) to be an accurate estimate of the error. We can now use (1.31) and the idea of error correction to improve the accuracy of T h [ f ] with the following approximation 2 S h [ f ] := T h [ f ] + 4 3 ( T h/ 2 [ f ] - T h [ f ] ) . (1.35) 1.2.5 Richardson Extrapolation We can view the error correction procedure as a way to eliminate the leading order (in h ) contribution to the error. Multiplying (1.29) by 4 and substracting (1.27) to the result we get I [ f ] = 4 T h/ 2 [ f ] - T h [ f ] 3 + 4 R ( h/ 2) - R ( h ) 3 (1.36) Note that S h [ f ] is exactly the first term in the right hand side of (1.36) and that the last term converges to zero faster than h 2 . This very useful and general procedure in which the leading order component of the asymptotic form of error is eliminated by a combination of two computations performed with two different values of h is called Richardson’s Extrapolation . Example 2. Consider again f ( x ) = e x in [0 , 1] . With h = 1 / 16 we get q 1 16 = T 1 / 32 [ e x ] - T 1 / 16 [ e x ] T 1 / 64 [ e x ] - T 1 / 32 [ e x ] 3 . 9998 (1.37) and the improved approximation is S 1 / 16 [ e x ] = T 1 / 16 [ e x ] + 4 3 ( T 1 / 32 [ e x ] - T 1 / 16 [ e x ] ) = 1 . 718281837561771 (1.38) which gives us nearly 8 digits of accuracy (error 9 . 1 × 10 - 9 ). S 1 / 32 gives us an error 5 . 7 × 10 - 10 . It decreased by approximately a factor of 1 / 16 . This would correspond to fourth order rate of convergence. We will see in Chapter 8 that indeed this is the case. 2 The symbol := means equal by definition.
12 CHAPTER 1. INTRODUCTION It appears that S h [ f ] gives us superior accuracy to that of T h [ f ] but at roughly twice the computational cost. If we group together the common terms in T h [ f ] and T h/ 2 [ f ] we can compute S h [ f ] at about the same compu- tational cost as that of T h/ 2 [ f ]: 4 T h/ 2 [ f ] - T h [ f ] = 4 h 2 " 1 2 f ( a ) + 2 N - 1 X j =1 f ( a + jh/ 2) + 1 2 f ( b ) # - h " 1 2 f ( a ) + N - 1 X j =1 f ( a + jh ) + 1 2 f ( b ) # = h 2 " f ( a ) + f ( b ) + 2 N - 1 X k =1 f ( a + kh ) + 4 N - 1 X k =1 f ( a + kh/ 2) # . Therefore S h [ f ] = h 6 " f ( a ) + 2 N - 1 X k =1 f ( a + kh ) + 4 N - 1 X k =1 f ( a + kh/ 2) + f ( b ) # . (1.39) The resulting quadrature formula S h [ f ] is known as the Composite Simpson’s Rule and, as we will see in Chapter 8, can be derived by approximating the integrand by quadratic polynomials. Thus, based on cost and accuracy, the Composite Simpson’s Rule would be preferable to the Composite Trapezoidal