Since f 00 is assumed continuous f 2 C 2 a b then by the Intermediate Value

Since f 00 is assumed continuous f 2 c 2 a b then by

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8 CHAPTER 1. INTRODUCTION But (1.20) gives us two important properties of the approximation method in question. First, (1.20) tell us that E h [ f ] ! 0 as h ! 0. That is, the quadrature rule T h [ f ] converges to the exact value of the integral as h ! 0 1 . Recall h = ( b - a ) /N , so as we increase N our approximation to the integral gets better and better. Second, (1.20) tells us how fast the approx- imation converges, namely quadratically in h . This is the approximation’s rate of convergence . If we double N (or equivalently halve h ) then the error decreases by a factor of 4. We also say that the error is order h 2 and write E h [ f ] = O ( h 2 ). The Big ‘O’ notation is used frequently in Numerical Analysis. Definition 1. We say that g ( h ) is order h , and write g ( h ) = O ( h ) , if there is a constant C and h 0 such that | g ( h ) | Ch for 0 h h 0 , i.e. for su ffi ciently small h . Example 1. Let’s check the Trapezoidal Rule approximation for an integral we can compute exactly. Take f ( x ) = e x in [0 , 1] . The exact value of the integral is e - 1 . Observe how the error | I [ e x ] - T 1 /N [ e x ] | decreases by a Table 1.1: Composite Trapezoidal Rule for f ( x ) = e x in [0 , 1]. N T 1 /N [ e x ] | I [ e x ] - T 1 /N [ e x ] | Decrease factor 16 1.718841128579994 5 . 593001209489579 10 - 4 32 1.718421660316327 1 . 398318572816137 10 - 4 0.250012206406039 64 1.718316786850094 3 . 495839104861176 10 - 5 0.250003051723810 128 1.718290568083478 8 . 739624432374526 10 - 6 0.250000762913303 factor of (approximately) 1/4 as N is doubled, in accordance to (1.20). 1.2.4 Error bounds and Computable Error Estimates We can get an upper bound for the error using (1.20) and that f 00 is bounded in [ a, b ], i.e. | f 00 ( x ) | M 2 for all x 2 [ a, b ] for some constant M 2 . Then | E h [ f ] | 1 12 ( b - a ) h 2 M 2 . (1.22) 1 Neglecting round-o errors introduced by finite precision number representation and computer arithmetic.
1.2. AN ILLUSTRATIVE EXAMPLE 9
10 CHAPTER 1.

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