Romberg2 - Romberg Integration James Keesling 1 The...

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Unformatted text preview: Romberg Integration James Keesling 1 The Trapezoidal Rule for Estimating the Integral A common way of estimating an integral is to use the Trapezoidal Rule. Let f ( x ) be the given function over the interval [ a,b ]. We want to estimate the integral R b a f ( x ) dx . Subdivide the interval [ a,b ] into n equal subintervals. Let x i = a + i b- a n for i = 0 , 1 , 2 ,...,n . The Trapezoidal Rule uses the following formula as the estimate of the integral. Z b a f ( x ) dx b- a 2 n f ( a ) + 2 n- 1 X i =1 f ( x i ) + f ( b ) ! In fact, the approximation can be shown to have an error that can be expressed as a power series in 1 n 2 . The form of this theoretical error is the basis for the Romberg algorithm. b- a 2 n f ( a ) + 2 n- 1 X i =1 f ( x i ) + f ( b ) ! = a + a 2 n 2 + a 4 n 4 + + a 2 k n 2 k + In the power series on the right hand side, a is the integral to be estimated. The rest of the power series is the theoretical error in estimating this integral. For largeof the power series is the theoretical error in estimating this integral....
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Romberg2 - Romberg Integration James Keesling 1 The...

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