Explanation Given information: R 33 of Romberg Integration. Concept used: The formula for Romberg Integration to approximate the definite integral ∫ a b f ( x ) d x are given by R 11 = h 1 2 [ f ( a ) + f ( b ) ] For j = 2 , 3 , 4 , … the formulae for step sizes h j and the approximating formulas R j 1 are given by, h j = 1 2 j − 1 ( b − a ) R j 1 = 1 2 R j − 1 , 1 + h j ∑ i = 1 2 j − 2 f ( a + ( 2 i − 1 ) h j ) Proof: For k = 2 , 3 , 4 , … , j the approximating formulas R j k are given by, R j k = 4 k − 1 R j , k − 1 − R j − 1 , k − 1 4 k − 1 − 1 Substitute k = 3 and j = 3 to obtain the approximating formula given by, R j k = 4 k − 1 R j , k − 1 − R j − 1 , k − 1 4 k − 1 − 1 R 33 = 4 2 R 3 , 2 − R 2 , 2 4 2 − 1 R 33 = 16 R 3 , 2 − R 2 , 2 15 ........... ( 1 ) To obtain the formula for R 3 , 2 , substitute k = 2 and j = 3 R 32 = 4 R 3 , 1 − R 2 , 1 4 − 1 = 4 R 3 , 1 − R 2 , 1 3 ........... ( 2 ) To obtain the formula for R 2 , 2 , substitute k = 2 and j = 2 R 22 = 4 R 2 , 1 − R 1 , 1 4 − 1 = 4 R 2 , 1 − R 1 , 1 3 ........... ( 3 ) Substitute the values of (2) and (3) in the formula (1) to obtain the following: R 33 = 16 ( 4 R 3 , 1 − R 2 , 1 3 ) − ( 4 R 2 , 1 − R 1 , 1 3 ) 15 = 16 ( 4 R 3 , 1 − R 2 , 1 ) − ( 4 R 2 , 1 − R 1 , 1 ) 45 = 64 R 3 , 1 − 16 R 2 , 1 − 4 R 2 , 1 + R 1 , 1 45 = 64 R 3 , 1 − 20 R 2 , 1 + R 1 , 1 45 Now to find R 21 and R 31 use the following formula: R j 1 = 1 2 R j − 1 , 1 + h j ∑ i = 1 2 j − 2 f ( a + ( 2 i − 1 ) h j ) R 21 = 1 2 R 2 − 1 , 1 + h 2 ∑ i = 1 2 2 − 2 f ( 0 + ( 2 i − 1 ) h 2 ) = 1 2 R 1 , 1 + h 2 ∑ i =

3rd Edition

ISBN: 9780134696454

Author: Sauer, Tim

Publisher: Pearson,

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Chapter 5.3, Problem 4E

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To show: R33 of Romberg Integration is expressed as Boole’s Rule with step size h3 .

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Chapter 5.3, Problem 3E

Chapter 5.3, Problem 5E

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R 33 of Romberg Integration is expressed as Boole’s Rule with step size h 3 .

Solution Summary: The author explains that R_33 of Romberg Integration is expressed as Boole’s Rule with step size.

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