final_solution - DCP3352 Numerical Methods Final Exam...

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Page 1 of 6 DCP3352 Numerical Methods Final Exam Solution 1. Please derive the first-order derivative formulas of a function f ( x ) and list the order of their error term used in (a) (5%) forward difference method (b) (5%) central difference method (Hint: consider the Taylor series expansion of f ( x ) at x = h and x = h ) Solution: (a) 2 0 0 0 ) ( " 2 1 ) ( ' ) ( ) ( h f h x f x f h x f h f h x f h x f x f ) ( " 2 1 ) ( ) ( ) ( ' 0 0 0 (b) 3 0 2 0 0 0 0 0 ) )( ( ' " ! 3 1 ) )( ( " 2 1 ) )( ( ' ) ( ) ( x x f x x x f x x x f x f x f 3 2 0 0 0 0 ) ( ' " ! 3 1 ) ( " 2 1 ) ( ' ) ( ) ( h f h x f h x f x f h x f (1) 3 2 0 0 0 0 ) ( " ' ! 3 1 ) ( " 2 1 ) ( ' ) ( ) ( h f h x f h x f x f h x f (2) Eq. 1 Eq. 2 3 0 0 0 ) ( ' " ! 3 2 ) ( ' 2 ) ( ) ( h f h x f h x f h x f 2 0 0 0 ) ( ' " ! 3 1 2 ) ( ) ( ) ( ' h f h h x f h x f x f 2. (15%) Use the method of undetermined coefficients to derive the following formula h f f f x f 2 3 4 ) ( ' 0 1 2 0 where ) 2 ( 0 2 h x f f , ) ( 0 1 h x f f , and ) ( 0 0 x f f . Solution: Let 2 2 1 1 0 0 0 ) ( ' f c f c f c x f and ) ( ) ( 2 x P x f Case 1: 1 ) ( 2 u P 1 2 1 0 f f f , 0 ) 0 ( ' ) ( ' 2 0 P x f Case 2: u u P ) ( 2 h f h f f 2 , , 0 2 1 0 , 1 ) 0 ( ' ) ( ' 2 0 P x f Case 3: 2 2 ) ( u u P 2 2 1 0 4 , 2 , 0 h f h f f , 0 ) 0 ( ' ) ( ' 2 0 P x f
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Page 2 of 6 From equations in case 1 to 3, we can obtain 0 1 0 4 0 2 0 1 1 1 2 1 0 2 2 c c c h h h h 1 4 3 2 1 2 1 0 h c c c Solve the system of equations for the coefficients in the quadrature rule. 3.
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This note was uploaded on 12/17/2011 for the course DCP 3352 taught by Professor Wen-chiehlin during the Spring '07 term at National Chiao Tung University.

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final_solution - DCP3352 Numerical Methods Final Exam...

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