hmwk_3_2010

# E how can you use the error term in step d to improve

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e) How can you use the error term in step d , to improve the Newton Forward interpolater established in step c? Plot the now 5 point fourth degree interpolator, comparing it to the 4 point cubic interpolator established in Step c. Problem 2 Develop a 3 point Newton Backward interpolation formula and the associated error. Consider points f 0 , f 1 and f 2 at locations x 0 , x 1 and x 2 . a) Step 1: Develop a 4 term Taylor series expanded about x 2 . The fifth term is the truncation term (this will be the next term) and will be . b) Step 2: Use the definition of to derive an expression for the first derivative evaluated at x 2 to third order. c) Step 3: Use the definition of to derive an expression second derivative evaluated at x 2 to second order. d) Step 4: Substitute into the general Taylor series in Step 1, the expressions you derived for the first and second derivatives evaluated at x 2 . Sort out the terms so that a sequence of higher order terms appears and gather all the appropriate terms in your error term. V 0 x ( ) V 1 x ( ) V 2 x ( ) V 3 x ( ) g x ( ) f x ( ) g x ( ) f x ( ) g x ( ) g x ( ) e x ( ) f x ( ) g x ( ) E x ( ) e x ( ) E x ( ) O x x 2 ( ) 4 f 2 2 f 2
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