lecture 13

# The results for the error estimates are the same

• Notes
• 27

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• The results for the error estimates are the same regardless of whether you: • Apply differentiation to , which represents the error estimate for . • Apply a Taylor series analysis to the differentiation formula you derived. • The error estimate for the interpolating function with is most precise in a formal sense (i.e. it includes all H.O.T. as well!). However there is a weak depen- dence of on and therefore some inaccuracies may be incurred when differenti- ating to obtain an error estimate for the corresponding finite difference approximation. • Practically, for estimating the error of the differentiating formula derived by esti- mating , we can apply the procedure used and examine an estimate of the error which does not depend on . This is equivalent to examining the leading order term in the truncated series. • Note that the derivative in the error formula on the previous page may also be esti- mated at e x   g x   g x   x e x   e 1   x   x i

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CE 30125 - Lecture 13 p. 13.23 Deriving a forward difference approximation to the second derivative and the associated error estimate Evaluating the second derivative of the function at : • Evaluate and substitute x o f 2   x x o = g 2   x o e 2   x o + = x 0 x 1 x 2 forward g 2   x   f 2   x x o = 1 2! ---- 2 f o h 2 ---------- 1 2! ---- 2 f o h 2 ---------- e 2   x   + + x x o = = f 2   x x o = 2 f o h 2 ---------- e 2   x o + = f 2   x x o = f 2 2 f 1 f o + h 2 --------------------------- e 2   x o + =
CE 30125 - Lecture 13 p. 13.24 Now evaluate using a non dependent expression for the error term and evalu- ating this expression at Substituting for results in: e 2   x   x o e 2   x o 1 3! ---- f 3   x o x x 1 x x 2 x x o x x 2 x x o x x 1 + + + + + x x o = e 2   x o 1 3! ---- f 3   x o x o x 1 x o x 2 x o x o x o x 2 x o x o x o x 1 + + + + + e 2   x o 1 3! ---- f 3   x o h 2 h 0 2 h 0 h + + e 2   x o h f 3   x o e 2   x o f 2   x x o = f 2 2 f 1 f o + h 2 --------------------------- hf 3   x o

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CE 30125 - Lecture 13 p. 13.25 Generalizing the node numbering: This results in the generic expression for a three node forward difference approximation to the second derivative with an appropriate error estimate x 0 x 1 x 2 i i+2 i+1 f 2   f i 2 + 2 f i 1 + f i h 2 --------------------------------------- E + = E hf 3   x i