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Unformatted text preview: COT4501 Fall2010 Midterm Solutions I: [25 points] Truncation Error and Rounding Error (a) (10 points) In the finite difference approximation of the first derivative f ′ ( x ) ≈ f ( x + h ) − f ( x ) h , why do the truncation error and rounding error go to zero and infinity respectively as h → ? The truncation error for the finite difference approximation of the first derivative is bounded by Mh 2 since we have via the Taylor series approximation that f ′ ( x ) = f ( x + h ) − f ( x ) h − h 2 f ′′ ( θ ) with θ ∈ [ x,x + h ] . When M is defined as M = sup θ ∈ [ x,x + h ]  f ′′ ( θ )  we see that the truncation error defined as  f ′ ( x ) − f ( x + h ) − f ( x ) h  ≤ Mh 2 which goes to zero as h → . The rounding error for the the finite difference approximation of the first derivative is bounded by 2 ϵ h where each function call is assumed to have a maximum rounding error of ϵ . Since the bound can be reached when the function call rounding errors are exactly ϵ and when the errors conspire to add up, the rounding error can go to infinity as h → . (b) (15 points) Derive the value of h at which the total error (truncation + rounding error) for the following approximation to the second derivative of the function f ( x ) is minimized: f ′′ ( x ) ≈ f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 . (1) You may assume a maximum absolute error of ϵ for each function call [ f ( x ) , f ( x + h ) and f ( x − h ) in (1) above]. Explain the significance of your result. From the Taylor series approximations, we have f ( x + h ) = f ( x ) + hf ′ ( x ) + h 2 2 f ′′ ( x ) + h 3 6 f ′′′ ( x ) + h 4 24 f (4) ( θ 1 ) , θ 1 ∈ [ x,x + h ] (2) and f ( x − h ) = f ( x ) − hf ′ ( x ) + h 2 2 f ′′ ( x ) − h 3 6 f ′′′ ( x ) + h 4 24 f (4) ( θ 2 ) , θ 2 ∈ [ x − h,x ] . (3) From (2) and (3), we get f ( x + h ) + f ( x − h ) − 2 f ( x ) = h 2 f ′′ ( x ) + h 4 24 f (4) ( θ 1 ) + h 4 24 f (4) ( θ 2 ) which can be further simplified as f ′′ ( x ) = f ( x + h ) − 2 f ( x ) + f ( x − h ) h 2 − h 2 24 f (4) ( θ 1 ) + h 2 24 f (4) ( θ 2 ) ....
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 Fall '10
 Rangarajan
 Linear Algebra, Numerical Analysis, ax, finite difference approximation, rounding error

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