Chapter-04 - Truncation Errors and the Taylor Series...

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Unformatted text preview: Truncation Errors and the Taylor Series Chapter 4 • Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values, derivatives, and integrals are computed. • Any smooth function can be approximated as a polynomial. Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point. x F(x) The approximation to F(x)=-0.1x 4-0.15x 3-0.5x 2-0.25x+1 at x=1, by 0 th , 1 st and 2 nd order Taylor expansion F(x i+1 ) = F(x i ) F(x i+1 )= F(x i )+F’(x i )h F(x i+1 )= F(x i )+F’(x i )h+1/2F’(x i )h 2 True value x i+1 =1 x i =0 h = x i+1- x i Example: To get the cos (x) for small x: If x=0.5 cos (0.5) =1-0.125+0.0026041-0.0000127+ … =0.877582 From the supporting theory, for this series, the error is no greater than the first omitted term. L +- +- = ! 6 ! 4 ! 2 1 cos 6 4 2 x x x x 0000001 . 5 . ! 8 8 = = ∴ ξ φορ ξ • Any smooth function can be approximated as a polynomial. f(x i+1 ) ≈ f(x i ) zero order approximation, only true if x i+1 and x i are very close to each other....
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This note was uploaded on 02/15/2010 for the course CHEN 320 taught by Professor Staff during the Spring '08 term at Texas A&M.

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Chapter-04 - Truncation Errors and the Taylor Series...

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