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Lecture04

# Lecture04 - Truncation Errors and the Taylor Series Lecture...

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2/4/10 1 Truncation Errors and the Taylor Series Lecture 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 . 0 5 . 0 ! 8 8 = = x for x 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. f(x i+1 ) f(x i ) + f (x i ) (x i+1 -x i ) first order approximation, in form of a straight line

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2/4/10 2 n n i i n i i i i i i i R x x n f x x f x x x f x f x f + + + +
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Lecture04 - Truncation Errors and the Taylor Series Lecture...

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