NumericalDifferentiation

NumericalDifferentiation - Numerical Differentiation From...

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Numerical Differentiation From Taylor’s series we have L + - + - + = 2 ) )( ( ) )( ( ) ( ) ( 2 ' ' ' a x a f a x a f a f x f If we then substitute i x a = and 1 + = i x x we get L + - + - + = + + + 2 ) )( ( ) )( ( ) ( ) ( 2 1 ' ' 1 ' 1 i i i i i i i i x x x f x x x f x f x f If we then truncate the second and higher terms and substitute i i x x h - = + 1 we get ) ( ) ( ) ( ) ( 2 ' 1 h O h x f x f x f i i i + + = + We then solve for ) ( ' i x f to get h x f x f x f i i i ) ( ) ( ) ( 1 ' - + with an error of ) ( h O . This is the forward difference approximation of the derivative. To find the backwards difference approximation we use we substitute i x a = and 1 - = i x x and get L + - + - + = - - - 2 ) )( ( ) )( ( ) ( ) ( 2 1 ' ' 1 ' 1 i i i i i i i i x x x f x x x f x f x f If we substitute 1 - - = i i x x h we get L + + - = - 2 ) ( ) ( ) ( ) ( 2 ' ' ' 1 h x f h x f x f x f i i i i Note that we must negate h since 1 - - = i i x x h and not i i x x h - = - 1 as in the equation. We then truncate the second and higher terms and get ) ( ) ( ) ( ) ( 2 ' 1 h O h x f x f x f i i i + - = -
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We then solve for ) ( ' i x f to get h x f x f x f i i i ) ( ) ( ) ( 1 ' - - with an error of ) ( h O . This is the backwards difference approximation of the derivative. For the center difference approximation of the derivative we subtract L + + - = - 2 ) ( ) ( ) ( ) ( 2 ' ' ' 1 h x f h x f x f x f i i i i From L + + + = + 2 ) ( ) ( ) ( ) ( 2 ' ' ' 1 h x f h x f x f x f i i i i To get L + + + = - + 3 ) ( ) ( 2 ) ( ) ( 3 ' ' ' ' 1 1 h x f h x f x f x f i i i i Note every term with an even power of h has been canceled out so we can truncate the third derivative term
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NumericalDifferentiation - Numerical Differentiation From...

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