lecture11 - Numerical Methods in Engineering ENGR 391...

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Unformatted text preview: Numerical Methods in Engineering ENGR 391 Faculty of Engineering and Computer Sciences Concordia University Numerical differentiation Know how to compute numerically a derivative of a function What is the velocity at different instance of time? 993 623 383 225 Distance (ft) 13 8 5 3 Time (s) A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table: Numerical Differentiation One of the idea would be to approximate f ( x ) by the Forward Difference : An alternative would be a Backward Difference : ) ( ) ( ) ( 1 h O h x f ) f(x x f i i i c ) ( ) ( ) ( 1 h O h x f ) f(x x f i i i c Numerical Differentiation ) ( ) ( ) ( ) ( 2 1 h O h x f x f x f i i i c ) ( ) ( ) ( ) ( 2 1 h O h x f x f x f i i i c Assume that y = f ( x ), suppose that we need information about the derivative of f ( x ) Taylors series expansion x i+1 = x i + h x i-1 = x i- h Averaging the forward and the backward leads to an expression is called Central Difference : ) ( 2 ) ( ) ( 2 1 1 h O h x f ) f(x x f i i i c c c c 2 1 ! 2 ) ( ) ( ) ( ) ( h x f h x f x f x f i i i i c 3 ) 3 ( 1 1 ! 3 ) ( 2 ) ( 2 ) ( ) ( h x f h x f x f x f i i i i c c c 2 1 ! 2 ) ( ) ( ) ( )...
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lecture11 - Numerical Methods in Engineering ENGR 391...

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