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Unformatted text preview: Lecture 27 Numerical Differentiation Approximating derivatives from data Suppose that a variable y depends on another variable x , i.e. y = f ( x ), but we only know the values of f at a finite set of points, e.g., as data from an experiment or a simulation: ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) . Suppose then that we need information about the derivative of f ( x ). One obvious idea would be to approximate f ( x i ) by the Forward Difference : f ( x i ) = y i y i +1- y i x i +1- x i . This formula follows directly from the definition of the derivative in calculus. An alternative would be to use a Backward Difference : f ( x i ) y i- y i 1 x i- x i 1 . Since the errors for the forward difference and backward difference tend to have opposite signs, it would seem likely that averaging the two methods would give a better result than either alone. If the points are evenly spaced, i.e. x i +1- x i = x i- x i 1 = h , then averaging the forward and backward differences leads to a symmetric expression called the Central Difference : f ( x i ) = y i y i +1- y i 1 2 h . Errors of approximation We can use Taylor polynomials to derive the accuracy of the forward, backward and central difference formulas. For example the usual form of the Taylor polynomial with remainder (sometimes calledformulas....
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- Fall '08