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HW9suggestions - Jacobs University Bremen School of...

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Jacobs University, Bremen School of Engineering and Science Prof. Dr. Lars Linsen, Orif Ibrogimov Spring Term 2010 Homework 9 120202: ESM4A - Numerical Methods Homework Problems 9.1. Least squares method Let f ( x ) = a ( x + 1) 2 + b sin πx + c cos πx 3 with f (0) = 0 , f ( 1 2 ) = 1 and f (1) = 1. (a) Derive the normal equations for the best approximative solution to a , b , and c in the least-squares sense. (b) Find the least-squares solution for a , b , and c . ( ? points ) 9.2. Differentation & Richardson extrapolation (a) Compute estimates for the derivative of function f ( x ) = ( x - 1)( x - 2)( x - 3)( x - 4)( x - 5) at x = 0 using forward and central differencing as well as using the estimate you obtain when applying two iterations of Richardson extrapolation to the central differencing estimate. Choose h = 0 . 1. Compare the absolute errors. (b) Derive an estimate for computing the second-order derivative of a continuous function f with error term being O ( h 4 ). (c) Derive the error term for the approximation: f ( x ) f ( x ) - 2 f ( x + h )+ f ( x +2 h ) h 2 (d) Derive a numerical differentation formula of order O ( h 4 ) by applying Richardson’s ex- trapolation to f ( x ) = f ( x + h ) - f ( x - h ) 2 h - h 2 6 f (3) ( x ) - h 4 4! f (5) ( x ) - . . . Give the error term of
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