NumericalAnalysis-683-2008jan - functions defined over the...

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MAT 683 Numerical Analysis January, 2008 NAME: 1. Find a polynomial p ( x ) of degree 2 that satisfies p ( x 0 ) = a, p 0 ( x 0 ) = b, p 0 ( x 1 ) = c, where a , b , c are given constants and x 0 , x 1 are two different points. 2. Let f (0), f ( h ) and f (2 h ) be the values of a real valued function f at x = 0, x = h and x = 2 h . (a) Derive the coefficients c 0 , c 1 and c 2 so that Df h ( x ) = c 0 f (0) + c 1 f ( h ) + c 2 f (2 h ) is as accurate an approximation to f 0 (0) as possible. (b) Derive the leading term of a truncation error estimate for the formula you derived in (a). 3. Consider the task of approximating a function f ( x ) by a linear combination of N function q k ( x ), k = 1 , 2 ,...,N , e.g. f ( x ) N X k =1 c k q k ( x ) for x [0 , 1] (a) Give the equations that determine the c k ’s so that k f ( x ) - N k =1 c k q k ( x ) k 2 is min- imized. (The norm is taken over the interval [0 , 1].) (b) How would you solve the resulting equations? 4. Consider the three point Legendre-Gauss integration formula Z 1 - 1 f ( x ) dx 5 9 f ˆ - r 3 5 ! + 8 9 f (0) + 5 9 f ˆ r 3 5 ! with the corresponding error expression Z 1 - 1 f ( x ) dx = 5 9 f ˆ - r 3 5 ! + 8 9 f (0) + 5 9 f ˆ r 3 5 ! + 1 525 f (4) ( ξ ) (1) for some point ξ , | ξ | < 1. (a) Derive the nodes and weights for the three point Legendre-Gauss quadrature for
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Unformatted text preview: functions defined over the interval [ a,b ]. (b) Derive an error expression similar to (1) for the formula you obtain in (a). 5. Consider the fixed point iteration x n +1 = F ( x n ) 1 (a) Assume that the fixed point iteration converges, explicitly derive the conditions on F that ensure a second order rate of convergence. (b) Using your result from (a), derive the condition on φ ( x ) so that the iteration x n +1 = x n + φ ( x n ) f ( x n ) will have a second order rate of convergence to a root α of the problem f ( x ) = 0. 6. Assume the points { x i } , for i = 1 , 2 ,...,n + 1, are distinct. Prove that the polynomial of degree ≤ n that interpolates the data { ( x i ,y i ) } is unique. 2...
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NumericalAnalysis-683-2008jan - functions defined over the...

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