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**Unformatted text preview: **1 MAT 683 Numerical Analysis Qualifying Exam Syracuse University August 24, Fall 2010 Ex 1. Let x < x 1 be two distinct real numbers. Let be such that 0 < < x 1- x . Find a polynomial p of degree 3 such that p ( x ) = p ( x + ) = 1 p ( x 1 ) = p ( x 1 + ) = 0 Let p ( x ) = lim → p ( x ). Find p ( x ), p ( x 1 ), p ( x ), p ( x 1 ) and p ( x 1 ). What is the Hermite interpolation problem that the polynomial p satisfies? Ex 2. Let α , 0 < α < 1 be a given real number and let x 1 =- 1, x 2 =- α , x 3 = α , x 4 = 1 and let w 1 ,w 2 ,w 3 ,w 4 be real numbers. We consider a quadrature Q ( f ) = 4 X j =1 w j f ( x j ) for any function f continuous on [- 1 , 1]. (a) Find w 1 ,w 2 ,w 3 ,w 4 as functions of α so that Q ( p ) = R 1- 1 p ( x ) d x for every polynomial p of degree ≤ 3. (b) Does there exist an α such that Q ( p ) = R 1- 1 p ( x ) d x for every polynomial p of degree r for r > 3 ? If yes what is the maximal value of r and what are the corresponding values values of α and w...

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