solexam2s11

solexam2s11 - Foundations of Computational Math II Exam 2...

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Unformatted text preview: Foundations of Computational Math II Exam 2 In-class Exam Open Notes, Textbook, Homework Solutions Only Calculators Allowed Monday April 11, 2011 Question Points Points Possible Awarded 1. Approximation 20 2. Newton Cotes 25 3. Extrapolation 25 4. Generalized Fourier Series 30 Total 100 Points Name: 1 Problem 1 (20 points) Let f ( x ) and g ( x ) be smooth functions and consider the discrete inner product ( f, g ) m = m summationdisplay j =0 f ( x j ) g ( x j ) and the induced norm defined by bardbl f ( x ) bardbl 2 m = ( f, f ) m where x < x 1 < < x m . Let p n ( x ) P n be the polynomial of degree n m that minimizes bardbl f ( x ) p n ( x ) bardbl 2 m Suppose { P i ( x ) } , i n is a set of polynomials that is a basis of the space P n such that ( P i , P j ) m braceleftBigg = 0 if i negationslash = j negationslash = 0 if i = j and we are interested in finding p n ( x ) in the form p n ( x ) = n summationdisplay i =0 i P i ( x ) What family of polynomials would you choose and what would the values of x , . . ., x m be so that p n ( x ) is the near-minimax approximation of f ( x ) when m = n ; and when m > n the coefficients , 1 , . . ., n can be determined by a matrix times vector product? 2 Solution: The polynomial p n ( x ) is given by p n ( x ) = argmin q n P n bardbl f ( x ) q n ( x ) bardbl 2 m Given p n ( x ) = n summationdisplay i =0 i P i ( x ) the coefficients, i , solve the linear least squares problem min g R n vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble f ( x ) f ( x 1 ) . . . f ( x m ) P ( x ) . . . P n ( x ) P ( x 1 ) . . . P n ( x 1 ) . . . . . . P ( x m ) . . . P n ( x m ) 1 . . . n vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble vextenddouble 2 g T = ( 1 . . . n ) The solution is to choose the polynomials as the appropriately scaled Chebyshev poly-...
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This note was uploaded on 11/10/2011 for the course MAD 5404 taught by Professor Gallivan during the Spring '11 term at FSU.

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solexam2s11 - Foundations of Computational Math II Exam 2...

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