Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2002

Problem 1

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Solution 1

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Problem 2

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Suppose there is a quadrature formula


 


which produces the exact integral whenever   is a polynomial of degree  . Here the nodes   are all distinct. Prove that the nodes lies in the open interval   and the weights   and   are positive.

Solution 2

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All nodes lies in (a,b)

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Let   be the nodes that lie in the interval  .


Let   which is a polynomial of degree  .


Let   which is a polynomial of degree  .


Then


 


since   is of one sign in the interval   since for  ,  


This implies   is of degree   since otherwise


 


from the orthogonality of  .

All weights positive

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Problem 3

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Solution 3

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