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Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2002
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Problem 1
Solution 1
Problem 2
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
All nodes lies in (a,b)
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
Problem 3
Solution 3