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

Problem 1Edit


Solution 1Edit

Problem 2Edit

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 2Edit

All nodes lies in (a,b)Edit

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 positiveEdit

Problem 3Edit


Solution 3Edit