Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2004

Problem 1Edit

To compute  , we consider the following Eudoxos iterations: starting with  , we set   followed by  . Then  


Problem 1aEdit

Explain the Eudoxos method in terms of the power method.

Solution 1aEdit

The iteration can be represented in matrix formulation as follows:


 


which can be written as


 


Thus the iteration is just the power method since each step is represented by a multiplication by the matrix  .


The power method converges to the eigenvector of the largest eigenvalue.


The eigenvalues of   are computed to be  . Hence the largest eigenvalue is  


The corresponding eigenvector is then


 


Then   as desired.


Problem 1bEdit

How many iterations are required for an error  


Solution 1bEdit

Since convergence is linear, 7 steps is required to achieve the error bound.

Problem 2Edit

Let   be a sequence of monic polynomials orthogonal on   with respect to the positive weight function   (   has degree  ). Show that   satisfy a three term recursion formula of the form


 


Give expressions for the coefficients   and  

Solution 2aEdit

First notice that   and therefore we can express it as a linear combination of the monic polynomials of degree   or less i.e.


 


Taking the inner product of both side of   with   yields from the orthogonality of the polynomials:


 


Rearranging terms then yields


 


Similarly, taking the inner product of both side of   with   yields from the orthogonality of the polynomials:


 


Notice that


 


Therefore,


 


Finally, taking inner product of both side of   with   yields,


 


Notice that


 


which implies   for  

Problem 3aEdit

Find   such that   is a polynomial of degree   and this set is orthogonal on   with respect to the weight function  


Solution 3aEdit

Using Gram Schmidt with inner product defined as


 


and power basis   as starting vectors, we get


 


 


 

Problem 3bEdit

Find the weights and nodes of the 2 point Gaussian formula


 


Note:  


Solution 3bEdit

Using test functions   and   and using the roots of   as nodes we find