Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2008

Problem 1Edit

Let   and  , with  , and let   be a scalar. Show how the constrained least squares problem,


 


can be reduced to solving a related unconstrained least squares problem. The algorithm should start by finding a Householder transformation   such that

 


and setting

 

Solution 1Edit

Since Householder Matrices are orthogonal and hermitian (i.e. symmetric if the matrix is real), we have


 


Then the constraint can be written


 


Hence, the first entry of the column vector  ,  (a scalar), represents our constraint.


Now we want to show that we can write   as a related unconstrained problem. Substituting for   in   we get,


 


Let   where   is the first column of   and   are the remaining columns.

Since,


 


block matrix multiplication yields,


 


So   , which is unconstrained.

Problem 2Edit

Prove or disprove the following interpolants exist for all values of   and all distinct values of  .

Problem 2aEdit

 

Solution 2aEdit

Substituting into this equation the pairs   gives rise to the following matrix equation:


 


Where A is the Vandermonde matrix, which is know to be non-singular. This proves existence and uniqueness of the coefficients  .

Problem 2bEdit

 

Solution 2bEdit

Now, substituting the pairs   gives:


 


Consider the unique set of points  

The system reduces to


 

Here, the matrix   is clearly singular.

More generally, the determinant of the matrix   is given by

  if   for any  

Problem 3Edit

Consider the linear system of equations   where   is a symmetric positive definite matrix of order  . The conjugate gradient method (CG) for solving this system is

Choose  , compute  
 Set  
 for   until convergence
  
  
  
 <Test for Convergence>
  
  
end

where   is the Euclidean inner product.

Let   be some other symmetric[1] positive-definite matrix of order  . We know that the forms

 

are inner products on  . In addition, a matrix   is symmetric with respect to the   inner product   if   for all   in  , and   is positive definite with respect to   if   for all nonzero  

Problem 3aEdit

Show that   is symmetric and positive-definite with respect to the  -inner product.

Solution 3aEdit

 

Problem 3bEdit

In light of this fact, CG can be used to solve the system   in an appropriate manner. Specify this algorithm and identify any extra costs required that would not be present with  .

Solution 3bEdit

Choose  
 
  :  solve  
 
for   until convergence
  
  
  
 <Test for Convergence>
   :  solve  
  
  
end


One additional cost is computing  .

Another one-time cost is multiplying   which has cost   and   which has cost  .

Problem 3cEdit

Use any facts you know about the conjugate gradient method to identify properties of   that would be desirable for computing the solution   efficiently.

Solution 3cEdit

We want   to have eigenvalues whose values are close together. This accelerates convergence. Furthermore, if there are only   distinct eigenvalues, the algorithm will terminate in   steps.

NotesEdit

  1. Omitted on the actual exam. This made the problem ambiguous and the word symmetric should have been included. (Howard Elman, 12/16/08)