Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2008
Let and , with , and let be a scalar. Show how the constrained least squares problem,
Sinceare 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.
block matrix multiplication yields,
So , which is unconstrained.
Prove or disprove the following interpolants exist for all values of and all distinct values of .
Substituting into this equation the pairs gives rise to the following matrix equation:
Where A is the , which is know to be non-singular. This proves existence and uniqueness of the coefficients .
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
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 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
Show that is symmetric and positive-definite with respect to the -inner product.
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 .
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 .
Use any facts you know about the conjugate gradient method to identify properties of that would be desirable for computing the solution efficiently.
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.
- Omitted on the actual exam. This made the problem ambiguous and the word symmetric should have been included. (Howard Elman, 12/16/08)