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
|
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
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Solution 3bEdit
Using test functions and and using the roots of as nodes we find