Numerical Methods Qualification Exam Problems and Solutions (University of Maryland)/August 2004
To compute , we consider the following Eudoxos iterations: starting with , we set followed by . Then
Explain the Eudoxos method in terms of the power method.
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.
How many iterations are required for an error
Since convergence is linear, 7 steps is required to achieve the error bound.
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
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:
Finally, taking inner product of both side of with yields,
which implies for
Find such that is a polynomial of degree and this set is orthogonal on with respect to the weight function
Using Gram Schmidt with inner product defined as
and power basis as starting vectors, we get
Find the weights and nodes of the 2 point Gaussian formula
Using test functions and and using the roots of as nodes we find