Evaluate the convergence of the algorithms, Mathematics

Evaluate the convergence of the algorithms:

From the convergence proof of power method, LR and QR algorithm for the computation of eigenvalues we see that the easiest case to proof convergence of these algorithms is when all eigenvalues of a matrix are distinct and their absolute values are also distinct.

Conversely, it is not difficult to imagine that the convergence can be difficult to obtain when several eigenvalues have similar absolute values or in the case of repeated eigenvalue. In this project, we attempt to examine some of these more challenging cases.

Algorithmic Analysis

(a) Show that for any real valued matrix A, if a complex number is an eigenvalue, the complex conjugate μ must also be an eigenvalue.

(b) Consider a matrix A with a complex eigenvalue with non-zero imaginary part. Consider the Jornal canonical form of matrix A obtained via similarity transformation. What are the relationships between elementary Jordan blocks associated with and ?

(c) When using the power method or the LR or QR algorithm, can the algorithm converge to an upper-triangular matrix?

(d) Propose a possible approach to compute complex eigenvalues of a real valued matrix A.

Computer Implementation

(a) Implement LR and QR for computation of eigenvalues including algorithm to first transform the input matrix to a Henssenberg matrix.

(b) Validate the correctness of your implementation.

(c) Evaluate the convergence of the algorithms in the case of matrix with complex eigenvalue.

Posted Date: 2/13/2013 12:13:55 AM | Location : United States







Related Discussions:- Evaluate the convergence of the algorithms, Assignment Help, Ask Question on Evaluate the convergence of the algorithms, Get Answer, Expert's Help, Evaluate the convergence of the algorithms Discussions

Write discussion on Evaluate the convergence of the algorithms
Your posts are moderated
Related Questions
The Dolphins football team gained 16 yards on their first play then lost 11 yards on the next play. Write an addition expression to represent this situation.Find the sum an explain

Rounding whole numbers List the order in which nancy, amy, ethel and cindy line up in single file to board the school bus. Then match the girls with their heights, which are fo

what is x(5x4)=26?

A function is an equation for which any x which can be plugged into the equation will yield accurately one y out of the equation. There it is. i.e. the definition of functions w

find the number of ways 17 employees can b chosen from 327


Metallic spheres of radii 6 centimetre, 8 centimetre and 10 centimetres respectively are melted to form a single solid sphere. Find the radius of the resulting sphere.

Optimization is required in situations that frequently arise in finance and other areas. Organizations would like to maximize their profits or minimize thei

We know that a factor is a quantity which divides the given quantity without leaving any remainder. Similar to LCM above we can find a highest common factor (HCF)

Q. Example of negative number? If you take an elevator 8 stories  down , what would be the opposite of this? The opposite would be that you take the elevator 8 stories  up .