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

Ask an Expert

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

Write your message here..

Related Questions

Quantitative, A lobster catcher spends $12 500 per month to maintain a lobs... A lobster catcher spends $12 500 per month to maintain a lobster boat. He plans to catch an average of 20 days per month during lobster season. For each day, he must allow approx

Gauss thm, statement of gauss thm statement of gauss thm

Intrgers, how to evaluate the sums how to evaluate the sums

Hypothesis testing procedure, Hypothesis Testing Procedure Whenever a b... Hypothesis Testing Procedure Whenever a business complaint comes up here is a recommended procedure for conducting a statistical test. The reason of such a test is to establish

Theorem on intervals of validity, Theorem Consider the subsequent IVP.... Theorem Consider the subsequent IVP. y′ =  p (t ) y = g (t )  y (t 0 )= y 0 If p(t) and g(t) are continuous functions upon an open interval a o , after that there i

Writing decimal place value in words, How will you write this in words 216.... How will you write this in words 216.9805

Indefinite integrals, Indefinite Integrals : In the past two chapters we'v... Indefinite Integrals : In the past two chapters we've been given a function, f ( x ) , and asking what the derivative of this function was.  Beginning with this section we are now

Fundamental theorem of calculus, Fundamental Theorem of Calculus, Part I ... Fundamental Theorem of Calculus, Part I As noted through the title above it is only the first part to the Fundamental Theorem of Calculus. The first part of this theorem us

Christie paid 5% sales tax purchase how much did she spend, Christie purcha... Christie purchased a scarf marked $15.50 and gloves marked $5.50. Both items were on sale for 20% off the marked price. Christie paid 5% sales tax on her purchase. How much did she

Trapezoid rule - approximating definite integrals, Trapezoid Rule - Approxi... Trapezoid Rule - Approximating Definite Integrals For this rule we will do similar set up as for the Midpoint Rule. We will break up the interval [a, b] into n subintervals of