Evaluate the convergence of the algorithms, Mathematics

Assignment Help:

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.


Related Discussions:- Evaluate the convergence of the algorithms

Solving an equation problems, Temperature: On one day in Fairfield, Montana...

Temperature: On one day in Fairfield, Montana the temperature dropped 80 degree fahrenheit from noon to midnight. If the temperature at midnight was -21 degree fahrenheit, write an

Triple integrals, Consider a circular disc of radius 1 and thickness 1 whic...

Consider a circular disc of radius 1 and thickness 1 which has a uniform density 10 ?(x, y, z) = 1. (a) Find the moment of inertia of this disc about its central axis (that is, the

Polynomials, find a quadratic polynomial whose zeroes are 2 and -6.verify t...

find a quadratic polynomial whose zeroes are 2 and -6.verify the relationship between the coefficients and zeroes of the polynomial

Nun, how do you identify area ??

how do you identify area ??

Find the number of students in the class, Students are made to stand in row...

Students are made to stand in rows. If one student is extra in a row there would be 2 rows less. If one student is less in a row there would be 3 rows more. Find the number of stud

Alegrabra, how do you do algebra with division

how do you do algebra with division

Stats, the automatic hopper loader is set to put 36 tons of coal in each ca...

the automatic hopper loader is set to put 36 tons of coal in each car. the actual weights of coal loaded into each car arw normally distributed with a mean of 36 tons and a standar

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd