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

What fraction of water flows out, A conical vessel of radius 6cm and height...

A conical vessel of radius 6cm and height 8cm is completely filled with water. A sphere is lowered into the water and its size is such that when it touches the sides, it is just im

Prove the parallelogram circumscribing a circle is rhombus, Prove that the ...

Prove that the parallelogram circumscribing a circle is rhombus. Ans   Given : ABCD is a parallelogram circumscribing a circle. To prove : - ABCD is a rhombus or AB

I want to learn mathematics, I was never really good at mathematics what is...

I was never really good at mathematics what is the best way? I am reading Math better explained but is there anything else I can do? I want to study advanced topics and get a good

Eometyr, Lines EF and GH are graphed on this coordinate plane. Which point ...

Lines EF and GH are graphed on this coordinate plane. Which point is the intersection of lines EF and GH?

One step ahead, how do we figure it out here is an example 3,4,6,9,_,_,_,_...

how do we figure it out here is an example 3,4,6,9,_,_,_,_,_,. please help

Find no. of non negative integral solutions, Find no. of non negative integ...

Find no. of non negative integral solutions x 1 +x 2 +x 3 +4x 4 =20 Solution)  140. Break them into prime factors . Put 4 = 2^2 and every variable will have factors in 2,3,5 with

What is the minimum number of students, Question 1: What is the minimum...

Question 1: What is the minimum number of students each of whom comes from one of the 50 different states, enrolled in a university to guarantee that there are at least 100 who

Pricing, what is skimming pricing?

what is skimming pricing?

Solve the inequality |x - 1| + |x - 2|, Solve the inequality |x - 1| + |x -...

Solve the inequality |x - 1| + |x - 2|≤ 3. Working Rule:    First of all measure the expression to zero whose modulus happens in the given inequation and from this search the va

Finite population correction factor or fpcf), Finite Population Correction ...

Finite Population Correction Factor Or Fpcf) If a specified population is relatively of small size and sample size is more than 5 percent of the population then the standard er

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