Numerical algorithms for finding an eigenvector

Click For Summary

Discussion Overview

The discussion revolves around numerical algorithms for finding eigenvectors of matrices, specifically exploring various methods and challenges associated with these algorithms. The conversation includes theoretical considerations, practical implementation issues, and references to literature on the topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that all matrices have at least one eigenvector and expresses interest in algorithms for finding them, mentioning a specific approach that maximizes a certain quantity related to eigenvectors.
  • The same participant reports encountering precision issues with their algorithm, suggesting that the shape of the optimization landscape affects the accuracy of the eigenvector found.
  • Another participant references a book that contains algorithms for finding eigenvectors, specifically mentioning the NR3 book and its predecessor.
  • A later reply indicates difficulties accessing the PDF files of the NR3 book, but later confirms successful access through an alternative method.
  • One participant expresses frustration with not being able to find a clear method for obtaining a Schur form from the Hessenberg form as described in the referenced book.
  • Another participant clarifies the accessibility of the NR3 book and suggests posting the question in a dedicated forum for further assistance.

Areas of Agreement / Disagreement

Participants express varying levels of success and understanding regarding the algorithms and resources mentioned, indicating that there is no consensus on the best approach or clarity on the transition from Hessenberg form to Schur form.

Contextual Notes

Some participants mention limitations in accessing resources and the clarity of explanations in the literature, which may affect their ability to implement the discussed algorithms effectively.

jostpuur
Messages
2,112
Reaction score
19
All matrices [itex]A\in\mathbb{C}^{n\times n}[/itex] have at least one eigenvector [itex]z\in\mathbb{C}^n[/itex]. I'm interested to know what kind of algorithms there are for the purpose of finding an eigenvector.

I noticed that

[tex] \frac{|z^{\dagger} A z|}{\|Az\|} = 1\quad\quad\quad\quad (1)[/tex]

holds only when [itex]z[/itex] is an eigenvector, so I succeeded writing one algorithm using this fact. I let [itex]z[/itex] move on the sphere [itex]\|z\|=1[/itex] when the program runs iterations so that the quantity (1) is maximized. Unfortunately I run into some precision problems. For some reason my function seems to give only 3 or 4 decimals right for the components of the eigenvector, even though floating point numbers in C-language could contain more accuracy. I believe that the inaccuracy problem rises from the fact that the quantity (1) is approximately a paraboloid in the environment of the eigenvector. If the peak was sharp, then its location would be easier to find with iterations, but paraboloid is not so sharp peak, which makes its location less precise.

Anyway, are there other kind of algorithms out there too?
 
Physics news on Phys.org
Those pdf files are locked somehow. My pdf viewer software asks for a password.

update:

I succeeded getting the book as a pdf file in an alternative way, and it's working now. So thank's for mentioning the book, anyway.
 
Last edited:
Actually I did not succeed in learning how to find an eigenvector from that book. For my purposes finding an eigenvector is approximately the same as finding a Schur form, so tried to look for that. I see that the book explains how the Householder transformations can be used to obtain a Hessenberg form, but I don't see how to proceed from Hessenberg form to a Schur form.
 
For clarity,

The pdf files for the NR3 book are locked. However, the older editions are free to view.

Also, you might want to post the question in the NR forum.
 

Similar threads

Challenge: Find the minimum amount of draws possible for a list of integers
  • · Replies 9 ·
Replies
9
Views
3K
Graduate Eigenvalues of block matrix/Related non-linear eigenvalue problem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad The algorithms involved in finding a solution with numerical methods
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Correct Upper/Lower limits for continuation of solutions for ODE
  • · Replies 0 ·
Replies
0
Views
4K
Graduate The eigenvalue power method for quantum problems
  • · Replies 2 ·
Replies
2
Views
2K
Graduate How to numerically diagonalize a Hamiltonian in a subspace?
  • · Replies 4 ·
Replies
4
Views
4K
Graduate Numerically Solving Scalar Propagation in Curved Spacetime
  • · Replies 1 ·
Replies
1
Views
2K
Numerical approximation of the eigenvalues and the eigenvector
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad An efficient way to find perfect squares?
  • · Replies 15 ·
Replies
15
Views
9K
Graduate What is the closed-form solution using ALS algorithm to optimize
  • · Replies 1 ·
Replies
1
Views
2K