Eigenpair closest to mu: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Eigenpair closest to mu (Eigenvalues (Iterative Methods))}} == Description == Given an $n \times n$ matrix $A$, find the eigenpair (eigenvalue and associated eigenvector) of $A$ with the eigenvalue closest to $\mu$. == Related Problems == Generalizations: Any Eigenpair Related: All Eigenvalues, Any Eigenvalue, All Eigenpairs, Eigenpair with the Largest Eigenvalue == Parameters == No parameters found. == Table of Algorithms ==...") |
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== Table of Algorithms == | == Table of Algorithms == | ||
{| class="wikitable sortable" style="text-align:center;" width="100%" | |||
== Time Complexity | ! Name !! Year !! Time !! Space !! Approximation Factor !! Model !! Reference | ||
|- | |||
| [[Inverse iteration (Eigenpair closest to mu; Any eigenpair; Any eigenvalue Eigenvalues (Iterative Methods))|Inverse iteration]] || 1921 || $O(n^{2})$ || $O(n^{2})$ || Exact || Deterministic || [https://onlinelibrary.wiley.com/doi/abs/10.1002/zamm.19210010104 Time] | |||
|- | |||
| [[LOBPCG algorithm (Eigenpair closest to mu; Any eigenpair; Any eigenvalue Eigenvalues (Iterative Methods))|LOBPCG algorithm]] || 1948 || $O(n^{2})$ || $O(n)$? || Exact || Deterministic || | |||
|- | |||
| [[Homotopy method (All eigenpairs; Eigenpair closest to mu; Any eigenpair; Any eigenvalue; All eigenvalues Eigenvalues (Iterative Methods))|Homotopy method]] || 1992 || $O(n^{2})$ || $O(n^{2})$?? || Exact || Deterministic || [https://www.scirp.org/(S(czeh2tfqyw2orz553k1w0r45))/reference/ReferencesPapers.aspx?ReferenceID=530065 Time] | |||
|- | |||
| [[Folded spectrum method (Eigenpair closest to mu; Any eigenpair; Any eigenvalue Eigenvalues (Iterative Methods))|Folded spectrum method]] || 1934 || $O(n^{2})$ || $O(n)$? || Exact || Deterministic || [https://journals.aps.org/pr/abstract/10.1103/PhysRev.46.828 Time] | |||
|- | |||
|} | |||
== Time Complexity Graph == | |||
[[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Time.png|1000px]] | [[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Time.png|1000px]] | ||
== Space Complexity | == Space Complexity Graph == | ||
[[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Space.png|1000px]] | [[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Space.png|1000px]] | ||
== Pareto | == Pareto Frontier Improvements Graph == | ||
[[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Pareto Frontier.png|1000px]] | [[File:Eigenvalues (Iterative Methods) - Eigenpair closest to mu - Pareto Frontier.png|1000px]] | ||
Revision as of 13:04, 15 February 2023
Description
Given an $n \times n$ matrix $A$, find the eigenpair (eigenvalue and associated eigenvector) of $A$ with the eigenvalue closest to $\mu$.
Related Problems
Generalizations: Any Eigenpair
Related: All Eigenvalues, Any Eigenvalue, All Eigenpairs, Eigenpair with the Largest Eigenvalue
Parameters
No parameters found.
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Inverse iteration | 1921 | $O(n^{2})$ | $O(n^{2})$ | Exact | Deterministic | Time |
| LOBPCG algorithm | 1948 | $O(n^{2})$ | $O(n)$? | Exact | Deterministic | |
| Homotopy method | 1992 | $O(n^{2})$ | $O(n^{2})$?? | Exact | Deterministic | Time |
| Folded spectrum method | 1934 | $O(n^{2})$ | $O(n)$? | Exact | Deterministic | Time |
Time Complexity Graph
_-_Eigenpair_closest_to_mu_-_Time.png)
Space Complexity Graph
_-_Eigenpair_closest_to_mu_-_Space.png)
Pareto Frontier Improvements Graph
_-_Eigenpair_closest_to_mu_-_Pareto_Frontier.png)