Toeplitz Matrix: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Toeplitz Matrix (Linear System)}} == Description == In this case, we restrict $A$ to be a Toeplitz matrix. == Related Problems == Generalizations: General Linear System Related: Sparse Linear System, Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Vandermonde Matrix == Parameters == <pre>n: number of variables and number of equations m: number of nonzero entries in matrix k: ratio between largest and smalle...") |
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== Parameters == | == Parameters == | ||
$n$: number of variables and number of equations | |||
m: number of nonzero entries in matrix | |||
k: ratio between largest and smallest eigenvalues | $m$: number of nonzero entries in matrix | ||
$k$: ratio between largest and smallest eigenvalues | |||
== Table of Algorithms == | == Table of Algorithms == | ||
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| [[Gaussian-Jordan Elimination (General Linear System; Positive Definite, Hermitian Matrix; Non-Definite, Symmetric Matrix; Toeplitz Matrix; Vandermonde Matrix Linear System)|Gaussian-Jordan Elimination]] || -150 || $O(n^{3})$ || $O(n^{2})$ || Exact || Deterministic || | |||
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| [[Levinson–Durbin recursion (Toeplitz Matrix Linear System)|Levinson–Durbin recursion]] || 1947 || $O(n^{2})$ || $O(n^{2})$ total || Exact || Deterministic || [https://onlinelibrary-wiley-com.ezproxy.canberra.edu.au/doi/abs/10.1002/sapm1946251261 Time] | | [[Levinson–Durbin recursion (Toeplitz Matrix Linear System)|Levinson–Durbin recursion]] || 1947 || $O(n^{2})$ || $O(n^{2})$ total || Exact || Deterministic || [https://onlinelibrary-wiley-com.ezproxy.canberra.edu.au/doi/abs/10.1002/sapm1946251261 Time] | ||
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== Time Complexity | == Time Complexity Graph == | ||
[[File:Linear System - Toeplitz Matrix - Time.png|1000px]] | [[File:Linear System - Toeplitz Matrix - Time.png|1000px]] | ||
== References/Citation == | == References/Citation == | ||
https://en.wikipedia.org/wiki/Levinson_recursion | https://en.wikipedia.org/wiki/Levinson_recursion | ||
Latest revision as of 09:05, 28 April 2023
Description
In this case, we restrict $A$ to be a Toeplitz matrix.
Related Problems
Generalizations: General Linear System
Related: Sparse Linear System, Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Vandermonde Matrix
Parameters
$n$: number of variables and number of equations
$m$: number of nonzero entries in matrix
$k$: ratio between largest and smallest eigenvalues
Table of Algorithms
| Name | Year | Time | Space | Approximation Factor | Model | Reference |
|---|---|---|---|---|---|---|
| Gaussian-Jordan Elimination | -150 | $O(n^{3})$ | $O(n^{2})$ | Exact | Deterministic | |
| Levinson–Durbin recursion | 1947 | $O(n^{2})$ | $O(n^{2})$ total | Exact | Deterministic | Time |
| Bareiss Algorithm | 1969 | $O(n^{2})$ | $O(n^{2})$ total | Exact | Deterministic | Time |
Time Complexity Graph
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