Vandermonde Matrix: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Vandermonde Matrix (Linear System)}} == Description == In this case, we restrict $A$ to be a Vandermonde matrix. == Related Problems == Generalizations: General Linear System Related: Sparse Linear System, Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Toeplitz Matrix == Parameters == <pre>n: number of variables and number of equations m: number of nonzero entries in matrix k: ratio between largest and sma...") |
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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 || | |||
|- | |||
| [[Bjorck-Pereyra (Vandermonde Matrix Linear System)|Bjorck-Pereyra]] || 1970 || $O(n^{2})$ || $O(n^{2})$ total || Exact || Deterministic || [https://www.jstor.org/stable/2004623?seq=1 Time] | | [[Bjorck-Pereyra (Vandermonde Matrix Linear System)|Bjorck-Pereyra]] || 1970 || $O(n^{2})$ || $O(n^{2})$ total || Exact || Deterministic || [https://www.jstor.org/stable/2004623?seq=1 Time] | ||
|- | |- | ||
|} | |} | ||
== Time Complexity | == Time Complexity Graph == | ||
[[File:Linear System - Vandermonde Matrix - Time.png|1000px]] | [[File:Linear System - Vandermonde Matrix - Time.png|1000px]] | ||
Latest revision as of 09:05, 28 April 2023
Description
In this case, we restrict $A$ to be a Vandermonde matrix.
Related Problems
Generalizations: General Linear System
Related: Sparse Linear System, Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Toeplitz 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 | |
| Bjorck-Pereyra | 1970 | $O(n^{2})$ | $O(n^{2})$ total | Exact | Deterministic | Time |
Time Complexity Graph
