General Linear System: Difference between revisions
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== Parameters == | == Parameters == | ||
n: number of variables and number of equations | $n$: number of variables and number of equations | ||
m: number of nonzero entries in matrix | $m$: number of nonzero entries in matrix | ||
k: ratio between largest and smallest eigenvalues | $k$: ratio between largest and smallest eigenvalues | ||
== 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 | ||
|- | |||
| [[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 || | |||
|- | |||
|} | |||
== Time Complexity Graph == | |||
[[File:Linear System - General Linear System - Time.png|1000px]] | [[File:Linear System - General Linear System - Time.png|1000px]] | ||
== Space Complexity | == Space Complexity Graph == | ||
[[File:Linear System - General Linear System - Space.png|1000px]] | [[File:Linear System - General Linear System - Space.png|1000px]] | ||
== | == Time-Space Tradeoff == | ||
[[File:Linear System - General Linear System - Pareto Frontier.png|1000px]] | [[File:Linear System - General Linear System - Pareto Frontier.png|1000px]] | ||
Latest revision as of 08:18, 10 April 2023
Description
A system of linear equations (or linear system) is a collection of one or more linear equations involving the same set of variables. This is typically written in the form $Ax=b$ where $A$ is a matrix and $x, b$ are vectors. In this case, we impose no restrictions on $A$.
Related Problems
Subproblem: Sparse Linear System, Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Toeplitz Matrix, Vandermonde Matrix
Related: Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Toeplitz 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 |
Time Complexity Graph
Space Complexity Graph
Time-Space Tradeoff
