Positive Definite, Hermitian Matrix: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Positive Definite, Hermitian Matrix (Linear System)}} == Description == In this case, we restrict $A$ to be positive definite and hermitian (or symmetric, if $A$ is real-valued). == Related Problems == Generalizations: General Linear System Related: Sparse Linear System, Non-Definite, Symmetric Matrix, Toeplitz Matrix, Vandermonde Matrix == Parameters == <pre>n: number of variables and number of equations m: number of nonzero e...") |
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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 || | |||
|- | |||
| [[Cholesky (Positive Definite, Hermitian Matrix Linear System)|Cholesky]] || 1940 || $O(n^{3})$ || $O(n^{2})$ || Exact || Deterministic || | | [[Cholesky (Positive Definite, Hermitian Matrix Linear System)|Cholesky]] || 1940 || $O(n^{3})$ || $O(n^{2})$ || Exact || Deterministic || | ||
|- | |- | ||
|} | |} | ||
== Time Complexity | == Time Complexity Graph == | ||
[[File:Linear System - Positive Definite, Hermitian Matrix - Time.png|1000px]] | [[File:Linear System - Positive Definite, Hermitian Matrix - Time.png|1000px]] | ||
== Space Complexity | == Space Complexity Graph == | ||
[[File:Linear System - Positive Definite, Hermitian Matrix - Space.png|1000px]] | [[File:Linear System - Positive Definite, Hermitian Matrix - Space.png|1000px]] | ||
== | == Time-Space Tradeoff == | ||
[[File:Linear System - Positive Definite, Hermitian Matrix - Pareto Frontier.png|1000px]] | [[File:Linear System - Positive Definite, Hermitian Matrix - Pareto Frontier.png|1000px]] | ||
Latest revision as of 08:18, 10 April 2023
Description
In this case, we restrict $A$ to be positive definite and hermitian (or symmetric, if $A$ is real-valued).
Related Problems
Generalizations: General Linear System
Related: Sparse Linear System, 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 | |
| Cholesky | 1940 | $O(n^{3})$ | $O(n^{2})$ | Exact | Deterministic |
Time Complexity Graph
Space Complexity Graph
Time-Space Tradeoff
