Positive Definite, Hermitian Matrix: Difference between revisions
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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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| [[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 || | ||
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== 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]] | ||
== Pareto | == Pareto Frontier Improvements Graph == | ||
[[File:Linear System - Positive Definite, Hermitian Matrix - Pareto Frontier.png|1000px]] | [[File:Linear System - Positive Definite, Hermitian Matrix - Pareto Frontier.png|1000px]] | ||
Revision as of 13:04, 15 February 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
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Space Complexity Graph
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Pareto Frontier Improvements Graph
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