Sparse Linear System: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Sparse Linear System (Linear System)}} == Description == In this case, we restrict $A$ to be sparse (i.e. $A$ only has $O(n)$ nonzero entries). == Related Problems == Generalizations: General Linear System Related: Positive Definite, Hermitian Matrix, Non-Definite, Symmetric Matrix, Toeplitz Matrix, Vandermonde Matrix == Parameters == <pre>n: number of variables and number of equations m: number of nonzero entries in matrix k:...") |
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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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| [[Harrow (Quantum) (Sparse Linear System Linear System)|Harrow (Quantum)]] || 2009 || $O(k^{2}* | | [[Harrow (Quantum) (Sparse Linear System Linear System)|Harrow (Quantum)]] || 2009 || $O(k^{2}*\log n)$ || $O(\log n)$ || Exact || Quantum || [https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.150502 Time] & [https://arxiv.org/pdf/0811.3171.pdf Space] | ||
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Latest revision as of 08:18, 10 April 2023
Description
In this case, we restrict $A$ to be sparse (i.e. $A$ only has $O(n)$ nonzero entries).
Related Problems
Generalizations: General Linear System
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 |
|---|---|---|---|---|---|---|
| Harrow (Quantum) | 2009 | $O(k^{2}*\log n)$ | $O(\log n)$ | Exact | Quantum | Time & Space |