Asymptotic Complexity of Gaussian Elimination using Complete Pivoting
$begingroup$
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
linear-algebra linear-solver complexity
edited 9 hours ago
Anton Menshov
3,84121663
asked 16 hours ago
IggIgg
132
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add a comment |
$begingroup$
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
linear-algebra linear-solver complexity
edited 9 hours ago
Anton Menshov
3,84121663
asked 16 hours ago
IggIgg
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
linear-algebra linear-solver complexity
edited 9 hours ago
Anton Menshov
3,84121663
asked 16 hours ago
IggIgg
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $O(n^3)$.
Is it the same for complete pivoting?
linear-algebra linear-solver complexity
linear-algebra linear-solver complexity
edited 9 hours ago
Anton Menshov
3,84121663
asked 16 hours ago
IggIgg
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 9 hours ago
Anton Menshov
3,84121663
asked 16 hours ago
IggIgg
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 9 hours ago
Anton Menshov
3,84121663
edited 9 hours ago
Anton Menshov
3,84121663
edited 9 hours ago
Anton Menshov
3,84121663
3,84121663
asked 16 hours ago
IggIgg
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 16 hours ago
IggIgg
132
asked 16 hours ago
IggIgg
132
132
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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2 Answers
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Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $O(n^2)$ to $O(n^3)$. While the total runtime is increased, the overall asymptotic complexity remains $O(n^3)$.
answered 15 hours ago
Carl ChristianCarl Christian
95748
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add a comment |
$begingroup$
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $O(n^2)$ steps to $O(n^3)$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$O$ notation should be doubly repeated, that omitting the constants can obscure important details.
The hidden problem here is that full pivoting appears to spoil the opportunity to use BLAS3 operations to do deferred updates on the trailing columns. Because any column might be holding the next pivot at any time, they always have to be to kept up to date using BLAS2 kernels.
In contrast, partial pivoting only requires BLAS2 to update a thin "leading panel" of upcoming columns, and can update all the trailing columns later using BLAS3 once the panel is done.
The bottom line of all this is that full pivoting runs vastly slower than partial pivoting on practical computers, despite having the same asymptotic complexity.
answered 9 hours ago
rchilton1980rchilton1980
2,228712
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $O(n^2)$ to $O(n^3)$. While the total runtime is increased, the overall asymptotic complexity remains $O(n^3)$.
answered 15 hours ago
Carl ChristianCarl Christian
95748
$endgroup$
add a comment |
$begingroup$
Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $O(n^2)$ to $O(n^3)$. While the total runtime is increased, the overall asymptotic complexity remains $O(n^3)$.
answered 15 hours ago
Carl ChristianCarl Christian
95748
$endgroup$
add a comment |
$begingroup$
Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $O(n^2)$ to $O(n^3)$. While the total runtime is increased, the overall asymptotic complexity remains $O(n^3)$.
answered 15 hours ago
Carl ChristianCarl Christian
95748
$endgroup$
Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $O(n^2)$ to $O(n^3)$. While the total runtime is increased, the overall asymptotic complexity remains $O(n^3)$.
answered 15 hours ago
Carl ChristianCarl Christian
95748
answered 15 hours ago
Carl ChristianCarl Christian
95748
answered 15 hours ago
Carl ChristianCarl Christian
95748
answered 15 hours ago
Carl ChristianCarl Christian
95748
95748
add a comment |
add a comment |
$begingroup$
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $O(n^2)$ steps to $O(n^3)$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$O$ notation should be doubly repeated, that omitting the constants can obscure important details.
The hidden problem here is that full pivoting appears to spoil the opportunity to use BLAS3 operations to do deferred updates on the trailing columns. Because any column might be holding the next pivot at any time, they always have to be to kept up to date using BLAS2 kernels.
In contrast, partial pivoting only requires BLAS2 to update a thin "leading panel" of upcoming columns, and can update all the trailing columns later using BLAS3 once the panel is done.
The bottom line of all this is that full pivoting runs vastly slower than partial pivoting on practical computers, despite having the same asymptotic complexity.
answered 9 hours ago
rchilton1980rchilton1980
2,228712
$endgroup$
add a comment |
$begingroup$
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $O(n^2)$ steps to $O(n^3)$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$O$ notation should be doubly repeated, that omitting the constants can obscure important details.
The hidden problem here is that full pivoting appears to spoil the opportunity to use BLAS3 operations to do deferred updates on the trailing columns. Because any column might be holding the next pivot at any time, they always have to be to kept up to date using BLAS2 kernels.
In contrast, partial pivoting only requires BLAS2 to update a thin "leading panel" of upcoming columns, and can update all the trailing columns later using BLAS3 once the panel is done.
The bottom line of all this is that full pivoting runs vastly slower than partial pivoting on practical computers, despite having the same asymptotic complexity.
answered 9 hours ago
rchilton1980rchilton1980
2,228712
$endgroup$
add a comment |
$begingroup$
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $O(n^2)$ steps to $O(n^3)$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$O$ notation should be doubly repeated, that omitting the constants can obscure important details.
The hidden problem here is that full pivoting appears to spoil the opportunity to use BLAS3 operations to do deferred updates on the trailing columns. Because any column might be holding the next pivot at any time, they always have to be to kept up to date using BLAS2 kernels.
In contrast, partial pivoting only requires BLAS2 to update a thin "leading panel" of upcoming columns, and can update all the trailing columns later using BLAS3 once the panel is done.
The bottom line of all this is that full pivoting runs vastly slower than partial pivoting on practical computers, despite having the same asymptotic complexity.
answered 9 hours ago
rchilton1980rchilton1980
2,228712
$endgroup$
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $O(n^2)$ steps to $O(n^3)$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$O$ notation should be doubly repeated, that omitting the constants can obscure important details.
The hidden problem here is that full pivoting appears to spoil the opportunity to use BLAS3 operations to do deferred updates on the trailing columns. Because any column might be holding the next pivot at any time, they always have to be to kept up to date using BLAS2 kernels.
In contrast, partial pivoting only requires BLAS2 to update a thin "leading panel" of upcoming columns, and can update all the trailing columns later using BLAS3 once the panel is done.
The bottom line of all this is that full pivoting runs vastly slower than partial pivoting on practical computers, despite having the same asymptotic complexity.
answered 9 hours ago
rchilton1980rchilton1980
2,228712
edited 9 hours ago
answered 9 hours ago
rchilton1980rchilton1980
2,228712
answered 9 hours ago
rchilton1980rchilton1980
2,228712
answered 9 hours ago
rchilton1980rchilton1980
2,228712
2,228712
add a comment |
add a comment |
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