Asymptotic Complexity of Gaussian Elimination using Complete Pivoting












2












$begingroup$


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







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    2












    $begingroup$


    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







    share|cite|improve this question









    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.







    $endgroup$















      2












      2








      2





      $begingroup$


      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







      share|cite|improve this question









      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






      share|cite|improve this question









      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.











      share|cite|improve this question









      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.









      share|cite|improve this question




      share|cite|improve this question








      edited 3 hours ago









      Anton Menshov

      3,84121663




      3,84121663






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      asked 9 hours ago









      IggIgg

      132




      132




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      Igg is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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          2 Answers
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          $begingroup$

          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 $O(n^3)$.






          share|cite|improve this answer









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            3












            $begingroup$

            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.






            share|cite|improve this answer











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              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              3












              $begingroup$

              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 $O(n^3)$.






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                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 $O(n^3)$.






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  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 $O(n^3)$.






                  share|cite|improve this answer









                  $endgroup$



                  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 $O(n^3)$.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 9 hours ago









                  Carl ChristianCarl Christian

                  95748




                  95748























                      3












                      $begingroup$

                      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.






                      share|cite|improve this answer











                      $endgroup$


















                        3












                        $begingroup$

                        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.






                        share|cite|improve this answer











                        $endgroup$
















                          3












                          3








                          3





                          $begingroup$

                          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.






                          share|cite|improve this answer











                          $endgroup$



                          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.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited 3 hours ago

























                          answered 3 hours ago









                          rchilton1980rchilton1980

                          2,228712




                          2,228712






















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