Complete Pivoting VS Partial Pivoting in Gauss Elimination












1














I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)






linear-algebra matrices numerical-methods gaussian-elimination







share|cite|improve this question
























  • Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
    – Arbo94
    Nov 18 '15 at 18:03










  • If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
    – JP McCarthy
    Nov 18 '15 at 18:06
















1














I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)






linear-algebra matrices numerical-methods gaussian-elimination







share|cite|improve this question
























  • Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
    – Arbo94
    Nov 18 '15 at 18:03










  • If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
    – JP McCarthy
    Nov 18 '15 at 18:06














1












1








1


0





I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)






linear-algebra matrices numerical-methods gaussian-elimination







share|cite|improve this question















I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the exact feature of a matrix that will tell us which one to choose?)






linear-algebra matrices numerical-methods gaussian-elimination




linear-algebra matrices numerical-methods gaussian-elimination






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 18 '15 at 15:54

























asked Nov 18 '15 at 15:51









Arbo94

234




234












  • Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
    – Arbo94
    Nov 18 '15 at 18:03










  • If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
    – JP McCarthy
    Nov 18 '15 at 18:06


















  • Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
    – Arbo94
    Nov 18 '15 at 18:03










  • If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
    – JP McCarthy
    Nov 18 '15 at 18:06
















Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03




Ok, I figured that if the matrix is diagonally dominant or if there is no zeros in the diagonal, no pivoting is needed. However, I'm not sure about this.
– Arbo94
Nov 18 '15 at 18:03












If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06




If the matrix is diagonally dominant then every time you go to pivot, you won't need to.
– JP McCarthy
Nov 18 '15 at 18:06










2 Answers
2






active

oldest

votes


















0














Gaussian Elimination can be used as long as you are not using decimal rounding.



If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.



I don't know without a Google when complete pivoting is necessary.






share|cite|improve this answer





















  • Thank you, but I didn't understand what you meant by rounding?
    – Arbo94
    Nov 18 '15 at 16:51










  • If instead of $1/3$ you use $0.333$ you are rounding.
    – JP McCarthy
    Nov 18 '15 at 16:53










  • ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
    – Arbo94
    Nov 18 '15 at 17:04










  • Are YOU going to use rounding in your row operations?
    – JP McCarthy
    Nov 18 '15 at 17:05










  • No, I'm not. because I want to run a matlab code.
    – Arbo94
    Nov 18 '15 at 17:13



















0















I have a hard time understanding that when and under what conditions
we can use Gauss elimination with complete pivoting, and when with
partial pivoting, and when with no pivoting? (I mean what is the exact
feature of a matrix that will tell us which one to choose?)




My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.



That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.



You won't be able to tell immediately but if you solve



$$ Ax=b tag{1} $$



$$ LUx =b tag{2}$$



For partial pivoting you'd have



$$ PA = LU implies P^{'}LUx b tag{3}$$



then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm



$$ | hat{x} - x | tag{4}$$



where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.



$$ frac{| hat{x} - x |}{|x|} tag{5}$$






share|cite|improve this answer























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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0














    Gaussian Elimination can be used as long as you are not using decimal rounding.



    If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.



    I don't know without a Google when complete pivoting is necessary.






    share|cite|improve this answer





















    • Thank you, but I didn't understand what you meant by rounding?
      – Arbo94
      Nov 18 '15 at 16:51










    • If instead of $1/3$ you use $0.333$ you are rounding.
      – JP McCarthy
      Nov 18 '15 at 16:53










    • ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
      – Arbo94
      Nov 18 '15 at 17:04










    • Are YOU going to use rounding in your row operations?
      – JP McCarthy
      Nov 18 '15 at 17:05










    • No, I'm not. because I want to run a matlab code.
      – Arbo94
      Nov 18 '15 at 17:13
















    0














    Gaussian Elimination can be used as long as you are not using decimal rounding.



    If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.



    I don't know without a Google when complete pivoting is necessary.






    share|cite|improve this answer





















    • Thank you, but I didn't understand what you meant by rounding?
      – Arbo94
      Nov 18 '15 at 16:51










    • If instead of $1/3$ you use $0.333$ you are rounding.
      – JP McCarthy
      Nov 18 '15 at 16:53










    • ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
      – Arbo94
      Nov 18 '15 at 17:04










    • Are YOU going to use rounding in your row operations?
      – JP McCarthy
      Nov 18 '15 at 17:05










    • No, I'm not. because I want to run a matlab code.
      – Arbo94
      Nov 18 '15 at 17:13














    0












    0








    0






    Gaussian Elimination can be used as long as you are not using decimal rounding.



    If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.



    I don't know without a Google when complete pivoting is necessary.






    share|cite|improve this answer












    Gaussian Elimination can be used as long as you are not using decimal rounding.



    If you are using rounding Gaussian Elimination can be very inaccurate and you should use partial pivoting in this case.



    I don't know without a Google when complete pivoting is necessary.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 18 '15 at 16:36









    JP McCarthy

    5,61712440




    5,61712440












    • Thank you, but I didn't understand what you meant by rounding?
      – Arbo94
      Nov 18 '15 at 16:51










    • If instead of $1/3$ you use $0.333$ you are rounding.
      – JP McCarthy
      Nov 18 '15 at 16:53










    • ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
      – Arbo94
      Nov 18 '15 at 17:04










    • Are YOU going to use rounding in your row operations?
      – JP McCarthy
      Nov 18 '15 at 17:05










    • No, I'm not. because I want to run a matlab code.
      – Arbo94
      Nov 18 '15 at 17:13


















    • Thank you, but I didn't understand what you meant by rounding?
      – Arbo94
      Nov 18 '15 at 16:51










    • If instead of $1/3$ you use $0.333$ you are rounding.
      – JP McCarthy
      Nov 18 '15 at 16:53










    • ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
      – Arbo94
      Nov 18 '15 at 17:04










    • Are YOU going to use rounding in your row operations?
      – JP McCarthy
      Nov 18 '15 at 17:05










    • No, I'm not. because I want to run a matlab code.
      – Arbo94
      Nov 18 '15 at 17:13
















    Thank you, but I didn't understand what you meant by rounding?
    – Arbo94
    Nov 18 '15 at 16:51




    Thank you, but I didn't understand what you meant by rounding?
    – Arbo94
    Nov 18 '15 at 16:51












    If instead of $1/3$ you use $0.333$ you are rounding.
    – JP McCarthy
    Nov 18 '15 at 16:53




    If instead of $1/3$ you use $0.333$ you are rounding.
    – JP McCarthy
    Nov 18 '15 at 16:53












    ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
    – Arbo94
    Nov 18 '15 at 17:04




    ok. let me clarify my question. I've got a matrix. how to know that if I should use guess partial pivoting or complete or naive? because I don't know if it is rounded.
    – Arbo94
    Nov 18 '15 at 17:04












    Are YOU going to use rounding in your row operations?
    – JP McCarthy
    Nov 18 '15 at 17:05




    Are YOU going to use rounding in your row operations?
    – JP McCarthy
    Nov 18 '15 at 17:05












    No, I'm not. because I want to run a matlab code.
    – Arbo94
    Nov 18 '15 at 17:13




    No, I'm not. because I want to run a matlab code.
    – Arbo94
    Nov 18 '15 at 17:13











    0















    I have a hard time understanding that when and under what conditions
    we can use Gauss elimination with complete pivoting, and when with
    partial pivoting, and when with no pivoting? (I mean what is the exact
    feature of a matrix that will tell us which one to choose?)




    My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.



    That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.



    You won't be able to tell immediately but if you solve



    $$ Ax=b tag{1} $$



    $$ LUx =b tag{2}$$



    For partial pivoting you'd have



    $$ PA = LU implies P^{'}LUx b tag{3}$$



    then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm



    $$ | hat{x} - x | tag{4}$$



    where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.



    $$ frac{| hat{x} - x |}{|x|} tag{5}$$






    share|cite|improve this answer




























      0















      I have a hard time understanding that when and under what conditions
      we can use Gauss elimination with complete pivoting, and when with
      partial pivoting, and when with no pivoting? (I mean what is the exact
      feature of a matrix that will tell us which one to choose?)




      My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.



      That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.



      You won't be able to tell immediately but if you solve



      $$ Ax=b tag{1} $$



      $$ LUx =b tag{2}$$



      For partial pivoting you'd have



      $$ PA = LU implies P^{'}LUx b tag{3}$$



      then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm



      $$ | hat{x} - x | tag{4}$$



      where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.



      $$ frac{| hat{x} - x |}{|x|} tag{5}$$






      share|cite|improve this answer


























        0












        0








        0







        I have a hard time understanding that when and under what conditions
        we can use Gauss elimination with complete pivoting, and when with
        partial pivoting, and when with no pivoting? (I mean what is the exact
        feature of a matrix that will tell us which one to choose?)




        My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.



        That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.



        You won't be able to tell immediately but if you solve



        $$ Ax=b tag{1} $$



        $$ LUx =b tag{2}$$



        For partial pivoting you'd have



        $$ PA = LU implies P^{'}LUx b tag{3}$$



        then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm



        $$ | hat{x} - x | tag{4}$$



        where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.



        $$ frac{| hat{x} - x |}{|x|} tag{5}$$






        share|cite|improve this answer















        I have a hard time understanding that when and under what conditions
        we can use Gauss elimination with complete pivoting, and when with
        partial pivoting, and when with no pivoting? (I mean what is the exact
        feature of a matrix that will tell us which one to choose?)




        My professor explained this to the class with an example which I can no longer specifically recall. He noted that the when you use the solver it will pretty much by default use partial pivoting as the error is too bad without it, however, the cost for using complete pivoting is not worth the improvement you get in precision typically compared to other decompositions.



        That is if you need complete pivoting you should probably just use the QR decomposition. You should be checking for the error you get when you solve the system of equations I believe but I've never seen how someone would implement everything.



        You won't be able to tell immediately but if you solve



        $$ Ax=b tag{1} $$



        $$ LUx =b tag{2}$$



        For partial pivoting you'd have



        $$ PA = LU implies P^{'}LUx b tag{3}$$



        then you'd back sub and front sub. Instead with pivoting you get a pivot matrix $P$. You'd like to get the norm



        $$ | hat{x} - x | tag{4}$$



        where $hat{x}$ is the solution vector you get and $x$ is the real solution . Alternatively the relative error.



        $$ frac{| hat{x} - x |}{|x|} tag{5}$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Oct 20 at 2:57

























        answered Oct 20 at 1:25









        Ryan Howe

        2,41911323




        2,41911323






























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