Simplify the matrix of a linear system knowing that some of the solutions are equal












2












$begingroup$


In order to improve the efficiency of my python program I'm trying to take advantage of some properties of a linear system I need to solve.



I have a linear system $Ax = b$ and I know beforehand that some of the solutions are equal, though I don't know their values before solving the system (i.e. $x_1 = x_{10}$, $x_3 = x_5$, ...).



Is there any way I can simplify my system ($A$ and b) for faster solving? My first thought was to eliminate one of the two unknowns for each pair and to do the sum of both columns associated with the elements of the pair, weighted by 1/2, but I'm not sure at all that this is mathematically correct.






linear-algebra matrices systems-of-equations







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Don't weight by $1/2$ !
    $endgroup$
    – Damien
    Dec 15 '18 at 18:38






  • 1




    $begingroup$
    You don't have to modify $b$
    $endgroup$
    – Damien
    Dec 15 '18 at 18:45
















2












$begingroup$


In order to improve the efficiency of my python program I'm trying to take advantage of some properties of a linear system I need to solve.



I have a linear system $Ax = b$ and I know beforehand that some of the solutions are equal, though I don't know their values before solving the system (i.e. $x_1 = x_{10}$, $x_3 = x_5$, ...).



Is there any way I can simplify my system ($A$ and b) for faster solving? My first thought was to eliminate one of the two unknowns for each pair and to do the sum of both columns associated with the elements of the pair, weighted by 1/2, but I'm not sure at all that this is mathematically correct.






linear-algebra matrices systems-of-equations







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Don't weight by $1/2$ !
    $endgroup$
    – Damien
    Dec 15 '18 at 18:38






  • 1




    $begingroup$
    You don't have to modify $b$
    $endgroup$
    – Damien
    Dec 15 '18 at 18:45














2












2








2





$begingroup$


In order to improve the efficiency of my python program I'm trying to take advantage of some properties of a linear system I need to solve.



I have a linear system $Ax = b$ and I know beforehand that some of the solutions are equal, though I don't know their values before solving the system (i.e. $x_1 = x_{10}$, $x_3 = x_5$, ...).



Is there any way I can simplify my system ($A$ and b) for faster solving? My first thought was to eliminate one of the two unknowns for each pair and to do the sum of both columns associated with the elements of the pair, weighted by 1/2, but I'm not sure at all that this is mathematically correct.






linear-algebra matrices systems-of-equations







share|cite|improve this question









$endgroup$




In order to improve the efficiency of my python program I'm trying to take advantage of some properties of a linear system I need to solve.



I have a linear system $Ax = b$ and I know beforehand that some of the solutions are equal, though I don't know their values before solving the system (i.e. $x_1 = x_{10}$, $x_3 = x_5$, ...).



Is there any way I can simplify my system ($A$ and b) for faster solving? My first thought was to eliminate one of the two unknowns for each pair and to do the sum of both columns associated with the elements of the pair, weighted by 1/2, but I'm not sure at all that this is mathematically correct.






linear-algebra matrices systems-of-equations




linear-algebra matrices systems-of-equations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 15 '18 at 17:08









FredVFredV

876




876








  • 1




    $begingroup$
    Don't weight by $1/2$ !
    $endgroup$
    – Damien
    Dec 15 '18 at 18:38






  • 1




    $begingroup$
    You don't have to modify $b$
    $endgroup$
    – Damien
    Dec 15 '18 at 18:45














  • 1




    $begingroup$
    Don't weight by $1/2$ !
    $endgroup$
    – Damien
    Dec 15 '18 at 18:38






  • 1




    $begingroup$
    You don't have to modify $b$
    $endgroup$
    – Damien
    Dec 15 '18 at 18:45








1




1




$begingroup$
Don't weight by $1/2$ !
$endgroup$
– Damien
Dec 15 '18 at 18:38




$begingroup$
Don't weight by $1/2$ !
$endgroup$
– Damien
Dec 15 '18 at 18:38




1




1




$begingroup$
You don't have to modify $b$
$endgroup$
– Damien
Dec 15 '18 at 18:45




$begingroup$
You don't have to modify $b$
$endgroup$
– Damien
Dec 15 '18 at 18:45










1 Answer
1






active

oldest

votes


















1












$begingroup$

Let us call $(c_i)_i$ the columns of$A$.



The equation can be written as:
$$b = sum_{i=1}^{n} x_ic_i $$



If for example we know that $x_1 = x_2$, the equation becomes:
$$b = x_2(c_1 + c_2) + sum_{i=3}^{n} x_ic_i $$



The procedure follows:




  • Replace $c_2$ by $c_1 + c_2$

  • Suppress $c_1$

  • Remove $x_1$ from the set of unknown values

  • Don't modify $b$






share|cite|improve this answer









$endgroup$













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    1 Answer
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    active

    oldest

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    1 Answer
    1






    active

    oldest

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    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    Let us call $(c_i)_i$ the columns of$A$.



    The equation can be written as:
    $$b = sum_{i=1}^{n} x_ic_i $$



    If for example we know that $x_1 = x_2$, the equation becomes:
    $$b = x_2(c_1 + c_2) + sum_{i=3}^{n} x_ic_i $$



    The procedure follows:




    • Replace $c_2$ by $c_1 + c_2$

    • Suppress $c_1$

    • Remove $x_1$ from the set of unknown values

    • Don't modify $b$






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      Let us call $(c_i)_i$ the columns of$A$.



      The equation can be written as:
      $$b = sum_{i=1}^{n} x_ic_i $$



      If for example we know that $x_1 = x_2$, the equation becomes:
      $$b = x_2(c_1 + c_2) + sum_{i=3}^{n} x_ic_i $$



      The procedure follows:




      • Replace $c_2$ by $c_1 + c_2$

      • Suppress $c_1$

      • Remove $x_1$ from the set of unknown values

      • Don't modify $b$






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        Let us call $(c_i)_i$ the columns of$A$.



        The equation can be written as:
        $$b = sum_{i=1}^{n} x_ic_i $$



        If for example we know that $x_1 = x_2$, the equation becomes:
        $$b = x_2(c_1 + c_2) + sum_{i=3}^{n} x_ic_i $$



        The procedure follows:




        • Replace $c_2$ by $c_1 + c_2$

        • Suppress $c_1$

        • Remove $x_1$ from the set of unknown values

        • Don't modify $b$






        share|cite|improve this answer









        $endgroup$



        Let us call $(c_i)_i$ the columns of$A$.



        The equation can be written as:
        $$b = sum_{i=1}^{n} x_ic_i $$



        If for example we know that $x_1 = x_2$, the equation becomes:
        $$b = x_2(c_1 + c_2) + sum_{i=3}^{n} x_ic_i $$



        The procedure follows:




        • Replace $c_2$ by $c_1 + c_2$

        • Suppress $c_1$

        • Remove $x_1$ from the set of unknown values

        • Don't modify $b$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 15 '18 at 19:50









        DamienDamien

        59714




        59714






























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