Matrix Factorization for $ADA^T$ for varying $D$

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Suppose we have a fixed, real $m times n$ matrix $A$, where $m < n$ and $mathrmrank(A) = m$, e.g. $A$ has full row-rank. I find myself needing to repeatedly solve equations of the form $$ ADA^T vecx = vecb $$ for varying matrices $D$, though $D$ will always be diagonal and possess strictly positive entries. Thus, it makes sense to do some up-front work on $A$ to make subsequent solves faster.



Currently, the best thing to do seems to be the Cholesky decomposition for each product $ADA^T=L_DL_D^T$; though I was wondering if either



  • there was some nice way to parameterize the factorization in terms of $D$ explicitly (e.g. $L_D = L(D)$) and reuse prior work, or


  • if another matrix decomposition was available and made more sense?




linear-algebra numerical-linear-algebra matrix-decomposition




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    Suppose we have a fixed, real $m times n$ matrix $A$, where $m < n$ and $mathrmrank(A) = m$, e.g. $A$ has full row-rank. I find myself needing to repeatedly solve equations of the form $$ ADA^T vecx = vecb $$ for varying matrices $D$, though $D$ will always be diagonal and possess strictly positive entries. Thus, it makes sense to do some up-front work on $A$ to make subsequent solves faster.



    Currently, the best thing to do seems to be the Cholesky decomposition for each product $ADA^T=L_DL_D^T$; though I was wondering if either



    • there was some nice way to parameterize the factorization in terms of $D$ explicitly (e.g. $L_D = L(D)$) and reuse prior work, or


    • if another matrix decomposition was available and made more sense?




    linear-algebra numerical-linear-algebra matrix-decomposition




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Suppose we have a fixed, real $m times n$ matrix $A$, where $m < n$ and $mathrmrank(A) = m$, e.g. $A$ has full row-rank. I find myself needing to repeatedly solve equations of the form $$ ADA^T vecx = vecb $$ for varying matrices $D$, though $D$ will always be diagonal and possess strictly positive entries. Thus, it makes sense to do some up-front work on $A$ to make subsequent solves faster.



      Currently, the best thing to do seems to be the Cholesky decomposition for each product $ADA^T=L_DL_D^T$; though I was wondering if either



      • there was some nice way to parameterize the factorization in terms of $D$ explicitly (e.g. $L_D = L(D)$) and reuse prior work, or


      • if another matrix decomposition was available and made more sense?




      linear-algebra numerical-linear-algebra matrix-decomposition




      share|cite|improve this question













      Suppose we have a fixed, real $m times n$ matrix $A$, where $m < n$ and $mathrmrank(A) = m$, e.g. $A$ has full row-rank. I find myself needing to repeatedly solve equations of the form $$ ADA^T vecx = vecb $$ for varying matrices $D$, though $D$ will always be diagonal and possess strictly positive entries. Thus, it makes sense to do some up-front work on $A$ to make subsequent solves faster.



      Currently, the best thing to do seems to be the Cholesky decomposition for each product $ADA^T=L_DL_D^T$; though I was wondering if either



      • there was some nice way to parameterize the factorization in terms of $D$ explicitly (e.g. $L_D = L(D)$) and reuse prior work, or


      • if another matrix decomposition was available and made more sense?





      linear-algebra numerical-linear-algebra matrix-decomposition





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 27 at 2:36









      Math Lover

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      12.3k21232









      asked Jul 27 at 2:32









      Jason

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