Computing Low-Rank Approximation of a Dense Matrix on Multicore CPUs with a GPU and Its Application to Solving a Hierarchically Semiseparable Linear System of Equations

Algorithm 2

QP3 factorization algorithm, where is the submatrix consisting of the th and th columns of .

setup: Compute the column norms

    for   , and .

While   do

   (1) panel factorization:

  for   do

     (1.1) pivoting:

       select the pivot column with the largest .

        and .

       .

        (update numerical rank).

     (1.2) left-looking update of pivot column:

       .

     (1.3) Householder matrix computation:

        such that .

     (1.4) Computation of the -auxiliary vector :

     (1.5) right-look update of pivot row:

     (1.6) norm downdate:

        for .

        (if norms must be recomputed then break)

  end for

   (2) trailing submatrix update:

  .

   (recompute norms if necessary).

end while