Scientific Programming / 2015 / Article / Alg 5

Research Article

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 5

QRP factorization and postprocessing algorithms, where condest   computes an estimation of the condition number of .
(a)
QPR factorization algorithm.
(1) QR factorization with restricted pivoting:
setup: and .
While   do
   (1.1) panel factorization:
   setup: initialize column norms within the window,
    (last column in window).
   for   do
   (1.1.1) restricted pivoting:
     swap th column with pivot column
        of the largest norm within the window.
    if condest    then
         and break.
     (update numerical rank).
   (1.1.2) Householder matrix computation:
    .
   (1.1.3) right-look update of window:
    .
   end for
   (1.2)  trailing submatrix update: ().
   compute matrix .
   update trailing submatrix right of current window.
   (1.3)  swapping rejected columns to end:
end while
(2) QR factorization of rejected columns
with column pivoting
(3) QR factorization of rejected columns
with no pivoting
(b)
QPR postprocessing algorithm.
setup: set to be the maximum column index
  such that condest  .
repeat
 (1) pivoting for task-1 and task-2
  repeat
      Golub-I
      Golub-I
      Chan-II
      Chan-II
  until no column pivot occurred
(2) convergence check
    condest
    condest
  if    and    then
      break
  else if    then
      , move to right
  else
      , move to left
  end if
end repeat

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