Partitioning Inverse Monte Carlo Iterative Algorithm for Finding the Three Smallest Eigenpairs of Generalized Eigenvalue Problem

Algorithm 3

Partitioning inverse  (𝑆,𝑛)

begin:

  𝑛=rank(𝑆);𝑝=𝑛/2

  𝐴=𝑆[1∶𝑝,1∶𝑝];𝐵=𝑆[1∶𝑝,𝑝+1∶𝑛]

  𝐶=𝑆[𝑝+1∶𝑛,1∶𝑝];𝐷=𝑆[𝑝+1∶𝑛,𝑝+1∶𝑛]

  𝑚= size (𝐴)

 if 𝑚≤ threshold

  𝐴𝐴  =  Monte Carlo procedure  (𝐴)

 else begin:

   𝐴𝐴 =  Partitioning  inverse  (𝐴,𝑚)

   𝑁 =  Partitioning  inverse  (𝐷−𝐶∗𝐴𝐴∗𝐵)

   𝑀=−𝑁∗𝐶∗𝐴𝐴;𝐿=−𝐴𝐴∗𝐵∗𝑁

   𝐾=𝐴𝐴−𝐴𝐴∗𝐵∗𝑀

   𝑆𝑆[1∶𝑝,1∶𝑝]=𝐾;𝑆𝑆[1∶𝑝,𝑝+1∶𝑛]=𝐿

   𝑆𝑆[𝑝+1∶𝑛,1∶𝑝]=𝑀;𝑆𝑆[𝑝+1∶𝑛,𝑝+1∶𝑛]=𝑁

 end

end