Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

Algorithm 3

DACG-Newton algorithm.

INPUT: Matrix 𝐴;

  number of sought eigenpairs 𝑛_eig;

  tolerance and maximum number of its for the outer iteration: 𝜏, ITMAX;

  tolerance for the initial eigenvector guess 𝜏_DACG;

  tolerance and maximum number of its for the inner iteration: 𝜏_PCG.

ITMAXPCG;

  parameters for the RFSAI preconditioner,

     𝛿_out,𝑑_out,𝜀_out, nband, for the 1st FSAI factor and,

     𝛿_in,𝑑_in,𝜀_in for the 2nd factor;

  maximum allowed rank-two update in the BFGS preconditioner: 𝑘_max.

𝑄∶=[].

Compute 𝑃₀, an RFSAI preconditioner for 𝐴,

for 𝑗∶=1 to 𝑛_eig

 (1) Choose 𝐱₀ such that 𝑄^⊤𝐱₀=0;

 (2) Compute 𝐮₀, an approximation to 𝐯_𝑗 by the DACG procedure with initial

  vector 𝐱₀, preconditioner 𝑃₀ and tolerance 𝜏_DACG;

  (3) 𝑘∶=0, 𝜃_𝑘∶=𝐮^⊤_𝑘𝐴𝐮_𝑘;

  (4) while 𝐴𝐮_𝑘−𝜃_𝑘𝐮_𝑘>𝜏𝜃_𝑘 and 𝑘<IMAX do

   (1) 𝑄∶=𝑄𝐮_𝑘.

   (2) Solve 𝐽_𝑘𝐬_𝑘=−𝐫_𝑘 for 𝐬_𝑘⟂𝑄 by the PCG method with preconditioner 𝑃_𝑘 and tolerance 𝜏_PCG.

    (3) 𝐮_𝑘+1∶=(𝐮_𝑘+𝐬_𝑘)/𝐮_𝑘+𝐬_𝑘, 𝜃_𝑘+1=𝐮^⊤_𝑘+1𝐴𝐮_𝑘+1.

    (4) 𝑘₁=𝑘 MOD 𝑘_max; 𝑉∗,2𝑘₁+1∶=𝑠_𝑘,𝑉(∗,2𝑘₁+2)∶=𝐫_𝑘.

    (5) 𝑘∶=𝑘+1

  (6) end while

  (7) Assume 𝐯_𝑗=𝐮_𝑘 and 𝜆_𝑗=𝜃_𝑘. Set 𝑄∶=𝑄𝐯_𝑗

end for