Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems

Algorithm 2

Computation of 𝐜=𝑃_𝑘𝐠_𝑙 for the BFGS preconditioner.

INPUT: Vector 𝐠_𝑙, scalar products 𝛼_𝑠=𝐬^⊤_𝑠𝐫_𝑠,𝑠=0,…,𝑘−1.

𝐰=𝐠_𝑙

for 𝑠∶=𝑘−1 to 0

  (1) 𝑎_𝑠∶=𝐬^⊤_𝑠𝐰/𝛼_𝑠

  (2) 𝐰∶=𝐰−𝑎_𝑠𝐫_𝑠

end for

𝑐=𝑃₀𝐰

for 𝑠∶=0 to 𝑘−1

  (1) 𝑏∶=𝐫^⊤_𝑠𝐜/𝛼_𝑠

  (2) 𝐜∶=𝐜−𝑎_𝑠+𝑏𝐬_𝑠

end for

𝐳∶=𝑄^⊤𝐜

𝐜∶=𝐜−𝑄𝐳