Journal of Applied Mathematics

Research Article

Parallel Rayleigh Quotient Optimization with FSAI-Based Preconditioning

Algorithm 1

DACG Algorithm.
Choose tolerance 𝜀 , set 𝑈 = 0 .
DO 𝑗 = 1 , 𝑠
  ( 1 ) Choose 𝐱 0 such that 𝑈 𝑇 𝐱 0 = 0 ; set 𝑘 = 0 , 𝛽 0 = 0 ;
  ( 2 )    𝐱 𝐴 0 = 𝐴 𝐱 0 , 𝛾 = 𝐱 𝑇 0 𝐱 𝐴 0 , 𝜂 = 𝐱 𝑇 0 𝐱 0 , 𝑞 0 ≡ 𝑞 ( 𝐱 0 ) = 𝛾 / 𝜂 , 𝐫 0 = 𝐱 𝐴 0 − 𝑞 0 𝐱 0 ;
  ( 3 ) REPEAT
    (3.1) 𝐠 𝑘 ≡ ∇ 𝑞 ( 𝐱 𝑘 ) = ( 2 / 𝜂 ) 𝐫 𝑘 ;
    (3.2) 𝐠 𝑀 𝑘 = 𝑀 𝐠 𝑘 ;
    (3.3) I F 𝑘 > 0 T H E N 𝛽 𝑘 = 𝐠 𝑇 𝑘 ( 𝐠 𝑀 𝑘 − 𝐠 𝑀 𝑘 − 1 ) / 𝐠 𝑇 𝑘 − 1 𝐠 𝑀 𝑘 − 1 ;
    (3.4) ̃ 𝐩 𝑘 = 𝐠 𝑀 𝑘 + 𝛽 𝑘 𝐩 𝑘 − 1 ;
    (3.5) 𝐩 𝑘 = ̃ 𝐩 𝑘 − 𝑈 ( 𝑈 𝑇 ̃ 𝐩 𝑘 )
    (3.6) 𝐩 𝐴 𝑘 = 𝐴 𝐩 𝑘 ,
    (3.7) 𝛼 𝑘 = a r g m i n 𝑡 { 𝑞 ( 𝐱 𝑘 + 𝑡 𝐩 𝑘 √ ) } = ( 𝜂 𝑑 − 𝛾 𝑏 + Δ ) / 2 ( 𝑏 𝑐 − 𝑎 𝑑 ) , with
      𝑎 = 𝐩 𝑇 𝑘 𝐱 𝐴 𝑘 , 𝑏 = 𝐩 𝑇 𝑘 𝐩 𝐴 𝑘 , 𝑐 = 𝐩 𝑇 𝑘 𝐱 𝑘 , 𝑑 = 𝐩 𝑇 𝑘 𝐩 𝑘 ,
      Δ = ( 𝜂 𝑑 − 𝛾 𝑏 ) 2 − 4 ( 𝑏 𝑐 − 𝑎 𝑑 ) ( 𝛾 𝑎 − 𝜂 𝑐 ) ;
    (3.8) 𝐱 𝑘 + 1 = 𝐱 𝑘 + 𝛼 𝑘 𝐩 𝑘 , 𝐱 𝐴 𝑘 + 1 = 𝐱 𝐴 𝑘 + 𝛼 𝑘 𝐩 𝐴 𝑘 ;
    (3.9) 𝛾 = 𝛾 + 2 𝑎 𝛼 𝑘 + 𝑏 𝛼 2 𝑘 , 𝜂 = 𝜂 + 2 𝑐 𝛼 𝑘 + 𝑑 𝛼 2 𝑘 ;
    (3.10) 𝑞 𝑘 + 1 ≡ 𝑞 ( 𝐱 𝑘 + 1 ) = 𝛾 / 𝜂 ;
    (3.11) 𝑘 = 𝑘 + 1 ;
    (3.12) 𝐫 𝑘 = 𝐱 𝐴 𝑘 − 𝑞 𝑘 𝐱 𝑘
  UNTIL ( 𝑞 𝑘 + 1 − 𝑞 𝑘 ) / 𝑞 𝑘 + 1 < 𝑡 𝑜 𝑙 ;
  ( 4 )    𝜆 𝑗 = 𝑞 𝑘 , 𝐮 𝑗 = 𝐱 𝑘 / √ 𝜂 , 𝑈 = [ 𝑈 , 𝐮 𝑗 ] .
END DO