A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

Algorithm 3

Neighborhood-Based aggregation, .

Input:

    : scaled problem matrix,

    : tolerance.

Output:

    : aggregation matrix.

    1st process: generate the initial aggregates

 (1) Set and .

 (2) for    do

 (3)  construct strong neighborhoods based on (19).

 (4)  if    then

 (5)   .

 (6)   

 (7)   .

 (8)  end if

 (9) end for

    2nd process: connect the remaining nodes to their most connected aggregates

 (10) while    do

 (11) pick and set .

 (12)  and .

 (13) end while

    3rd process: construct the aggregation matrix  

 (14) for    do

 (15) if  ,   then

 (16)  .

 (17) else

 (18)  .

 (19) end if

 (20) end for

 (21) return  ;