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

Algorithm 4

Modified Neighborhood-Based aggregation, .

Input:

    : scaled problem matrix,

    : tolerance.

Output:

    : aggregation matrix.

    1st process: generate the initial aggregates

 (1) Set and .

 (2) for    do

 (3)  compute based on (25).

 (4) end for

 (5) Reorder by increasing values:

 (6) new reordered .

 (7) for    do

 (8)  construct strong neighborhoods based on (19)

 (9)  if    then

 (10)   .

 (11)   .

 (12)   .

 (13)   end if

 (14) end for

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

 (15) while    do

 (16)  Pick and set according to (26)

 (17)  

 (18)  Set .

 (19)  if    then

 (20)   .

 (21)     and .

 (22)  else

 (23)    and .

 (24)   

 (25)  end if

 (26) end while

     3rd process: construct the aggregation matrix  

 (27) for    do

 (28)  if    then

 (29)   .

 (30)  else

 (31)   .

 (32)  end if

 (33) end for

 (34) return  ;