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my @realval = (
    0.994704478,
...
my @row_index = qw( 1 2 3 4 5 1 3 1 2 1 2 1 2);
my @col_index =  qw(1 2 3 4 5 5 3 2 1 3 3 4 4);
my @mat_dim  = (5,5); # M always equal to N

 my @actual_mat = (

...
# the corresponding (@row,@index) as coordinate;
# e.g. to assign $real_value[-1] into  
# $actual_mat[$row_index[-1]-1,$col_index[-1]-1);

my @p = (0.4,0.2,0.2,0.2,0.2);

my $result = [
     0.3989409,
...
     0.2000000,
     0.2000000,
];