my @realval = (
    0.994704478,
    0.989459074,
    0.994717023,
    1.000000000, 
    1.000000000,
    0.002647761,
    0.005282977, 
    0.000882587, 
    0.005270463, 
    0.000882587, 
    0.002635231, 
    0.000882587,
    0.002635231,);

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

##</code><code>##

 my @actual_mat = (

   [ 0.994704478, 0.000882587, 0.000882587, 0.000882587, 0.002647761],
   [ 0.005270463, 0.989459074, 0.002635231, 0.002635231, 0.000000000],
   [ 0.000000000, 0.000000000, 0.005282977, 0.000000000, 0.000000000].
   [ 0.000000000, 0.000000000, 0.000000000, 1.000000000, 0.000000000],
   [ 0.000000000, 0.000000000, 0.000000000, 0.000000000, 1.000000000]
);

# It is simply done by assigning the value in @real_value given
# the corresponding (@row,@index) as coordinate;
# e.g. to assign $real_value[-1] into  
# $actual_mat[$row_index[-1]-1,$col_index[-1]-1);

##</code><code>##

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

##</code><code>##

my $result = [
     0.3989409,
     0.2010541,
     0.2000000,
     0.2000000,
     0.2000000,
];