sub add_vectors
{
    my @left = @{ shift };
    my @right = @{ shift };
    my @result = ();
    my $index = $#left > $#right ? @left : @right;
    while( --$index >= $[ )
    {
        $left[$index]   = 0 if not defined $left[$index];
        $right[$index]  = 0 if not defined $right[$index];
        $result[$index] = $left[$index] + $right[$index];
    }
    return \@result;
}