How to vectorize this for loop

Hello.Here's my codes from a program.I want to improve its efficiency,so I try to vectorize for loop.But I'm just a beginner in coding.I can't figure it out.Hope anyone who mastering in Matlab can help me to solve this.Thanks.

 K = 10;
 N = 10;
 M = 10;
 X(1:10,1:10) = 1;
 seed_num = 123;
 rng(seed_num);
 B = abs(randi([1,100],M,N))
 a = [];
 m = [];
 for l = 1:K        
     for j = 1:N   
        for k = 1:M
            num = X(l,k)*B(k,j); 
            a = [a,num];
        end
            m = [m,max(a)];    
            a = [];
     end
 end

 S = sum(m);

Answer 1

[Stephen23]

Method one: If X only contains ones (like in your example)

S = K*sum(max(B,[],1))

Method two: If X contains values other than one, then you could use this:

M = max(bsxfun(@times,B,permute(X,[2,3,1])),[],1);
S = sum(M(:))

Explanation:

The original code is very inefficient, and it needs a lot of revision. For example, this loop:

a = []
...
      for k = 1:M
          num = X(l,k)*B(k,j); 
          a = [a,num];
      end

just puts the j-th column into a variable a (note all X are one), so this is equivalent to

a = B(:,j);

But then the only operation on a is to have max applied, so there is no point doing this or each column separately because we can do this all at once using

max(B,[],1)

Taking the sum is then trivial. The original code repeats this K times, inefficiently concatenating new max value onto m each time, so we can just multiply that sum by K to get exactly the same output.