Using basic programming (for loops, while loops, and if statements), write two matlab functions, both taking as input:
- dimension n;
- n x n matrix a;
- n x n matrix b;
-n × 1 vector x.
have the first function compute abx through (ab)x and the second compute abx through a(bx). have both output:
- the number of flops used.
(a) print out or write out the first function.
(b) print out or write out the second function.
(c) apply both your functions to the case with random matrices and vectors for n = 100 and print out or write out the results. do the same for 200 x 200, 400 x 400, and 800 x 800. which approach of computing abx is faster?

For n = 100

For n = 200

For n = 400

For n = 800

The faster approach is A(Bx)

Explanation step by step functions:

A(Bx) is faster because requires fewer interactions to find a result: for (AB)x you have (n*n)+n interactions while for A(Bx) you have n+n, to understand why please see the step by step:

a) Function for (AB)x:

function loopcount1 = FirstAB(A,B,x)  

 n = size(A)(1);

 AB = zeros(n,n);

 ABx = zeros(n,1);

 loopcount1 = 0;  

 for i = 1:n

   for j = 1:n

     AB(i,j) = A(i,:)*B(:,j);

     loopcount1 += 1;

   end

 end

 for k = 1:n

   ABx(k) = AB(k,:)*x;

   loopcount1 += 1;

 end

end

b) Function for A(Bx):

function loopcount2 = FirstBx(A,B,x)

 n = size(A)(1);

 Bx = zeros(n,1);

 ABx = zeros(n,1);

 loopcount2 = 0;  

 for i = 1:n

   Bx(i) = B(i,:)*x;

   loopcount2 += 1;

 end

 for j = 1:n

   ABx(j) = A(j,:)*Bx;

   loopcount2 += 1;

 end

end

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