matlab - How to code a vectorized expression that depends on other vectorized expression? -

if example have 3 expressions: a, b , c follows:

a(i+1) = a(i) + c(i).k b(i+1) = b(i) + a(i).h c(i+1) = a(i) + b(i) 

where k , h constants , m , n desired size of c. i previous obtained value, i+1 next value. now, if use for loop, can code as:

a(1)= 2; b(1)= 5; c(1)= 3; i=1:10     a(i+1) = a(i) + c(i)*2;     b(i+1) = b(i) + a(i)*3;     c(i+1) = a(i) + b(i); end 

and works fine. want code in vector form, in without having use loop. problem not know how around dependency of:

• a on previous value , previous c value
• b on previous values , previous c value of a
• c on previous values of a , b

here's matrix-based way obtain n-th value of [a;b;c] vector. wouldn't call vectorization, speed things considerably you:

[a,b,c] = deal(zeros(11,1)); a(1)= 2; b(1)= 5; c(1)= 3;  %% // original method k=1:10   a(k+1) = a(k) + c(k)*2;   b(k+1) = b(k) + a(k)*3;   c(k+1) = a(k) + b(k); end  %% // matrix method: %// [ ]     [1  0  2][ ] %// | b |  =  |3  1  0|| b | %// [ c ]     [1  1  0][ c ] %//      i+1                %//  %// [ ]     [1  0  2][ ]        [1  0  2]   ( [1  0  2][ ] ) %// | b |  =  |3  1  0|| b |    =   |3  1  0| * ( |3  1  0|| b | ) %// [ c ]     [1  1  0][ c ]        [1  1  0]   ( [1  1  0][ c ] ) %//      i+2                i+1                                  %// thus, coefficient matrix taken n-th power, multiplied input  %// vector yield values of a(n+1), b(n+1), , c(n+1): m = [1 0 2      3 1 0      1 1 0]; isequal(m^10*[a(1);b(1);c(1)],[a(11);b(11);c(11)]) 

in reality can use m appropriate power (positive or negative) obtain [a,b,c]_n [a,b,c]_k ...